Differentially Private Gomory-Hu Trees

Anders Aamand BARC, University of Copenhagen. aamand@mit.edu. Supported by VILLUM Foundation grant 16582 and DFF-International Postdoc Grant 0164-00022B from the Independent Research Fund Denmark. Justin Y. Chen Massachusetts Institute of Technology. justc@mit.edu. Supported by an NSF Graduate Research Fellowship under Grant No. 17453. Part of this work was conducted while the author was visiting the Simons Institute for the Theory of Computing. Mina Dalirrooyfard Morgan Stanley. minad@mit.edu. Slobodan Mitrović UC Davis. smitrovic@ucdavis.edu. Supported by the Google Research Scholar and NSF Faculty Early Career Development Program No. 2340048. Part of this work was conducted while the author was visiting the Simons Institute for the Theory of Computing. Yuriy Nevmyvaka Morgan Stanley. yuriy.nevmyvaka@morganstanley.com. Sandeep Silwal UW-Madison. silwal@cs.wisc.edu. Yinzhan Xu Massachusetts Institute of Technology. xyzhan@mit.edu. Supported by NSF Grant CCF-2330048 and a Simons Investigator Award.

Abstract

Given an undirected, weighted $n$ -vertex graph $G=(V,E,w)$ , a Gomory-Hu tree $T$ is a weighted tree on $V$ such that for any pair of distinct vertices $s,t\in V$ , the Min- $s$ - $t$ -Cut on $T$ is also a Min- $s$ - $t$ -Cut on $G$ . Computing a Gomory-Hu tree is a well-studied problem in graph algorithms and has received considerable attention. In particular, a long line of work recently culminated in constructing a Gomory-Hu tree in almost linear time [Abboud, Li, Panigrahi and Saranurak, FOCS 2023].

We design a differentially private (DP) algorithm that computes an approximate Gomory-Hu tree. Our algorithm is $\varepsilon$ -DP, runs in polynomial time, and can be used to compute $s$ - $t$ cuts that are $\tilde{O}(n/\varepsilon)$ -additive approximations of the Min- $s$ - $t$ -Cuts in $G$ for all distinct $s,t\in V$ with high probability. Our error bound is essentially optimal, as [Dalirrooyfard, Mitrović and Nevmyvaka, NeurIPS 2023] showed that privately outputting a single Min- $s$ - $t$ -Cut requires $\Omega(n)$ additive error even with $(1,0.1)$ -DP and allowing for a multiplicative error term. Prior to our work, the best additive error bounds for approximate all-pairs Min- $s$ - $t$ -Cuts were $O(n^{3/2}/\varepsilon)$ for $\varepsilon$ -DP [Gupta, Roth and Ullman, TCC 2012] and $O(\sqrt{mn}\cdot\text{polylog}(n/\delta)/\varepsilon)$ for $(\varepsilon,\delta)$ -DP [Liu, Upadhyay and Zou, SODA 2024], both of which are implied by differential private algorithms that preserve all cuts in the graph. An important technical ingredient of our main result is an $\varepsilon$ -DP algorithm for computing minimum Isolating Cuts with $\tilde{O}(n/\varepsilon)$ additive error, which may be of independent interest.

1 Introduction

Given an undirected, weighted graph $G=(V,E,w)$ with positive edge weights, a cut is a bipartition of vertices $(U,V\setminus U)$ , and the value of the cut is the total weights of edges crossing the bipartition. Given a pair of distinct vertices $s,t\in V$ , the Min- $s$ - $t$ -Cut is a minimum-valued cut where $s\in U$ and $t\in V\setminus U$ . Min- $s$ - $t$ -Cut is dual to the Max- $s$ - $t$ -Flow problem, and the celebrated max-flow min-cut theorem states that the value of the Min- $s$ - $t$ -Cut equals the value of the Max- $s$ - $t$ -Flow [FF56, EFS56]. Finding a Min- $s$ - $t$ -Cut (or equivalently Max- $s$ - $t$ -Flow) is a fundamental problem in algorithmic graph theory, which has been studied for over seven decades [Dan51, HR55, FF56, EFS56], and has inspired ample algorithmic research (e.g., see a survey in [CKL⁺22, Appendix A]). It also has a wide range of algorithmic applications, including edge connectivity [Men27], bipartite matching (see, e.g., [CLRS22]), minimum Steiner cut [LP20], vertex-connectivity oracles [PSY22], among many others.

A natural and well-studied variant of Min- $s$ - $t$ -Cut is the All-Pairs Min Cut (APMC) problem. Given an input graph, the goal is to output the Min- $s$ - $t$ -Cut for all the pairs of vertices $s$ and $t$ in $V$ . In a seminal paper, Gomory and Hu [GH61] showed that there is a tree representation for this problem, called GH-tree or cut tree, that takes only $n-1$ Min- $s$ - $t$ -Cut (Max Flow) calls to compute. Consequently, there are only $n-1$ different max flow / minimum cut values in an arbitrary graph with positive edge weights. There has been a long line of research in designing faster GH-tree algorithms (e.g. [HKPB07, BSWN15, AKT20, Zha21, LP21, AKT21, AKT22, ALPS23], also see the survey [Pan08]), culminating an almost linear time algorithm for computing the GH-tree [ALPS23]. Beyond theoretical considerations, the GH-Tree has a long list of applications in a variety of research areas, such as networks [Hu74], image processing [WL93], optimization such as the $b$ -matching problem [PR82] and webpage segmenting [LLT11]. Furthermore, the global Min Cut of a graph is easily obtainable from the GH-tree.

In practice, algorithms are often applied to large data sets containing sensitive information. It is well understood by now that even minor negligence in handling users’ data can have severe consequences on user privacy; see [BDK07, NS08, Kor10, SSSS17, CRB⁺19] for a few examples. Differential privacy (DP), introduced by Dwork, McSherry, Nissim, and Smith in their seminal work [DMNS06], is a widely adopted standard for formalizing the privacy guarantees of algorithms. Informally, an algorithm is deemed differentially private if the outputs for two given neighboring inputs are statistically indistinguishable. The notion of neighboring inputs is application-dependent. In particular, when the underlying input belongs to a family of graphs, arguably the most studied notion of DP is with respect to edges, in which two neighboring graphs differ in only one edge; if the graphs are weighted, two neighboring graphs are those whose weights differ by at most one and in a single edge¹¹1Algorithms satisfying this notion are often also private for graphs whose vector of $\binom{n}{2}$ edge weights differ by $1$ in $\ell_{1}$ distance.. This is the setting for all the of the private cut problems we discuss in this work. In a related notion called vertex DP, two graphs are called neighboring if they differ in the existence of one vertex and the edges incident to that vertex. Two qualitative notions of DP that formalize the meaning of being “statistically indistinguishable” are pure and approximate DP. In pure DP, which is the focus of this paper, privacy is measured by a parameter $\varepsilon$ , and we use $\varepsilon$ -DP to refer to an algorithm having pure DP. In approximate DP, the privacy is expressed using two parameters, $\varepsilon$ and $\delta$ where an individual’s data can be leaked with a small probability corresponding to $\delta$ . Pure DP implies approximate DP and is a qualitatively stronger privacy guarantee. We formally define these notions in Section 2.

Over the last two decades, the design of DP graph algorithms has gained significant attention [MT07, HLMJ09, RHMS09, GLM⁺10, KRSY11, GRU12, BBDS13, KNRS13, BLR13, BNSV15, Sea16, AU19, US19, HKM⁺20, EKKL20, BEK21, RSSS21, NV21, BKM⁺22, FHS22, DLR⁺22, FLL22, CGK⁺23, DLL23, DKLV23, ELRS23, WRRW23, LUZ24b, JKR⁺24]. Several of those works focus on approximating graph cuts in the context of DP. For instance, Gupta et al. [GLM⁺10] gave an $\varepsilon$ -DP algorithm for global Min Cut with additive error $O(\log{n}/\varepsilon)$ . The authors also showed that there does not exist an $\varepsilon$ -DP algorithm for global Min Cut incurring less than $\Omega{(\log{n})}$ additive error.

Gupta, Roth, and Ullman [GRU12] and, independently, Blocki Blum, Datta, and Sheffet [BBDS12] focused on preserving all graph cuts in a DP manner. Given a graph $G$ , their algorithms output a synthetic graph $H$ such that each cut-value in $H$ is the same as the corresponding cut-value in $G$ up to an additive error of $O\left(n^{1.5}/\varepsilon\right)$ . The former result is pure DP while the latter is approximate DP. Eliáš, Kapralov, Kulkarni and Lee [EKKL20] improved on these results for sparse, unweighted (or small total weight) graphs, achieving error $\tilde{O}\left(\sqrt{mn/\varepsilon}\right)$ ²²2In this work, the notation $\tilde{O}(x)$ stands for $O(x\cdot\operatorname{polylog}x)$ . with approximate DP. The authors show that this error is essentially tight. In a follow-up work, Liu, Upadhyay, and Zou [LUZ24b] extended these results to weighted graphs and gave an algorithm to release a synthetic graph with $\tilde{O}\left(\sqrt{mn}/\varepsilon\right)$ error. A recent paper [LUZ24a] gives an algorithm for releasing a synthetic graph with worse error $\tilde{O}(m/\varepsilon)$ but which runs in near-linear time.

For a different problem, Dalirrooyfard, Mitrović and Nevmyvaka [DMN23] gave an $\varepsilon$ -DP algorithm for the Min- $s$ - $t$ -Cut problem with additive error $O(n/\varepsilon)$ and showed an essentially matching $\Omega(n)$ lower bound, even with approximate DP and both multiplicative and additive error.

Since the algorithm by [LUZ24b] approximately preserves all the cuts, it can also be used to solve APMC in a DP manner, albeit with approximate DP and with the additive error of $\tilde{O}\left(\sqrt{mn}/\varepsilon\right)$ . Additionally, the $\Omega(n)$ lower bound for Min- $s$ - $t$ -Cut shown by [DMN23] also applies to APMC, as the latter is a harder problem. While the additive error for computing global Min Cut [GLM⁺10], Min- $s$ - $t$ -Cut [DMN23], and all-cuts [EKKL20, LUZ24b] while ensuring DP is tightly characterized up to $\operatorname{polylog}n$ and $1/\varepsilon$ factors, there still remains a gap of roughly $\sqrt{m/n}$ between the best known lower and upper bound for solving DP APMC.³³3We remark that for the mentioned problems there still remain several interesting questions. For instance, finding a pure DP global Min Cut with $O(\log n/\varepsilon)$ additive error in polynomial time is unknown. The pure DP algorithm presented in [GLM⁺10] runs in exponential time, and can be improved to polynomial time only at the expense of approximate DP. This motivates the following question:

Can we obtain tight bounds on the additive error for APMC with differential privacy?

1.1 Our Contribution

Problem

Additive Error

Output

Runtime

Global Min Cut [GLM⁺10]

\Theta(\log(n)/\varepsilon)

\cellcolor[HTML]9AFF99Pure

\cellcolor[HTML]9AFF99Cut

\cellcolor[HTML]FFCCC9Exponential

Global Min Cut [GLM⁺10]

\Theta(\log(n)/\varepsilon)

\cellcolor[HTML]FFFC9EApprox

\cellcolor[HTML]9AFF99Cut

\cellcolor[HTML]9AFF99Polynomial

Min-

s

t

-Cut [DMN23]

O(n/\varepsilon)

and

\Omega(n)

\cellcolor[HTML]9AFF99 Pure

\cellcolor[HTML]9AFF99Cut

\cellcolor[HTML]9AFF99Polynomial

All Cuts [GRU12]

O(n^{3/2}/\varepsilon)

\cellcolor[HTML]9AFF99Pure

\cellcolor[HTML]9AFF99Synthetic Graph

\cellcolor[HTML]9AFF99Polynomial

All Cuts [EKKL20, LUZ24b]

\tilde{O}(\sqrt{mn}/\varepsilon)

and

\Omega(\sqrt{mn/\varepsilon})

\cellcolor[HTML]FFFC9EApprox

\cellcolor[HTML]9AFF99Synthetic Graph

\cellcolor[HTML]9AFF99Polynomial

APMC Values (Trivial)

O(n/\varepsilon)

\cellcolor[HTML]FFFC9EApprox

\cellcolor[HTML]FFFC9EValues Only

\cellcolor[HTML]9AFF99Polynomial

APMC (Our Work)

\tilde{O}(n/\varepsilon)

\cellcolor[HTML]9AFF99Pure

\cellcolor[HTML]9AFF99GH-Tree

\cellcolor[HTML]9AFF99Polynomial

Table 1: State-of-the-art bounds for various cut problems with differential privacy. Dependencies on the approximate DP parameter

\delta

are hidden. The APMC values result with approximate DP follows from advanced composition by adding Lap

(O(n/\varepsilon))

random noise to all

\binom{n}{2}

true values. For APMC, the lower bound of

\Omega(n)

error [DMN23] also applies as APMC generalizes Min-

s

t

-Cut.

Our main contribution is an $\varepsilon$ -DP algorithm for APMC with $\tilde{O}(n/\varepsilon)$ additive error. Up to $\operatorname{polylog}(n)$ factors, our algorithm privately outputs all the Min- $s$ - $t$ -Cuts while incurring the same error required to output an Min- $s$ - $t$ -Cut for a single pair of vertices $s$ and $t$ . This closes the aforementioned gap in the literature in private cut problems and shows that the error required to privatize the APMC problem is closer to that of the Min- $s$ - $t$ -Cut problem than to the synthetic graph/all-cuts problem.

To achieve this result, we introduce a DP algorithm that outputs an approximate Gomory-Hu tree (GH-tree). Gomory and Hu [GH61] showed that for any undirected graph $G$ , there exists a tree $T$ defined on the vertices of graph $G$ such that for all pairs of vertices $s,t$ , the Min- $s$ - $t$ -Cut in $T$ is also a Min- $s$ - $t$ -Cut in $G$ . We develop a private algorithm for constructing such a tree.

Theorem 1.1.

Given a weighted graph $G=(V,E,w)$ with positive edge weights and a privacy parameter $\varepsilon>0$ , there exists an $\varepsilon$ -DP algorithm that outputs an approximate GH-tree $T$ with additive error $\tilde{O}(n/\varepsilon)$ : for any $s\neq t\in V$ , the Min- $s$ - $t$ -Cut on $T$ and the Min- $s$ - $t$ -Cut on $G$ differ in $\tilde{O}(n/\varepsilon)$ in cut value with respect to edge weights in $G$ . The algorithm runs in time $\tilde{O}(n^{2})$ , and the additive error guarantee holds with high probability.

Since the GH-tree output by Theorem 1.1 is private, any post-processing on this tree is also private. This yields the following corollary.

Corollary 1.1.

Given a weighted graph $G$ with positive edge weights and a privacy parameter $\varepsilon>0$ , there exists an $\varepsilon$ -DP algorithm that outputs a cut for all the pairs of vertices $s$ and $t$ whose value is within $\tilde{O}(n/\varepsilon)$ from the value of the Min- $s$ - $t$ -Cut with high probability.

Note that Corollary 1.1 is tight up to $\operatorname{polylog}(n)$ and $\frac{1}{\epsilon}$ factors since any (pure or approximate) DP algorithm outputting the Min- $s$ - $t$ -Cut for fixed pair of vertices $s$ and $t$ requires $\Omega{(n)}$ additive error [DMN23]. Another corollary of Theorem 1.1 is a polynomial time pure DP algorithm for the global Min-Cut problem.

Corollary 1.2.

Given a weighted graph $G$ with positive edge weights and a privacy parameter $\varepsilon>0$ , there exists an $\varepsilon$ -DP algorithm that outputs an approximate global Min-Cut of $G$ in $\tilde{O}(n^{2})$ time and has additive error $\tilde{O}(n/\varepsilon)$ with high probability.

Note that [GLM⁺10] obtained an exponential time pure DP algorithm and a polynomial time approximate DP algorithm for global Min-Cut. It remains an intriguing question whether pure DP global Min-Cut can be found with only $\operatorname{polylog}n/\varepsilon$ additive error and in polynomial time.

Lastly, we note an application to the minimum $k$ -cut problem. Here, the goal is to partition the vertex set into $k$ pieces and the cost of any partitioning is weight of all edges that cross partitions. We wish to find the smallest cost solution. It is known that simply removing the smallest $k-1$ edges of an exact GH tree gives us a solution to the minimum $k$ -cut problem with a multiplicative approximation of $2$ [SV95]. Since we compute an approximate GH tree with additive error $\tilde{O}(n/\varepsilon)$ , we can obtain a solution to minimum $k$ -cut with multiplicative error $2$ and additive error $\tilde{O}(nk/\varepsilon)$ . We give the proof of the following corollary in Appendix A.

Corollary 1.3.

Given a weighted graph $G$ with positive edge weights and a privacy parameter $\varepsilon>0$ , there exists an $\varepsilon$ -DP algorithm that outputs a solution to the minimum $k$ -cut problem on $G$ in $\tilde{O}(n^{2})$ time with multiplicative error $2$ and additive error $\tilde{O}(nk/\varepsilon)$ with high probability.

The only prior DP algorithm for the minimum $k$ -cut problem is given in [CDFZ24]. It requires approximate DP and has the same multiplicative error $2$ but additive error $\tilde{O}(k^{1.5}/\varepsilon)$ . [CDFZ24] also give a pure DP algorithm which only handles unweighted graphs and requires exponential time. To the best of our knowledge, there are no prior pure DP algorithms which can compute the minimum $k$ -cut on weighted graphs. It is an interesting question to determine the limits of pure DP and efficient (polynomial time) algorithms for the minimum $k$ -cut problem.

1.2 Technical Overview

A greatly simplified view of a typical approach to designing a DP algorithm is to start with its non-DP version and then privatize it. The main challenge is finding the right way to privatize an algorithm, assuming a way exists in the first place, and then proving that DP guarantees are indeed achieved. To provide a few examples, Gupta et al. [GLM⁺10] employ Karger’s algorithm [Kar93] to produce a set of cuts and then use the Exponential Mechanism [MT07] to choose one of those cut. This simple but clever approach results in an $(\varepsilon,\delta)$ -DP algorithm for global Min Cut, for $\delta=O(1/\operatorname{poly}(n))$ , with $O(\log{n}/\varepsilon)$ additive error. Dalirrooyfard et al. [DMN23] show that the following simple algorithm yields $\varepsilon$ -DP Min- $s$ - $t$ -Cut with $O(n/\varepsilon)$ additive error: for each vertex $v$ , add an edge from $v$ to $s$ and from $v$ to $t$ with their weights chosen from the exponential distribution with parameter $1/\varepsilon$ ; return the Min- $s$ - $t$ -Cut on the modified graph. In the rest of this section, we first describe why directly privatizing some of the existing non-private algorithms does not yield the advertised additive error. Then, we describe our approach.

1.2.1 Obstacles in privatizing Gomory-Hu Trees

The All-Pairs Min-Cut (APMC) problem outputs cuts for $\Theta(n^{2})$ pairs but there are known to be only $O(n)$ distinct cuts. This property was leveraged in the pioneering work by Gomory and Hu [GH61], who introduced the Gomory-Hu tree (GH-tree), a structure that succinctly represents all the $\binom{n}{2}$ Min- $s$ - $t$ -Cuts in a graph.

A Gomory-Hu tree is constructed through a recursive algorithm that solves the Min- $s$ - $t$ -Cut problem at each recursion step. The algorithm can be outlined as follows: (1) Start with one supernode in a tree $T$ , consisting of all the vertices of the input graph $G$ . (2) At each step, select an arbitrary supernode $S$ where $|S|>1$ and choose two arbitrary vertices $s$ and $t$ within it. (3) Consider the graph where all supernodes, except $S$ , are each contracted into a single vertex. Compute the Min- $s$ - $t$ -Cut in this contracted graph. (4) Using the result of this Min- $s$ - $t$ -Cut and the submodularity of cuts (see Lemma 2.1), update the tree $T$ to ensure $s$ and $t$ are in different supernodes. The GH-tree efficiently represents all pairwise min-cuts in the graph by iterating through these steps.

One can replace the Min- $s$ - $t$ -Cut procedures with the private Min- $s$ - $t$ -Cut algorithm of [DMN23] to privatize this algorithm. Several obstacles arise when attempting to turn this idea into an error-efficient algorithm. Firstly, the algorithm of [DMN23] changes the graph. It is unclear whether we need to keep these changes for each run of the Min- $s$ - $t$ -Cut or revert back to the original graph. Secondly, the depth of this recursion could be $O(n)$ , which means that using Basic Composition [DKM⁺06], we obtain an $\varepsilon n$ -DP algorithm with additive error $O(n^{2}/\varepsilon)$ . Even if we apply Advanced Composition [DR14], which will result in an approximate DP algorithm, the resulting algorithm would be $(\varepsilon\sqrt{n},\delta)$ -DP with additive error of $O(n^{2}\log(1/\delta)/\varepsilon)$ . This is significantly worse then prior work preserving all cuts.

1.2.2 DP Approximate Min Isolating Cuts (Section 3)

The above approach suggests attempting to privatize a GH-tree algorithm with low recursion depth. There has been recent interest in low-depth recursion GH-tree algorithms (e.g., [AKT21, AKL⁺22, LPS22, ALPS23]), intending to construct GH-tree using fewer than $O(n)$ Min- $s$ - $t$ -Cut, i.e., Max-Flow, computations. Most of these works use an important sub-routine, called Min Isolating Cuts introduced by [LP20, AKT21]. For a set $U$ of vertices in $G$ , let $w(U)$ be the sum of the weights of the edges in $G$ with exactly one endpoint in $U$ .

Definition 1.1 (Min Isolating Cuts [LP20, AKT21]).

Given a set of terminals $R\subseteq V$ , the Min Isolating Cuts problem asks to output a collection of sets $\{S_{v}\subseteq V:v\in R\}$ such that for each vertex $v\in R$ , the set $S_{v}$ satisfies $S_{v}\cap R=\{v\}$ , and it has the minimum value of $w(S^{\prime}_{v})$ over all sets $S^{\prime}_{v}$ that $S^{\prime}_{v}\cap R=\{v\}$ . In other words, each $S_{v}$ is the minimum cut separating $v$ from $R\setminus\{v\}$ .

[LP20] and [AKT21] independently introduce the Isolating Cuts Lemma, showing how to solve the Min Isolating Cuts problem in $O(\log{R})$ many Min- $s$ - $t$ -Cut runs. [LP20] use it to find the global Min Cut in poly logarithmically many invocations of Max Flow and [AKT21] use it to compute GH-tree in simple graphs. Subsequent algorithms for GH-tree also use the Isolating Cuts Lemma, including the almost linear time algorithm for weighted graphs [ALPS23]. The Isolating Cuts Lemma has been extended to obtain new algorithms for finding the non-trivial minimizer of a symmetric submodular function and solving the hypergraph minimum cut problem [MN21, CQ21]. We develop the first differentially private algorithm for finding Min Isolating Cuts.

Theorem 1.2.

There is an $\varepsilon$ -DP algorithm that given a graph $G$ and a set of terminals $R$ , outputs sets $\{S_{v}:v\in R\}$ , such that for each vertex $v\in R$ , the set $S_{v}$ satisfies $S_{v}\cap R=\{v\}$ , and $w(S_{v})\leq w(S^{*}_{v})+\tilde{O}(n/\varepsilon)$ with high probability, where $\{S^{*}_{v}:v\in R\}$ are the Min Isolating Cuts for $R$ .

Our overall algorithm builds on a reduction first shown by [AKT20] from GH-tree to single-source Min Cuts, where we need to compute the $s$ - $t$ min cut values for a fixed $s$ and every $t\in V$ . Recent almost linear time algorithms for GH-tree [AKL⁺22, ALPS23] also utilize this reduction. We will mostly follow an exposition by Li [Li21].

We now provide a high-level overview of the algorithm of [Li21]. The approach of [Li21] is a recursive algorithm with depth $\operatorname{polylog}(n)$ . A generalized version of GH-tree called Steiner GH-Tree was considered, where instead of looking for Min- $s$ - $t$ -Cut for all pairs of vertices $s$ and $t$ in $G$ , we only focus on pairs $s,t\in U$ where $U$ is a subset of vertices. This especially helps in recursive algorithms, preventing the algorithm from computing the Min- $s$ - $t$ -Cut of the pairs of vertices $s,t$ that appear in multiple instances of the problem multiple times.

Each recursive step receives a set of terminals $U$ , a source terminal $s\in U$ , and a graph $G$ . The goal of each recursive step is to decompose the graph into subgraphs of moderate sizes, enabling further recursion on each subgraph. To achieve this, the algorithm utilizes the Min Isolating Cuts Lemma. Specifically, it applies the Min Isolating Cuts Lemma to random subsets $U^{\prime}\subseteq U$ . For each $U^{\prime}$ and $v\in U^{\prime}$ , a Min Isolating Cut $S_{v}$ is computed, which separates $v$ from all terminals in $U^{\prime}$ . Note that vertices in $U\setminus U^{\prime}$ may be included in $S_{v}$ . The algorithm considers a set $S_{v}$ to be “good” if $w(S_{v})$ equals the Min- $s$ - $v$ -Cut and $S_{v}$ contains at most $|U|/2$ terminals. This criterion ensures that, during the recursion, the number of terminals decreases exponentially, resulting in the recursion depth being $\operatorname{polylog}(n)$ .

[Li21] shows that in one recursive step, one can obtain a collection of disjoint “good” sets $S_{v}$ . Let $S_{\text{large}}$ be the set of vertices that are not in any $S_{v}$ . The authors show that $S_{\text{large}}$ has at most $|U|\cdot(1-1/\operatorname{polylog}(n))$ many terminals in $U$ . The recursion branches out as follows: for each $S_{v}$ , contract all the vertices that are not in $S_{v}$ , and recurse on this graph $G_{v}$ . For $S_{\text{large}}$ , contract each $S_{v}$ into a vertex, and recurse on this graph $G_{\text{large}}$ . It is similar to the original GH-tree algorithm [GH61] to aggregate the output of these recursions into a GH-tree.⁴⁴4Much of the effort in the recent works on constructing GH-tree is directed on obtaining Min- $s$ - $v$ -Cut values from $s$ to all $v$ in the recursive step efficiently. From the point of view of privacy and error, this step is trivial as single source minimum cuts can be released with $O(n/\varepsilon)$ error via basic composition as there are $n$ values.

1.2.3 DP Gomory-Hu Tree (Sections 4 and 5)

We now outline the main challenges and how we overcome them in obtaining a differentially private GH-tree.

Approximately good sets.

We will follow the high-level strategy of using a Min Isolating Cuts algorithm (now via our DP algorithm described above) to find sets $S_{v}$ which are approximately Min- $s$ - $t$ -Cuts. In defining which approximate Min Isolating Cuts are “good” and which our algorithm will recurse on, we want to accomplish three goals:

1.

The sets $S_{v}$ we return are $\tilde{O}(n/\varepsilon)$ -additive approximate Min- $s$ - $v$ -Cuts.
2.

Each set $S_{v}$ does not contain more than a constant fraction of $U$ in order to bound the recursion depth on $S_{v}$ .
3.

The union of sets $S_{v}$ we return contain a $1/\operatorname{polylog}(n)$ fraction of the vertices in $U$ , this ensures that recursing on $S_{\text{large}}$ has reasonably bounded depth.

The first goal can be achieved by privately estimating the value of each Min- $s$ - $v$ -Cut and ensuring that $w(S_{v})$ is close to this value up to approximation errors due to privacy.

To address the second goal, we only require a good set $S_{v}$ to have at most $0.9|U|$ terminals in $U$ . To bias the algorithm towards selecting smaller sets, we use the following idea. Consider a graph $G$ with terminals $s$ and $t$ , and a subset of vertices $U$ . We aim to obtain a DP Min- $s$ - $t$ -Cut such that the $s$ -side of the cut contains a small number of vertices in $U$ , without significantly sacrificing accuracy. To achieve this, we add edges from $t$ to every vertex in $U$ with a certain weight, penalizing the placement of vertices in $U$ on the $s$ -side of the cut. By increasing the error of our Min Isolating Cuts algorithm by only a constant factor, we enforce that if a true minimum isolating cut of size $|U|/2$ exists, we will output an isolating of size at most $0.9|U|$ .

A subtlety in addressing the final goal is that in the original analysis of [Li21], an argument is made that randomly sampling $U^{\prime}\subset U$ will, with reasonable probability, mean that for some $v$ , its Min Isolating Cut $S_{v}$ will be the same as its Min- $s$ - $v$ -Cut $S^{*}_{v}$ . In particular, this will be true if $v$ is the only vertex sampled on its side of the Min- $s$ - $v$ -Cut. For the right sampling rate, this happens with probability $O(1/|S^{*}|)$ but contributes $|S^{*}|$ to the output size. So, in this case, each vertex in $U$ contributes constant expectation to the output size (up to $\log$ factors coming from choosing the right sampling rate and enforcing the second goal). Unfortunately, even though the Min Isolating Cuts output by our DP algorithm have small additive error, it is possible for them to be much smaller than the optimal $|S^{*}_{v}|$ even if we sample the right terminals to get a set $S_{v}$ with approximately the same cost as $S^{*}_{v}$ . In order to make the analysis work with approximation, we give up on comparing to $S^{*}_{v}$ and instead compare ourselves to the smallest cardinality set $\tilde{S}_{v}$ which is an approximate Min- $s$ - $v$ -Cut. For the full argument, we adjust our notion of approximation based on the size of $\tilde{S}_{v}$ , allowing for weaker approximations for smaller cardinality sets. This only degrades our approximation with respect to the first goal by an extra logarithmic factor.

The privacy guarantee.

The second challenge we face is controlling the privacy budget. We need to demonstrate that for two neighboring graphs, the distribution of the outputs of $\operatorname{polylog}(n)$ recursive layers differs by at most a factor of $e^{\varepsilon}$ . To achieve this, we split our privacy budget across $\operatorname{polylog}(n)$ recursive sub-instances. We point out that the recursion depth of $\operatorname{polylog}(n)$ does not directly imply that the privacy budget is split across $\operatorname{polylog}(n)$ instances. We describe this challenge in more detail and the way our algorithm bypasses it.

To understand this more clearly, first consider two neighboring instances, $G$ and $G^{\prime}$ , that differ by the edge $xy$ . Suppose that in the first step of the algorithm, the good sets $S_{v}$ and $S_{\text{large}}$ are the same in both graphs $G$ and $G^{\prime}$ . If both $x$ and $y$ are in one of the good sets $S_{v}$ outputted by the first step, or if they are both in $S_{\text{large}}$ , all respective recursion instances are exactly the same except for one. For example, if $x,y\in S_{v}$ , then in all instances where $S_{v}$ is contracted, the resulting recursion graph does not depend on the edge $xy$ and is, therefore, the same in both instances.

On the other hand, suppose that $x\in S_{v}$ and $y\in S_{u}$ . In this case, the edge $xy$ influences multiple recursion instances: specifically, in $G_{v}$ , $G_{u}$ , and $G_{\text{large}}$ (similarly $G^{\prime}_{v}$ , $G^{\prime}_{u}$ , and $G^{\prime}_{\text{large}}$ ). This poses a challenge because the computation in multiple branches of the same recursion call depends on the same edge. Consequently, the privacy guarantee is not solely affected by the recursion depth.

To overcome this, we add a simple step before recursion on some of the new instances. For instances obtained by contracting all vertices in $V\setminus S_{v}$ to a single vertex and recursing on the resulting graph, we first add an edge from the contracted vertex to every vertex in $S_{v}$ with weights drawn from $\text{Lap}(\operatorname{polylog}(n)/\varepsilon)$ . Then we recurse on this altered graph. The intuition is that in neighboring instances, if $x\in S_{v}$ and $y\notin S_{v}$ , these noisy edges cancel the influence of $xy$ , preventing its influence from propagating further down this branch of the recursion. Thus, $xy$ only impacts $G_{\text{large}}$ . Fortunately, the additional added noise only contributes $\tilde{O}(n/\varepsilon)$ to the final additive error as noise is only added on $O(n)$ edges.

1.3 Organization

We provide necessary definitions and basic lemmas in Section 2. We present our DP Min Isolating Cuts algorithm in Section 3 and prove Theorem 1.2. In Section 4 we present the recursive step we use in our DP GH-tree algorithm, and in Section 5 we present our final algorithm and prove Theorem 1.1.

1.4 Open Problems

Our algorithm outputs an $\tilde{O}(n/\varepsilon)$ -approximate Gomory-Hu Tree from which we can recover an approximate Min- $s$ - $t$ -Cut for any two distinct vertices $s$ and $t$ . The additive error is necessary up to $\operatorname{polylog}n$ and $1/\varepsilon$ factors by the lower bound in [DMN23] for just releasing a single Min- $s$ - $t$ -Cut.

It is an interesting open question whether one can get a better additive error or prove lower bounds in the setting where we are only required to release approximate values of the cuts and not the cuts themselves. To our knowledge, the best achievable error is $O(n/\varepsilon)$ under approximate DP via the trivial algorithm, which adds $\text{Lap}(n/\varepsilon)$ noise to each true value (this is private via “advanced composition” [DR14]). Prior to our work, the best algorithm in the pure DP setting was to solve the all-cuts/synthetic graph problem, incurring $O(n^{3/2}/\varepsilon)$ error [GRU12]. Our work yields $\tilde{O}(n/\varepsilon)$ error for this problem also with pure DP, but no non-trivial lower bound is known for releasing APMC values. The $\Omega(n)$ lower bound of [DMN23] is for releasing an approximate Min- $s$ - $t$ -Cut, but releasing a single cut value can be done trivially with $O(1/\varepsilon)$ error via the Laplace mechanism. This question parallels the all-pairs shortest-paths distances problem studied in [Sea16, FLL22, CGK⁺23, BDG⁺24] for which sublinear additive error is possible.

Another open problem is whether there is a polynomial time $\varepsilon$ -DP algorithm for the global Min Cut problem with error below $\tilde{O}(n/\varepsilon)$ . As mentioned below Corollary 1.2, the polynomial time algorithm from [GLM⁺10] is only approximate DP (but can be made pure DP if the runtime is exponential). Lastly, the same question can be asked for the minimum $k$ -cut problem (see Corollary 1.3): what are the limits of efficient, i.e., polynomial time, algorithms that are also pure DP?

2 Preliminaries

2.1 Notation

We will denote a weighted, undirected graph by $G=(V,E,w)$ . For a subset of vertices $S\subseteq V$ , we will use $\partial_{G}S$ , or simply $\partial S$ , to denote the set of edges between $S$ and $V\setminus S$ . For a set of edges $Q\subseteq E$ , we will use $w(Q)$ to denote the sum of the weights of the edges in $Q$ . For a set of vertices $S\subseteq V$ , we will use $w(S)$ to denote $w(\partial S)$ , for simplicity. For $s,t\in V$ , we use $\lambda_{G}(s,t)$ to denote the Min- $s$ - $t$ -Cut value in $G$ . We will assume that all Min- $s$ - $t$ -Cuts or Min Isolating Cuts are unique. For $n>0$ , $\lg(n)$ is logarithm base- $2$ and $\ln(n)$ is logarithm base- $e$ .

2.2 Graph Cuts

In our algorithms, we use the notion of vertex contractions which we formally define here.

Definition 2.1 (Vertex contractions).

Let $X\subseteq V$ be a subset of vertices of the graph $G=(V,E,w)$ . Contracting the set $X$ into a vertex is done as follows: we add a vertex $x$ to the graph and remove all the vertices in $X$ from the graph. Then for every vertex $v\in V\setminus X$ , we add an edge from $x$ to $v$ with weight $\sum_{x^{\prime}\in X}w(x^{\prime}v)$ . Note that if none of the vertices in $X$ has an edge to $v$ , then there is no edge from $x$ to $v$ .

We use the submodularity property of cuts in many of our proofs.

Lemma 2.1 (Submodularity of Cuts [Cun85]).

For any graph $G=(V,E,w)$ , and any two subsets $S,T\subseteq V$ ,

w(S)+w(T)\geq w(S\cup T)+w(S\cap T).

Recall the definition of Min Isolating Cuts problem (see Definition 1.1). We use the following simple fact.

Fact 2.1 ([LP20]).

There always exists a minimum isolating cuts solution where the solution $\{S_{v}:v\in R\}$ are disjoint.

Definition 2.2 (Gomory-Hu Steiner tree [Li21]).

Given a graph $G=(V,E,w)$ and a set of terminals $U\subseteq V$ , the Gomory-Hu Steiner tree is a weighted tree $T$ on the vertices $U$ , together with a function $f:V\rightarrow U$ , such that For all $s,t\in U$ , consider the minimum-weight edge $uv$ on the unique $s$ - $t$ path in $T$ . Let $U_{0}$ be the vertices of the connected component of $T-uv$ containing $s$ . Then, the set $f^{-1}(U_{0})\subseteq V$ is a Min- $s$ - $t$ -Cut, and its value is $w_{T}(uv)$ .

Note that for $U=V$ and $f(v)=v$ the Gomory-Hu Steiner tree equals the Gomory-Hu tree.

2.3 Differential Privacy

Definition 2.3 (Edge-Neighboring Graphs).

Graphs $G=(V,E,w)$ and $G^{\prime}=(V,E^{\prime},w^{\prime})$ are called edge-neighboring if there is $uv\in V^{2}$ such that $|w_{G}(uv)-w_{G^{\prime}}(uv)|\leq 1$ and for all $u^{\prime}v^{\prime}\neq uv$ , $u^{\prime}v^{\prime}\in V^{2}$ , we have $w_{G}(u^{\prime}v^{\prime})=w_{G^{\prime}}(u^{\prime}v^{\prime})$ .

Definition 2.4 (Differential Privacy [Dwo06]).

A (randomized) algorithm $\mathcal{A}$ is $(\varepsilon,\delta)$ -private (or $(\varepsilon,\delta)$ -DP) if for any neighboring graphs $G$ and $G^{\prime}$ and any set of outcomes $O\subset Range(\mathcal{A})$ it holds

\operatorname*{\mathbb{P}}\left[\mathcal{A}(G)\in O\right]\leq e^{\varepsilon}\operatorname*{\mathbb{P}}\left[\mathcal{A}(G^{\prime})\in O\right]+\delta.

When $\delta=0$ , algorithm $\mathcal{A}$ is pure differentially private, or $\varepsilon$ -DP.

Theorem 2.1 (Basic composition [DMNS06, DL09]).

Let $\varepsilon_{1},\ldots,\varepsilon_{t}>0$ and $\delta_{1},\ldots,\delta_{t}\geq 0$ . If we run $t$ (possibly adaptive) algorithms where the $i$ -th algorithm is $(\varepsilon_{i},\delta_{i})$ -DP, then the entire algorithm is $(\varepsilon_{1}+\ldots+\varepsilon_{t},\delta_{1}+\ldots+\delta_{t})$ -DP.

Theorem 2.2 (Laplace mechanism [DR14]).

Consider any function $f$ which maps graphs $G$ to $\mathbb{R}^{d}$ with the property that for any two neighboring graphs $G,G^{\prime}$ , $|f(G)-f(G^{\prime})|\leq\Delta$ . Then, releasing

f(G)+(X_{1},\ldots,X_{d})

where each $X_{i}$ is i.i.d. with $X_{i}\sim\text{Lap}(\Delta/\varepsilon)$ satisfies $\varepsilon$ -DP.

Theorem 2.3 (Private Min- $s$ - $t$ -Cut [DMN23]).

Fix any $\varepsilon>0$ . There is an $(\varepsilon,0)$ -DP algorithm $\text{PrivateMin-s-t-Cut}(G=(V,E,w),s,t)$ for $s\neq t\in V$ that reports an $s$ - $t$ cut for $n$ -vertex weighted graphs that is within $O(\frac{n}{\varepsilon})$ additive error from the Min- $s$ - $t$ -Cut with high probability.

By standard techniques, we can also use Theorem 2.3 to design an $(\varepsilon,0)$ -DP algorithm for computing an approximate min- $S$ - $T$ -cut for two disjoint subsets $S,T\subseteq V$ that is within $O(\frac{n}{\varepsilon})$ additive error from the actual min- $S$ - $T$ -cut (e.g., by contracting all vertices in $S$ and all vertices in $T$ to two supernodes). Furthermore, our final algorithm is recursive with many calls to Private Min- $S$ - $T$ -Cut for graphs with few vertices, and it is not enough to succeed with high probability with respect to $n$ . The error analysis of [DMN23] shows that the error is bounded by the sum $O(n)$ random variables distributed as $\text{Exp}(\varepsilon)$ . Using Theorem 2.4 yields the following corollary:

Corollary 2.1 (Private Min- $S$ - $T$ -Cut).

Fix any $\varepsilon>0$ , there exists an $(\varepsilon,0)$ -DP algorithm
$\text{PrivateMin-S-T-Cut}(G=(V,E,w),S,T,\varepsilon)$ for disjoint $S,T\subseteq V$ that reports a set $C\subseteq V$ where $S\subseteq C$ and $C\cap T=\emptyset$ , and $w(\partial C)$ is within $O\MT@delim@Auto\paren@star{\frac{n+\log(1/\beta)}{\varepsilon}}$ additive error from the min- $S$ - $T$ -cut with probability at least $1-\beta$ .

2.4 Concentration Inequalities

Theorem 2.4 (Sums of Exponential Random Variables (Theorem 5.1 of [Jan17])).

Let $X_{1},\ldots,X_{N}$ be independent random variables with $X_{i}\sim\text{Exp}(a_{i})$ . Let $\mu=\sum_{i=1}^{N}\frac{1}{a_{i}}$ be the expectation of the sum of the $X_{i}$ ’s and let $a^{*}=\min_{i}a_{i}$ . Then, for any $\lambda\geq 1$ ,

\operatorname*{\mathbb{P}}\left[\sum_{i\in S}X_{i}\geq\lambda\mu\right]\leq\frac{1}{\lambda}\exp\MT@delim@Auto\brack@star{-a^{*}\mu(\lambda-1-\ln\lambda)}

3 Private Min Isolating Cuts

In this section we prove Theorem 1.2. In fact we prove a stronger version denoted in Lemma 3.1. The steps in the algorithm that differ meaningfully from the non-private version are in color.

Algorithm 1 PrivateMinIsolatingCuts

(G=(V,E,w),R,U,\varepsilon,\beta)

1:Initialize

W_{r}\leftarrow V

for every

r\in R

2:Identify

R

with

\{0,\ldots,|R|-1\}

3:for

i

from

0

\lfloor\lg(|R|-1)\rfloor

A_{i}\leftarrow\{r\in R:r\bmod 2^{i+1}<2^{i}\}

C_{i}\leftarrow

\text{PrivateMin-S-T-Cut}(G,A_{i},R\setminus A_{i},\varepsilon/(\lg|R|+2))

W_{r}\leftarrow W_{r}\cap C_{i}

for every

r\in A_{i}

W_{r}\leftarrow W_{r}\cap(V\setminus C_{i})

for every

r\in R\setminus A_{i}

8:end for

9:for

r\in R

10: Let

H_{r}

G

with all vertices in

V\setminus W_{r}

contracted, and let

t_{r}

be the contracted vertex

11: In

H_{r}

, add weight

B_{\text{H}}{}\cdot\frac{(n+\lg(1/\beta))\lg^{2}(|R|)\}}{\varepsilon|U|}

for every edge from vertex in

W_{r}\cap U

t_{r}

for some sufficiently large constant

B_{\text{H}}{}

12:end for

13:

{\mathcal{H}}\leftarrow\bigcup_{r\in R}H_{r}

14:

{\mathcal{C}}\leftarrow

\text{PrivateMin-S-T-Cut}({\mathcal{H}},R,\{t_{r}\}_{r\in R},\varepsilon/(\lg|R|+2))

15:return

\{{\mathcal{C}}\cap W_{r}\}_{r\in R}

Lemma 3.1.

On a graph $G$ with $n$ vertices, a set of terminals $R\subseteq V$ , another set of vertices $U\subseteq V$ , and a privacy parameter $\varepsilon$ , there is an $(\varepsilon,0)$ -DP algorithm PrivateMinIsolatingCuts $(G,R,U,\varepsilon,\beta)$ that returns a set of Isolating Cuts over terminals $R$ . The total cut values of the Isolating Cuts is within additive error $O((n+\lg(1/\beta))\lg^{2}(|R|)/\varepsilon)$ from the Min Isolating Cuts with probability $1-\beta$ .

Furthermore, if the Min Isolating Cut for any terminal $r\in R$ contains at most $0.5|U|$ vertices from $U$ , then the Isolating Cut for $r$ returned by the algorithm will contain at most $0.9|U|$ vertices from $U$ , with high probability.

Proof.

The algorithm is presented in Algorithm 1. On a high level, the algorithm follows the non-private Min Isolating Cuts algorithm by [LP20, AKT21], but replacing all calls to Min- $S$ - $T$ -Cut with private Min- $S$ - $T$ -Cut from Corollary 2.1. One added step is 11, which is used to provide the guarantee that if the Min Isolating Cut for terminal $r$ contains a small number of vertices in $U$ , then the isolating cut for terminal $r$ returned by the algorithm also does.

Next, we explain the algorithm in more detail. For every $r\in R$ , we maintain a set $W_{r}$ that should contain the $r$ -side of a $(r,R\setminus\{r\})$ cut obtained in the algorithm. In each of the $\lfloor\lg(|R|-1)\rfloor+1$ iterations, we find a subset $A_{i}\subseteq R$ , and find a cut that separates $A_{i}$ from $R\setminus A_{i}$ . Let $C_{i}$ be the side of the cut containing $A_{i}$ . Then for every $r\in A_{i}$ , we update $W_{r}$ with $W_{r}\cap C_{i}$ ; for every $r\in R\setminus A_{i}$ , we update $W_{r}$ with $W_{r}\cap(V\setminus C_{i})$ . The choice of $A_{i}$ is so that every pair $r_{1},r_{2}\in R$ are on different sides of the Min- $S$ - $T$ Cut in at least one iteration; as a result, $W_{r}\cap R=\{r\}$ for every $r\in R$ after all iterations.

Next, for every $r\in R$ , the algorithm aims to compute a cut separating $r$ from $R\setminus\{r\}$ , where the side containing $r$ is inside $W_{r}$ . This can be done by contracting all vertices outside of $W_{r}$ to a vertex $t_{r}$ and computing private Min- $r$ - $t_{r}$ Cut. To incentivize cuts that contain fewer vertices in $U$ on the side containing $r$ , the algorithm adds a positive weight from every vertex in $W_{r}\cap U$ to $t_{r}$ . Finally, these private Min- $r$ - $t_{r}$ Cut instances can be solved at once by combining them into a single graph ${\mathcal{H}}$ .

Privacy analysis.

The only parts of the algorithm that depend on the edges or edge weights are the calls to PrivateMin-S-T-Cut. Each call to PrivateMin-S-T-Cut is $(\varepsilon/(\lg|R|+2),0)$ -DP, and the number of calls is $\lfloor\lg(|R|-1)\rfloor+2$ , so the overall algorithm is $\varepsilon$ -DP via basic composition (Theorem 2.1).

Error analysis.

First, we analyze the error introduced by the for loop starting at 3. Let $\{S_{r}\}_{r\in R}$ be the (non-private) Min Isolating Cuts for terminals in $R$ , these are only used for analysis purposes. Take an iteration $i$ of the for loop and let $\{W_{r}\}_{r\in R}$ be the values of $W_{r}$ ’s before the start of the iteration, and let $\{W_{r}^{\prime}\}_{r\in R}$ denote the value of $W_{r}$ ’s at the end of the iteration. We show the following claim:

Claim 1.

With high probability, $\sum_{r\in R}w\left(W^{\prime}_{r}\cap S_{r}\right)\leq\sum_{r\in R}w\left(W_{r}\cap S_{r}\right)+O(n\lg(|R|)/\varepsilon)$ .

Proof.

We first show $\sum_{r\in A_{i}}w\left(W^{\prime}_{r}\cap S_{r}\right)\leq\sum_{r\in A_{i}}w\left(W_{r}\cap S_{r}\right)+O(n\lg(|R|)/\varepsilon)$ . Let $S^{\prime}_{A_{i}}:=\bigcup_{r\in A_{i}}(S_{r}\cap W_{r})$ . By Lemma 2.1,

w(S^{\prime}_{A_{i}})+w(C_{i})\geq w(S^{\prime}_{A_{i}}\cup C_{i})+w(S^{\prime}_{A_{i}}\cap C_{i}).

(1)

Recall that with probability $1-\beta/\lg|R|$ , $C_{i}$ is within $O((n+\lg(\lg|R|/\beta))\lg(|R|)/\varepsilon)=O((n+\lg(1/\beta))\lg(|R|)/\varepsilon)$ of the minimum cut separating $A_{i}$ and $R\setminus A_{i}$ , by the guarantee of Corollary 2.1, and note that $S^{\prime}_{A_{i}}\cup C_{i}$ is also a cut separating $A_{i}$ and $R\setminus A_{i}$ . Therefore,

w(C_{i})\leq w(S^{\prime}_{A_{i}}\cup C_{i})+O((n+\lg(1/\beta))\lg(|R|)/\varepsilon).

(2)

Combining Equations 1 and 2, we get that

w(S^{\prime}_{A_{i}}\cap C_{i})\leq w(S^{\prime}_{A_{i}})+O((n+\lg(1/\beta))\lg(|R|)/\varepsilon).

(3)

Therefore,

$\displaystyle\sum_{r\in A_{i}}w\left(W^{\prime}_{r}\cap S_{r}\right)$	$\displaystyle=\sum_{r\in A_{i}}w\left(W_{r}\cap S_{r}\cap C_{i}\right)$
	$\displaystyle=w\left(\bigcup_{r\in A_{i}}\left(W_{r}\cap S_{r}\cap C_{i}\right)\right)+\sum_{r_{1}\neq r_{2}\in A_{i}}w\left(E\cap\left(\left(W_{r_{1}}\cap S_{r_{1}}\cap C_{i}\right)\times\left(W_{r_{2}}\cap S_{r_{2}}\cap C_{i}\right)\right)\right)$
	$\displaystyle\leq w(S^{\prime}_{A_{i}}\cap C_{i})+\sum_{r_{1}\neq r_{2}\in A_{i}}w\left(E\cap\left(\left(W_{r_{1}}\cap S_{r_{1}}\right)\times\left(W_{r_{2}}\cap S_{r_{2}}\right)\right)\right)$
	$\displaystyle\leq w(S^{\prime}_{A_{i}})+O((n+\lg(1/\beta))\lg(\|R\|)/\varepsilon)+\sum_{r_{1}\neq r_{2}\in A_{i}}w\left(E\cap\left(\left(W_{r_{1}}\cap S_{r_{1}}\right)\times\left(W_{r_{2}}\cap S_{r_{2}}\right)\right)\right)$	(by Equation 3)
	$\displaystyle=\sum_{r\in A_{i}}w\left(W_{r}\cap S_{r}\right)+O((n+\lg(1/\beta))\lg(\|R\|)/\varepsilon).$

By an analogous argument, we can show $\sum_{r\in R\setminus A_{i}}w\left(W^{\prime}_{r}\cap S_{r}\right)\leq\sum_{r\in R\setminus A_{i}}w\left(W_{r}\cap S_{r}\right)+O(n\lg(|R|)/\varepsilon)$ . Summing up the two inequalities gives the desired claim. $\Box$

By applying 1 repeatedly, we can easily show the following claim:

Claim 2.

At the end of the for loop starting at 3, $\sum_{r\in R}w(W_{r}\cap S_{r})\leq\sum_{r\in R}w(S_{r})+O((n+\lg(1/\beta))\lg^{2}(|R|)/\varepsilon)$ , with probability $1-\beta$ .

The following claim is a simple observation:

Claim 3.

At the end of the for loop starting at 3, $r\in W_{r}$ for every $r\in R$ and distinct $W_{r}$ ’s are disjoint.

Next, we show that the Min- $r$ - $t_{r}$ Cut values in $H_{r}$ are close to the Min Isolating Cut values:

Claim 4.

With probability $1-\beta$ , $\sum_{r\in R}\lambda_{H_{r}}(r,t_{r})\leq\sum_{r\in R}w_{G}(S_{r})+O((n+\lg(1/\beta))\lg^{2}(|R|)/\varepsilon)$ .

Proof.

We have that

	$\displaystyle\sum_{r\in R}\lambda_{H_{r}}(r,t_{r})$	$\displaystyle\leq\sum_{r\in R}w_{H_{r}}(W_{r}\cap S_{r})$
		$\displaystyle=\sum_{r\in R}\left(w_{G}(W_{r}\cap S_{r})+\|W_{r}\cap S_{r}\cap U\|\cdot O((n+\lg(1/\beta))\lg^{2}(\|R\|)/(\varepsilon\|U\|))\right)$
		$\displaystyle\leq\sum_{r\in R}w_{G}(W_{r}\cap S_{r})+O((n+\lg(1/\beta))\lg^{2}(\|R\|)/\varepsilon)$
		$\displaystyle\leq\sum_{r\in R}w_{G}(S_{r})+O((n+\lg(1/\beta))\lg^{2}(\|R\|)/\varepsilon),$

where the last step is by 2. $\Box$

Because of 4 (and note that $|V({\mathcal{H}})|=O(n)$ ), with high probability, the final cuts ${\mathcal{C}}\cap W_{r}$ returned by the algorithm will have the property that

	$\displaystyle\sum_{r\in R}w_{H_{r}}({\mathcal{C}}\cap W_{r})$	$\displaystyle\leq\sum_{r\in R}\lambda_{H_{r}}(r,t_{r})+O((n+\lg(1/\beta))\lg(\|R\|)/\varepsilon)$
		$\displaystyle\leq\sum_{r\in R}w_{G}(S_{r})+O((n+\lg(1/\beta))\lg^{2}(\|R\|)/\varepsilon).$

Furthermore, $w_{G}({\mathcal{A}}\cap W_{r})\leq w_{H_{r}}({\mathcal{A}}\cap W_{r})$ as we only add positive weights to $H_{r}$ compared to $G$ , we further get

\sum_{r\in R}w_{G}({\mathcal{C}}\cap W_{r})\leq\sum_{r\in R}w_{G}(S_{r})+O((n+\lg(1/\beta))\lg^{2}(|R|)/\varepsilon),

which is the desired error bound.

Additional guarantee.

Finally, we need to show that if $|S_{r}\cap U|\leq 0.5U$ for some $r\in R$ , then with high probability, the returned isolating cut by the algorithm ${\mathcal{C}}\cap W_{r}$ has $|{\mathcal{C}}\cap W_{r}\cap U|\leq 0.9U$ . By Corollary 2.1,

w_{{\mathcal{H}}}({\mathcal{C}})\leq w_{{\mathcal{H}}}(({\mathcal{C}}\setminus W_{r})\cup(S_{r}\cap W_{r}))+O((n+\lg(1/\beta))(\lg(|R|))/\varepsilon),

with high probability. By removing cut values contributed by $H_{r^{\prime}}$ for $r^{\prime}\neq r$ from both sides, we get that

w_{H_{r}}({\mathcal{C}}\cap W_{r})\leq w_{H_{r}}(S_{r}\cap W_{r})+O((n+\lg(1/\beta))(\lg(|R|))/\varepsilon).

Rewriting the cut values in terms of the edge weights of $G$ instead of $H_{r}$ , the above becomes

	$\displaystyle w({\mathcal{C}}\cap W_{r})+(B_{\text{H}}{}\cdot(n+\lg(1/\beta))\lg^{2}(\|R\|)/(\varepsilon\|U\|))\|{\mathcal{C}}\cap W_{r}\cap U\|$
	$\displaystyle\leq w(S_{r}\cap W_{r})+(B_{\text{H}}{}\cdot(n+\lg(1/\beta))\lg^{2}(\|R\|)/(\varepsilon\|U\|))\|S_{r}\cap W_{r}\cap U\|+O((n+\lg(1/\beta))(\lg(\|R\|))/\varepsilon)$
	$\displaystyle\leq w(S_{r})+(B_{\text{H}}{}\cdot(n+\lg(1/\beta))\lg^{2}(\|R\|)/(\varepsilon\|U\|))\|S_{r}\cap W_{r}\cap U\|+O((n+\lg(1/\beta))(\lg^{2}(\|R\|))/\varepsilon).$		(By 2)

Because $w({\mathcal{C}}\cap W_{r})\geq w(S_{r})$ , the above implies

$\displaystyle\|{\mathcal{C}}\cap W_{r}\cap U\|$	$\displaystyle\leq\frac{O((n+\lg(1/\beta))(\lg^{2}(\|R\|))/\varepsilon)}{B_{\text{H}}{}\cdot(n+\lg(1/\beta))\lg^{2}(\|R\|)/(\varepsilon\|U\|)}+\|S_{r}\cap W_{r}\cap U\|$
	$\displaystyle\leq 0.4\|U\|+\|S_{r}\cap W_{r}\cap U\|$	(By setting $B_{\text{H}}{}$ large enough)
	$\displaystyle\leq 0.4\|U\|+\|S_{r}\cap U\|$
	$\displaystyle\leq 0.9\|U\|,$

as desired. $\Box$

4 Core Recursive Step

Algorithm 2 PrivateGHTreeStep

(G=(V,E,w),s,U,{\color[rgb]{0.8,0.25,0.33}{\varepsilon,\beta}})

{\Gamma_{\text{iso}}}\leftarrow O\MT@delim@Auto\paren@star{\frac{(n+\lg(1/\beta))\lg^{3}(|U|)}{\varepsilon}}

and

{\Gamma_{\text{values}}}\leftarrow O\MT@delim@Auto\paren@star{\frac{|U|\lg(|U|/\beta)}{\varepsilon}}

\hat{\lambda}(s,v)\leftarrow\lambda(s,v)+\text{Lap}\MT@delim@Auto\paren@star{\frac{4(|U|-1)}{\varepsilon}}

for all

v\in U\setminus\{s\}

3:Initialize

R^{0}\leftarrow U

and

D\leftarrow\emptyset

4:for

i

from

0

\MT@delim@Auto\floor@star{\lg|U|}

5: Call PrivateMinIsolatingCuts

\MT@delim@Auto\paren@star{G,R^{i},\frac{\varepsilon}{2(\MT@delim@Auto\floor@star{\lg U}+1)},\frac{\beta}{\MT@delim@Auto\floor@star{\lg U}+1}}

(Algorithm 1) obtaining disjoint sets

{\hat{S}}_{v}^{i}

{\hat{w}}({\hat{S}}_{v}^{i})\leftarrow w({\hat{S}}_{v}^{i})+\text{Lap}\MT@delim@Auto\paren@star{\frac{8(\MT@delim@Auto\floor@star{\lg U}+1)}{\varepsilon}}

for each

v\in R^{i}\setminus\{s\}

7: Let

D^{i}\subseteq U

be the union of

{\hat{S}}_{v}^{i}\cap U

over all

v\in R^{i}\setminus\{s\}

satisfying

{\hat{w}}({\hat{S}}_{v}^{i})\leq\hat{\lambda}(s,v)+\MT@delim@Auto\paren@star{2(\MT@delim@Auto\floor@star{\lg|U|}-i)+1}{\Gamma_{\text{iso}}}+{\Gamma_{\text{values}}}

and

|{\hat{S}}_{v}^{i}\cap U|\leq(9/10)|U|

R^{i+1}\leftarrow

sample of

U

where each vertex in

U\setminus\{s\}

is sampled independently with probability

2^{-i+1}

, and

s

is sampled with probability

1

9:end for

10:return the largest set

D^{i}

, the corresponding terminals

v\in R^{i}\setminus\{s\}

and sets

{\hat{S}}_{v}^{i}

satisfying the conditions on 7

We now describe a key subroutine, outlined as Algorithm 2, used to compute a DP Gomory-Hu tree. The high-level goal is to use Min Isolating Cuts to find minimum cuts that cover a large fraction of vertices in the graph. The overall structure of this algorithm follows that of the prior work [Li21] with several key changes to handle additive approximations and privacy. The inputs to Algorithm 2 are the weighted graph, a source vertex $s$ , a set of active vertices $U\subseteq V$ , a privacy parameter $\varepsilon$ , and a failure probability $\beta$ . The steps that differ meaningfully from the non-private version developed in [Li21] are in color. To obtain a DP version of this method, Algorithm 2 invokes DP Min- $s$ - $t$ -Cut and DP Min Isolating Cuts algorithm; the latter primitive is developed in this work in Section 3. In the original non-private algorithm, isolating cuts $S_{v}^{i}$ are included in $D^{i}$ if the set $S_{v}^{i}$ corresponds to the $v$ side of a Min- $s$ - $v$ -Cut, i.e., $w(S^{i}_{v})=\lambda(s,v)$ . The analysis in prior work relies on this equality, i.e., on $w(S^{i}_{v})$ and $\lambda(s,v)$ being the same, in a crucial way. Informally speaking, it enables the selection of many Min Isolating Cuts of the right size. In our case, since the cuts and their values are released privately by random perturbations, it is unclear how to test that condition with equality. On the other hand, we still would like to ensure that many isolating cuts have “the right” size. Among our key technical contributions is relaxing that condition by using a condition which changes from iteration to iteration of the for-loop. The actual condition we use is

{\hat{w}}({\hat{S}}_{v}^{i})\leq\hat{\lambda}(s,v)+\MT@delim@Auto\paren@star{2(\MT@delim@Auto\floor@star{\lg U}-i)+1}{\Gamma_{\text{iso}}}+{\Gamma_{\text{values}}}

(4)

on 7 of Algorithm 2 where ${\Gamma_{\text{iso}}}$ and ${\Gamma_{\text{values}}}$ are upper bounds on the additive errors of the approximate Min Isolating Cuts and the approximate Min- $s$ - $v$ -Cut values, respectively.

When using Equation 4, we also have to ensure that significant progress can still be made, i.e., to ensure that both (a) we will find a large set $D^{i}$ which is the union of approximate Min Isolating Cuts ${\hat{S}}_{v}^{i}$ satisfying the condition above and (b) none of the individual ${\hat{S}}$ which we return are too large as we will recurse within each of these sets. A new analysis uses this changing inequality to show that the former is true. For the latter, we utilize the special property of our PrivateMinIsolatingCuts in Section 3 which forces an approximate isolating cut to contain at most $0.9|U|$ terminals if there exists an exact isolating cut of size at most $|U|/2$ . We now turn to the analysis.

4.1 Correctness

As in prior work [Li21, AKL⁺22], let $D^{*}\subseteq U\setminus\{s\}$ be the set of vertices $v$ such that if $S_{v}^{*}$ is the $v$ side of the Min- $s$ - $v$ -Cut, $|S_{v}^{*}\cap U|\leq|U|/2$ .

Lemma 4.1.

PrivateGHTreeStep $(G,U,s,\varepsilon,\beta)$ (Algorithm 2) has the following properties:

•

Let ${\Gamma_{\text{iso}}}=C_{1}(n+\lg(1/\beta))\lg^{3}(|U|)/\varepsilon$ and ${\Gamma_{\text{values}}}=C_{2}|U|\lg(|U|/\beta)/\varepsilon$ for large enough constants $C_{1},C_{2}$ . Let $\{S_{v}^{i}\}_{v\in R^{i}}$ be optimal Min Isolating Cuts for terminals $R^{i}$ . Then, with probability at least $1-O(\beta)$ , the sets $\{{\hat{S}}_{v}^{i}:v\in R^{*}\}$ returned by the algorithm are approximate Min Isolating Cuts and approximate Min- $v$ - $s$ -Cuts:

\sum_{v\in R^{*}}w({\hat{S}}_{v}^{i})-w(S_{v}^{i})\leq{\Gamma_{\text{iso}}},

and, for all $v\in R^{*}$ ,

w({\hat{S}}_{v}^{i})-\lambda(s,v)\leq 2(\MT@delim@Auto\floor@star{\lg U}+1){\Gamma_{\text{iso}}}+2{\Gamma_{\text{values}}}.

•

With probability at least $1/2$ , $D^{i}$ returned by the algorithm satisfies

$|D^{i}|=\Omega\left(\frac{|D^{*}|}{\lg|U|}\right).$

To prove this, we will need the following helpful definition and lemma. Let $X_{v}^{i}$ be a random variable for the number of vertices in $U$ added to $D^{i}$ by a set ${\hat{S}}_{v}^{i}$ :

X_{v}^{i}=\begin{cases}|{\hat{S}}_{v}^{i}\cap U|\qquad&\text{ if }v\in R^{i}\text{ and }|{\hat{S}}_{v}^{i}\cap U|\leq(9/10)|U|\text{ and }\\ &\quad{\hat{w}}({\hat{S}}_{v}^{i})\leq\hat{\lambda}(s,v)+\MT@delim@Auto\paren@star{2(\MT@delim@Auto\floor@star{\lg|U|}-i)+1}{\Gamma_{\text{iso}}}+{\Gamma_{\text{values}}}\\ 0\qquad&\text{o.w. }\end{cases}.

(5)

Lemma 4.2.

Consider a vertex $v\in D^{*}$ . Let $S_{v}^{i}$ be the $v$ part of an optimal solution to Min Isolating Cuts at stage $i$ . Assume that $|\lambda(s,v)-\hat{\lambda}(s,v)|\leq{\Gamma_{\text{values}}}$ and $|w({\hat{S}}_{v}^{i})-{\hat{w}}({\hat{S}}_{v}^{i})|+|w(S_{v}^{i})-w({\hat{S}}_{v}^{i})|\leq{\Gamma_{\text{iso}}}$ for all $i\in\{0,\ldots\MT@delim@Auto\floor@star{\lg|U|}\}$ . Then, there exists an $i^{\prime}\in\{0,\ldots\MT@delim@Auto\floor@star{\lg|U|}\}$ such that

{\mathbb{E}}\left[X_{v}^{i^{\prime}}\right]=O(1).

Proof.

Consider a specific sampling level $i\in\{0,\ldots,\lfloor\lg|U|\rfloor\}$ . We say that $i$ is “active” if there exists a set $\tilde{S}_{v}^{i}\subset U$ containing $v$ and not $s$ such that $|\tilde{S}_{v}^{i}\cap U|\in[2^{i},2^{i+1})$ and

w(\tilde{S}_{v}^{i})\leq\hat{\lambda}(s,v)+2(\MT@delim@Auto\floor@star{\lg(|U|)}-i){\Gamma_{\text{iso}}}+{\Gamma_{\text{values}}}.

(6)

Note that this is a deterministic property regarding the existence of such a set $\tilde{S}_{v}^{i}$ independent of the randomness used to sample terminals or find private Min Isolating Cuts.

Let $i^{\prime}$ be the smallest active $i$ . Let $S^{*}_{v}$ be the $v$ side of the true min $s\text{-}v$ cut, and let $i^{*}=\lfloor\lg|S^{*}_{v}\cap U|\rfloor$ . As $w(S^{*}_{v})=\lambda(s,v)\leq\hat{\lambda}(s,v)+{\Gamma_{\text{values}}}$ , $i^{*}$ must be active, so $i^{\prime}$ is well-defined and $i^{\prime}\leq i^{*}$ . As $i^{\prime}$ is active, there exists a set $\tilde{S}_{v}^{i^{\prime}}$ with $|\tilde{S}_{v}^{i^{\prime}}\cap U|\in[2^{i^{\prime}},2^{i^{\prime}+1})$ and with cost within $2(\MT@delim@Auto\floor@star{\lg(|U|)}-i^{\prime}){\Gamma_{\text{iso}}}+{\Gamma_{\text{values}}}$ of $\hat{\lambda}(s,v)$ . On the other hand, as $i^{\prime}$ is the smallest active level, there is no set of size less than $2^{i^{\prime}}$ with cost within $2(\MT@delim@Auto\floor@star{\lg(|U|)}-(i^{\prime}-1)){\Gamma_{\text{iso}}}+{\Gamma_{\text{values}}}$ of $\hat{\lambda}(s,v)$ .

Consider the event that in $R^{i^{\prime}}$ , we sample $v$ but no other vertices in $\tilde{S}_{v}^{i^{\prime}}$ as terminals, i.e., $R^{i^{\prime}}\cap\tilde{S}_{v}^{i^{\prime}}=\{v\}$ . Then, $\tilde{S}_{v}^{i^{\prime}}$ would be a valid output of a call to isolating cuts. By the assumed guarantee of the error of the private Min Isolating Cuts algorithm, the actual cut we output has approximated cost:

	$\displaystyle{\hat{w}}({\hat{S}}_{v}^{i^{\prime}})$	$\displaystyle\leq w(\tilde{S}_{v}^{i^{\prime}})+\|w(\tilde{S}_{v}^{i^{\prime}})-{\hat{w}}({\hat{S}}_{v}^{i^{\prime}})\|$
		$\displaystyle\leq w(\tilde{S}_{v}^{i^{\prime}})+\|w(\tilde{S}_{v}^{i^{\prime}})-w({\hat{S}}_{v}^{i^{\prime}})\|+\|w({\hat{S}}_{v}^{i^{\prime}})-{\hat{w}}({\hat{S}}_{v}^{i^{\prime}})\|$
		$\displaystyle\leq w(\tilde{S}_{v}^{i^{\prime}})+{\Gamma_{\text{iso}}}$
		$\displaystyle\leq\hat{\lambda}(s,v)+\MT@delim@Auto\paren@star{2(\MT@delim@Auto\floor@star{\lg(\|U\|)}-i^{\prime})+1}{\Gamma_{\text{iso}}}+{\Gamma_{\text{values}}}.$

For sake of contradiction, consider the case that $|{\hat{S}}_{v}^{i^{\prime}}\cap U|<2^{i^{\prime}}$ . Using the fact that all solutions of this size have large cost, we can conclude that

	$\displaystyle{\hat{w}}({\hat{S}}_{v}^{i^{\prime}})$	$\displaystyle\geq w({\hat{S}}_{v}^{i^{\prime}})-{\Gamma_{\text{iso}}}$
		$\displaystyle>\hat{\lambda}(s,v)+2(\MT@delim@Auto\floor@star{\lg(\|U\|)}-(i^{\prime}-1)){\Gamma_{\text{iso}}}+{\Gamma_{\text{values}}}-{\Gamma_{\text{iso}}}$
		$\displaystyle>\hat{\lambda}(s,v)+\MT@delim@Auto\paren@star{2(\MT@delim@Auto\floor@star{\lg(\|U\|)}-i^{\prime}))+1}{\Gamma_{\text{iso}}}+{\Gamma_{\text{values}}}.$

This contradicts the previous inequality that shows that ${\hat{w}}({\hat{S}}_{v}^{i^{\prime}})$ is upper bounded by this quantity, so $|{\hat{S}}_{v}^{i^{\prime}}\cap U|\geq 2^{i^{\prime}}$ as long as the sampling event occurs.

Next, we show that $|{\hat{S}}_{v}^{i^{\prime}}\cap U|\leq(9/10)|U|$ . As $v\in D^{*}$ , the true minimum cut $S^{*}_{v}$ has the property $|S^{*}_{v}\cap U|\leq|U|/2$ . Furthermore, by the isolating cuts lemma of [LP20], the minimum isolating cut solution for any set of terminals including $v$ and $s$ will have that the $v$ part $S_{v}$ is a subset of $S^{*}_{v}$ (this is the basis for the isolating cuts algorithm). So, if $v$ is sampled in $R^{i}$ , there will exist an optimal isolating cuts solution $S_{v}^{i}$ with $|S_{v}^{i}\cap U|\leq|U|/2$ . By the guarantee of Lemma 3.1, $|{\hat{S}}_{v}^{i^{\prime}}\cap U|\leq(9/10)|U|$ .

Overall, we can bound the contribution of ${\hat{S}}_{v}^{i^{\prime}}$ to $D^{i^{\prime}}$ as

	$\displaystyle{\mathbb{E}}\left[X_{v}^{i^{\prime}}\right]$	$\displaystyle\geq\|S_{v}^{i^{\prime}}\cap U\|\cdot\operatorname*{\mathbb{P}}\left[\text{$v$ is the only vertex sampled in }\tilde{S}_{v}^{i^{\prime}}\text{ under sampling probability }2^{-i^{\prime}}\right]$
		$\displaystyle\geq 2^{i^{\prime}}\left(2^{-i^{\prime}}\right)\left(1-2^{-i^{\prime}}\right)^{\|\tilde{S}_{v}^{i^{\prime}}\|-1}$
		$\displaystyle\geq\left(1-2^{-i^{\prime}}\right)^{2^{i^{\prime}+1}-2}.$

If $i^{\prime}=0$ , this evaluates to $1$ . Otherwise, if $i^{\prime}\geq 1$ ,

{\mathbb{E}}\left[X_{v}^{i^{\prime}}\right]\geq\left(1-2^{-i^{\prime}}\right)^{2^{i^{\prime}+1}}=\MT@delim@Auto\paren@star{\frac{1}{1+\frac{2^{-i^{\prime}}}{1-2^{-i^{\prime}}}}}^{2^{i^{\prime}+1}}\geq\MT@delim@Auto\paren@star{\frac{1}{e^{\frac{2^{-i^{\prime}}}{1-2^{-i^{\prime}}}}}}^{2^{i^{\prime}+1}}=e^{-\frac{2}{1-2^{-i^{\prime}}}}\geq e^{-4}.

$\Box$

We are now ready to prove the main lemma of this section.

Proof of Lemma 4.1.

The first step of the proof will be to show that ${\Gamma_{\text{iso}}}$ and ${\Gamma_{\text{values}}}$ upper bound the error of the approximate isolating cuts and min cut values used in the algorithm with probability $1-O(\beta)$ . Applying the guarantee of Lemma 3.1 and union bounding over all $i$ , with probability $1-\beta$ , we get the following guarantee of the quality of ${\hat{S}}_{v}^{i}$ . If $\{S_{v}^{i}\}$ are optimal Min Isolating Cuts for terminals $R^{i}$ :

\sum_{v\in R^{i}}w({\hat{S}}_{v}^{i})-w(S_{v}^{i})\leq O\MT@delim@Auto\paren@star{\frac{(n+\lg(1/\beta))\lg^{3}(|R^{i}|)}{\varepsilon}}.

Furthermore, via the tail of the Laplace distribution and a union bound over all $i$ , each of the approximated weights satisfies the following inequality with probability $1-\beta$ :

|{\hat{w}}({\hat{S}}_{v}^{i})-w({\hat{S}}_{v}^{i})|\leq O\MT@delim@Auto\paren@star{\frac{\lg|U|(\lg(|U|/\beta))}{\varepsilon}}.

Therefore, there exists a choice of ${\Gamma_{\text{iso}}}=O\MT@delim@Auto\paren@star{\frac{(n+\lg(1/\beta))\lg^{3}(n)}{\varepsilon}}$ such that with probability $1-O(\beta)$ , for all $i\in\{0,\ldots,\MT@delim@Auto\floor@star{\lg|U|}\}$ ,

\sum_{v\in R^{i}}|w(S_{v}^{i})-w({\hat{S}}_{v}^{i})|+|w({\hat{S}}_{v}^{i})-{\hat{w}}({\hat{S}}_{v}^{i})|\leq{\Gamma_{\text{iso}}}.

This satisfies the approximate Min Isolating Cuts guarantee of the lemma.

For the approximate minimum cut values, by the tail of the Laplace distribution and a union bound, each $\hat{\lambda}(s,v)$ satisfies

|\lambda(s,v)-\hat{\lambda}(s,v)|\leq{\Gamma_{\text{values}}}=O\MT@delim@Auto\paren@star{\frac{|U|\lg(|U|/\beta)}{\varepsilon}}

with probability $1-\beta$ . We will condition on these events going forward.

Sets ${\hat{S}}_{v}^{i}$ are only included in our output if they are close to the min cut value $\lambda(s,v)$ . Specifically, for any set returned by our algorithm:

{\hat{w}}({\hat{S}}_{v}^{i})\leq\hat{\lambda}(s,v)+\MT@delim@Auto\paren@star{2(\MT@delim@Auto\floor@star{\lg|U|}-i)+1}{\Gamma_{\text{iso}}}+{\Gamma_{\text{values}}}.

Applying the error guarantees for ${\hat{w}}({\hat{S}}_{v}^{i})$ , ${\hat{S}}_{v}^{i}$ , and $\hat{\lambda}(s,v)$ ,

	$\displaystyle w({\hat{S}}_{v}^{i})$	$\displaystyle\leq{\Gamma_{\text{iso}}}+{\hat{w}}({\hat{S}}_{v}^{i})$
		$\displaystyle\leq{\Gamma_{\text{iso}}}+\hat{\lambda}(s,v)+\MT@delim@Auto\paren@star{2(\MT@delim@Auto\floor@star{\lg\|U\|}-i)+1}{\Gamma_{\text{iso}}}+{\Gamma_{\text{values}}}$
		$\displaystyle\leq{\Gamma_{\text{iso}}}+(\lambda(s,v)+{\Gamma_{\text{values}}})+\MT@delim@Auto\paren@star{2(\MT@delim@Auto\floor@star{\lg\|U\|}-i)+1}{\Gamma_{\text{iso}}}+{\Gamma_{\text{values}}}$
		$\displaystyle\leq\lambda(s,v)+2(\MT@delim@Auto\floor@star{\lg\|U\|}+1){\Gamma_{\text{iso}}}+2{\Gamma_{\text{values}}}.$

This completes the first part of the proof concerning the error of the returned sets. In the remainder, we focus on the cardinality of the output.

By definition of $X_{v}^{i}$ , the size of $D^{i}$ is given by a sum over $X_{v}^{i}$ :

|D^{i}|=\sum_{v\in U\setminus\{s\}}X_{v}^{i}

By linearity of expectation and as $D^{*}\subseteq U\setminus\{s\}$ ,

{\mathbb{E}}\left[\sum_{i=0}^{\lfloor\lg|U|\rfloor}|D^{i}|\right]\geq\sum_{i=0}^{\lfloor\lg|U|\rfloor}\sum_{v\in D^{*}}{\mathbb{E}}\left[X_{v}^{i}\right].

As we output the largest $D^{i}$ across all $i$ , the output of our algorithm will have expected size at least

\frac{1}{\lfloor\lg|U|\rfloor+1}\sum_{i=0}^{\lfloor\lg|U|\rfloor}\sum_{v\in D^{*}}{\mathbb{E}}\left[X_{v}^{i}\right].

Via Lemma 4.2 (note that the error condition holds with probability $1-O(\beta)$ from the first part of this proof), the expected output size will be at least

\Omega\MT@delim@Auto\paren@star{\frac{|D^{*}|}{\lg|U|}}.

The final result follows from Markov’s inequality. $\Box$

4.2 Privacy

Lemma 4.3.

PrivateGHTreeStep (Algorithm 2) is $\varepsilon$ -DP.

Proof.

The algorithm PrivateGHTreeStep interacts with the sensitive edges only through calculations of approximate min cut values $\hat{\lambda}(s,v)$ , approximate isolating cut values ${\hat{w}}({\hat{S}}_{v}^{i})$ , and calls to PrivateMinIsolatingCuts (Algorithm 1). Otherwise, all computation only deals with the vertices of the graph which are public. The calculation of each of the $|U|-1$ cut values is $\frac{\varepsilon}{4(|U|-1)}$ -DP via the Laplace mechanism (Theorem 2.2). Via the privacy of PrivateMinIsolatingCuts, each call to that subroutine is $\frac{\varepsilon}{2(\MT@delim@Auto\floor@star{\lg|U|}+1)}$ -DP. At any sampling level $i$ , the approximate isolating cuts ${\hat{S}}_{v}^{i}$ are disjoint. So, calculation of $w({\hat{S}}_{v}^{i})$ for all $v\in R^{i}\setminus\{s\}$ has sensitivity $2$ as each edge can cross at most two sets. By the Laplace mechanism, calculation of all ${\hat{w}}({\hat{S}}_{v}^{i})$ for any $i$ is $\frac{\varepsilon}{4(\MT@delim@Auto\floor@star{\lg|U|}+1)}$ -DP. Summing over all sampling levels, the total privacy via basic composition (Theorem 2.1) is:

(|U|-1)\frac{\varepsilon}{4(|U|-1)}+(\MT@delim@Auto\floor@star{\lg|U|}+1)\MT@delim@Auto\paren@star{\frac{\varepsilon}{{2(\MT@delim@Auto\floor@star{\lg|U|}+1)}}+\frac{\varepsilon}{4(\MT@delim@Auto\floor@star{\lg|U|}+1)}}=\frac{\varepsilon}{4}+\frac{\varepsilon}{2}+\frac{\varepsilon}{4}=\varepsilon.

$\Box$

5 Final Algorithm

Algorithm 3 PrivateGHTree

(G=(V,E,w),\varepsilon)

(T,f)\leftarrow{\color[rgb]{0.8,0.25,0.33}{\text{PrivateGHSteinerTree}{}(G,V,\varepsilon/2,0,n)}}

2:Add

\text{Lap}\MT@delim@Auto\paren@star{\frac{2(n-1)}{\varepsilon}}

noise to each edge in

T

3:return

T

Algorithm 4 PrivateGHSteinerTree

(G=(V,E,w),U,{\color[rgb]{0.8,0.25,0.33}{\varepsilon,t,n_{\max}}})

t_{\max}\leftarrow\Omega(\lg^{2}n_{\max})

2:if

t>t_{\max}

then

3: return abort // the privacy budget is exhausted

4:end if

s\leftarrow

uniformly random vertex in

U

6:Call PrivateGHTreeStep

(G,s,U,\frac{\varepsilon}{4t_{\max}},\frac{1}{n_{\max}^{3}})

to obtain

D,R\subseteq U

and disjoint sets

{\hat{S}}_{v}

(where

D=\bigcup_{v\in R}{\hat{S}}_{v}\cap U

)

7:for each

v\in R

8: Let

G_{v}

be the graph with vertices

V\setminus{\hat{S}}_{v}

contracted to a single vertex

x_{v}

9: Add edges with weight

\text{Lap}\MT@delim@Auto\paren@star{\frac{8t_{\max}}{\varepsilon}}

from

x_{v}

to every other vertex in

G_{v}

, truncating resulting edge weights to be at least

0

10:

U_{v}\leftarrow{\hat{S}}_{v}\cap U

11: If

|U_{v}|>1

, recursively set

(T_{v},f_{v})\leftarrow{\color[rgb]{0.8,0.25,0.33}{\text{PrivateGHSteinerTree}{}(G_{v},U_{v},\varepsilon,t+1,n_{\max})}}

12:end for

13:Let

G_{\text{large}}

be the graph

G

with (disjoint) vertex sets

{\hat{S}}_{v}

contracted to single vertices

y_{v}

for all

v\in D

14:

U_{\text{large}}\leftarrow U\setminus D

15:If

|U_{\text{large}}|>1

, recursively set

(T_{\text{large}},f_{\text{large}})\leftarrow{\color[rgb]{0.8,0.25,0.33}{\text{PrivateGHSteinerTree}{}(G_{\text{large}},U_{\text{large}},\varepsilon,t+1,n_{\max})}}

16:return Combine

((T_{\text{large}},f_{\text{large}}),\{(T_{v},f_{v}):v\in R\},\{w({\hat{S}}_{v}):v\in R\})

// The weights used in this step are not private

Algorithm 5 Combine

((T_{\text{large}},f_{\text{large}}),\{(T_{v},f_{v}):v\in R\},\{w({\hat{S}}_{v}):v\in R)\}

1:Construct

T

by starting with the disjoint union

T_{\text{large}}\cup\bigcup_{v\in R}T_{v}

and for each

v\in R

, adding an edge between

f_{v}(x_{v})\in U_{v}

and

f_{\text{large}}(y_{v})\in U_{\text{large}}

with weight

w({\hat{S}}_{v})

2:Construct

f:V\to U=U_{\text{large}}\cup\bigcup_{v\in R}U_{v}

f(v^{\prime})=f_{\text{large}}(v^{\prime})

v^{\prime}\in V\setminus\bigcup_{v\in R}{\hat{S}}_{v}

and

f(v^{\prime})=f_{v}(v^{\prime})

v^{\prime}\in{\hat{S}}_{v}

for some

v\in R

In this section, we present the algorithm PrivateGHTree (Algorithm 3) for constructing an $\varepsilon$ -DP approximate Gomory-Hu tree and analyze its approximation error and privacy guarantees. The steps that differ meaningfully from the non-private version developed in [Li21] are in color. As in [Li21], we construct the slightly more general structure of an Gomory-Hu Steiner tree as an intermediate step in Algorithm 4.

Definition 5.1.

Let $G=(V,E,w)$ be a weighted graph and $U$ a set of terminals. A $\Gamma$ -approximate Gomory-Hu Steiner tree is a weighted spanning tree $T$ on $U$ with a function $f:V\to U$ such that $f|_{U}$ is the identity and

•

for all distinct $s,t\in U$ , if $(u,v)$ is the minimum weight edge on the unique path between $s$ and $t$ , in $T$ , and if $U^{\prime}$ is the connected component of $T\setminus\{(u,v)\}$ containing $s$ , then $f^{-1}(U^{\prime})$ is a $\Gamma$ -approximate Min- $s$ - $t$ -Cut with $\lambda_{G}(s,t)\leq w_{T}(u,v)=w_{G}(f^{-1}(U^{\prime}))\leq\lambda_{G}(s,t)+\Gamma$ .

To construct the final approximate Gomory-Hu tree, we make a call to PrivateGHSteinerTree (Algorithm 4) with $U=V$ , the entire vertex set. The algorithm PrivateGHSteinerTree is a private version of the GHTree algorithm in [Li21]. It computes several (approximate) min cuts from a randomly sampled vertex $s\in U$ by making a call to PrivateGHTreeStep (Algorithm 2) to obtain $D,R\subseteq U$ and disjoint sets ${\hat{S}}_{v}$ (where $D=\bigcup_{v\in R}{\hat{S}}_{v}\cap U$ ). For each of these cuts ${\hat{S}}_{v}$ it constructs recursive sub-instances $(G_{v},U_{v})$ where $G_{v}$ is obtained by contracting $V\setminus{\hat{S}}_{v}$ to a single vertex $x_{v}$ and $U_{v}\leftarrow{\hat{S}}_{v}\cap U$ . Moreover, it creates a sub-instance $(G_{\text{large}},U_{\text{large}})$ by contracting each of ${\hat{S}}_{v}$ to a single vertex $y_{v}$ for $y\in D$ and setting $U_{\text{large}}=U\leftarrow D$ .

Notably, on 9, where the algorithm recurses on the graph $G_{v}$ with $V\setminus{\hat{S}}_{v}$ contracted to a single vertex $x_{v}$ , we add noisy edges from $x_{v}$ to all other vertices of the graph. This ensures the privacy of any actual edge from $x_{v}$ in the entire recursive subtree of that instance without incurring too much error. This will imply that for any edge and any instance during the recursion, there is at most one sub-instance where the edge does not receive this privacy guarantee. If $t$ is the depth of the recursion tree, this allows us to apply basic composition over only $O(t)$ computations of the algorithm. Essentially, there is only one path down the recursion tree on which we need to track privacy for any given edge in the original graph. We enforce $t<t_{\max}$ , and as we will show, the algorithm successfully terminates with depth less than $t_{\max}$ with high probability.

To combine the solutions to the recursive sub-problems, we use the Combine algorithm (Algorithm 5) from [Li21], which in turn is similar to the original Gomory-Hu combine step except that it combines more than two recursive sub-instances.

Finally, Algorithm 3 calls Algorithm 4 with privacy budget $\varepsilon/2$ . To be able to output weights of the tree edges, it simply adds Laplace noise $\text{Lap}(\frac{1}{2(n-1)\varepsilon})$ to each of them, hence incurring error $O(\frac{n\lg n}{\varepsilon})$ with high probability. This also has privacy loss $\varepsilon/2$ by basic composition, so the full algorithm is $\varepsilon$ -differentially private.

5.1 Correctness

In this section, we analyze the approximation guarantee of our algorithm. The main lemma states that Algorithm 4 outputs an $O(n\operatorname{polylog}(n))$ -approximate Gomory-Hu Steiner tree.

Lemma 5.1.

Let $t_{\max}=C\lg^{2}n$ for a sufficiently large constant $C$ . PrivateGHSteinerTree $(G,V,\varepsilon,0,n)$ outputs an $O(\frac{n\lg^{8}n}{\varepsilon})$ -approximate Gomory-Hu Steiner tree $T$ of $G$ with high probability.

We start by proving a lemma for analyzing a single recursive step of the algorithm. It is similar to [Li21, Lemma 4.5.4] but its proof requires a more careful application of the submodularity lemma.

Lemma 5.2.

With high probability, for any distinct vertices $p,q\in U_{\text{large}}$ , we have that $\lambda_{G}(p,q)\leq\lambda_{G_{\text{large}}}(p,q)\leq\lambda_{G}(p,q)+O(\frac{n\lg^{5}n}{\varepsilon})$ . Also with high probability, for any $v\in R$ and distinct vertices $p,q\in U_{v}$ , we have that $\lambda_{G}(p,q)\leq\lambda_{G_{v}}(p,q)\leq\lambda_{G}(p,q)+O(\frac{n\lg^{6}n}{\varepsilon})$ .

Proof.

Let us start by upper bounding how close the cuts ${\hat{S}}_{v}$ are to being Min- $s$ - $v$ -Cuts and how close the approximate min cut values ${\hat{w}}({\hat{S}}_{v})$ are to the true sizes $w({\hat{S}}_{v})$ . Algorithm 4 calls Algorithm 2 with privacy parameter $\varepsilon_{1}=\frac{\varepsilon}{4t_{\max}}=\Theta\MT@delim@Auto\paren@star{\frac{\varepsilon}{\lg^{2}n}}$ . By Lemma 4.1 with privacy $\varepsilon_{1}$ , it follows that (a) the sets $\{{\hat{S}}_{v}\}$ are approximate minimum isolating cuts with total error $O\MT@delim@Auto\paren@star{\frac{n\lg^{5}n}{\varepsilon}}$ and (b) each set ${\hat{S}}_{v}$ is an approximate Min- $s$ - $v$ -Cut with error $O\MT@delim@Auto\paren@star{\frac{n\lg^{6}n}{\varepsilon}}$ . On any given call to Algorithm 2, these error bounds hold with probability $1-n_{\max}^{-3}$ where $n_{\max}$ is the number of vertices in the original graph. As each call to Algorithm 2 ultimately contributes an edge to the final Gomory-Hu tree via Algorithm 5, there can be at most $n_{\max}-1$ calls throughout the entire recursion tree, resulting in failure probability of $n_{\max}^{-2}$ overall after a union bound.

The fact that $\lambda_{G}(p,q)\leq\lambda_{G_{\text{large}}}(p,q)$ follows since $G_{\text{large}}$ is a contraction of $G$ . To prove the second inequality, let $S$ be one side of the true Min- $p$ - $q$ -Cut. Let $R_{1}=R\cap S$ and $R_{2}=R\setminus S$ . We show that the cut $S^{*}=(S\cup\bigcup_{v\in R_{1}}{\hat{S}}_{v})\setminus(\bigcup_{v\in R_{2}}{\hat{S}}_{v})$ is an $O(n\lg^{6}n/\varepsilon)$ -approximate min cut. Since $S^{*}$ is also a cut in $G_{\text{large}}$ , the desired bound on $\lambda_{G_{\text{large}}}(p,q)$ follows.

Let $v_{1},\dots,v_{|R_{1}|}$ be the vertices of $R_{1}$ in an arbitrary order. By $|R_{1}|$ applications of the submodularity lemma (Lemma 2.1),

w\left(S\cup\bigcup_{v\in R_{1}}{\hat{S}}_{v}\right)\leq w(S)+\sum_{i=1}^{|R_{1}|}\left(w({\hat{S}}_{v_{i}})-w\left(\left(S\cup\bigcup_{j<i}{\hat{S}}_{v_{j}}\right)\cap{\hat{S}}_{v_{i}}\right)\right).

Note that $S\cup\bigcup_{v\in R_{1}}{\hat{S}}_{v}$ is still a $(p,q)$ -cut as $p,q\in U_{\text{large}}$ and the sets ${\hat{S}}_{v_{i}}$ are each disjoint from $U_{\text{large}}$ . Moreover, for each $i$ , $S$ contains $v_{i}$ and so $\left(S\cup\bigcup_{j<i}{\hat{S}}_{v_{j}}\right)\cap{\hat{S}}_{v_{i}}$ isolates $v_{i}$ from all vertices in $V\setminus{\hat{S}}_{v_{i}}$ . The ${\hat{S}}_{v_{i}}$ are the outputs of the private isolating cuts algorithm (Algorithm 1). Using the fact that $\{{\hat{S}}_{v}\}$ are approximate minimum isolating cuts, the sum in the RHS above can be upper bounded by $O(\frac{n\lg^{5}n}{\varepsilon})$ . Letting $S^{\prime}=S\cup\bigcup_{v\in R_{1}}{\hat{S}}_{v}$ , and $S^{\prime\prime}=(V\setminus S^{\prime})\cup\bigcup_{v\in R_{2}}{\hat{S}}_{v}$ a similar argument but applied to $V\setminus S^{\prime}$ , shows that

w(S^{\prime\prime})=w\left((V\setminus S^{\prime})\cup\bigcup_{v\in R_{2}}{\hat{S}}_{v}\right)\leq w(V\setminus S^{\prime})+O\MT@delim@Auto\paren@star{\frac{n\lg^{5}n}{\varepsilon}}.

But $S^{*}=V\setminus S^{\prime\prime}$ , so we get that

w(S^{*})=w(S^{\prime\prime})\leq w(S^{\prime})+O\MT@delim@Auto\paren@star{\frac{n\lg^{5}n}{\varepsilon}}\leq w(S)+O\MT@delim@Auto\paren@star{\frac{n\lg^{5}n}{\varepsilon}},

as desired.

For the case of $p,q\in U_{v}$ for some $v$ , again the bound $\lambda_{G}(p,q)\leq\lambda_{G_{u}}(p,q)$ is clear. Thus, it suffices to consider the upper bound on $\lambda_{G_{v}}(p,q)$ . Let $S$ be the side of a Min- $p$ - $q$ -Cut in $G$ which does not contain $v$ . Assume first that $s\notin S$ . By the submodularity lemma (Lemma 2.1)

w(S\cap{\hat{S}}_{v})\leq w(S)+w({\hat{S}}_{v})-w(S\cup{\hat{S}}_{v}).

By the approximation guarantees of Algorithm 2, $w({\hat{S}}_{v})\leq\lambda_{G}(s,v)+O(\frac{n\lg^{6}n}{\varepsilon})$ . Moreover, note that $S\cup{\hat{S}}_{v}$ is an $(s,v)$ -cut of $G$ , so $w(S\cup{\hat{S}}_{v})\geq\lambda_{G}(s,v)$ . Thus,

w(S\cap{\hat{S}}_{v})\leq w(S)+O\MT@delim@Auto\paren@star{\frac{n\lg^{6}n}{\varepsilon}}=\lambda_{G}(p,q)+O\MT@delim@Auto\paren@star{\frac{n\lg^{6}n}{\varepsilon}}.

Since $S\cap{\hat{S}}_{v}$ is a $(p,q)$ -cut of $G_{v}$ , we must have that $w(S\cap{\hat{S}}_{v})\geq\lambda_{G_{v}}(p,q)$ , so in conclusion $\lambda_{G_{v}}(p,q)\leq\lambda_{G}(p,q)+O(\frac{n\lg^{6}n}{\varepsilon})$ ignoring the added noisy edges to $G_{v}$ in 9 of Algorithm 4.

Adding the noisy edges can only increase the cost by $O(\frac{n\lg^{3}n}{\varepsilon})$ with high probability via Laplace tail bounds (note that there can be at most $n-1$ noisy edges added as each time a noisy edge is added an edge is added to the final approximate Gomory Hu Steiner tree). This finishes the proof in the case $s\notin S$ . A similar argument handles the case where $s\in S$ but here we relate the value $w(V\setminus S)$ to $(w(V\setminus S)\cap{\hat{S}}_{v})$ . $\Box$

To bound the error of the algorithm, we need a further lemma bounding the depth of its recursion. The argument is similar to that of [Li21].

Lemma 5.3.

If $t_{\max}=C\lg^{2}n$ for a sufficiently large constant $C$ , then, with high probability, no recursive call to Algorithm 4 from PrivateGHTree $(G,\varepsilon)$ aborts.

Proof.

Each of the recursive instances, $(G_{v},U_{v})$ has $|U_{v}|\leq\frac{9}{10}|U|$ by the way they are picked in 7 of Algorithm 2. Moreover, by a lemma from [AKT20], if $s$ is picked uniformly at random from $U$ , then ${\mathbb{E}}\left[D^{*}\right]=\Omega(|U|-1)$ . By Lemma 4.1, the expected size of $\bigcup_{v\in R}{\hat{S}}_{v}$ returned by a call to Algorithm 2 when picking $s$ at random from $|U|$ is then at least $\Omega\MT@delim@Auto\paren@star{\frac{|U|-1}{\lg n}}$ with constant probability. It follows that, ${\mathbb{E}}\left[|U_{\text{large}}|\right]\leq|U|(1-\Omega(1/\lg n))$ when $|U|\geq 1$ . By Markov’s inequality and union bound, all sub-instances have $|U|=1$ within $O(\lg^{2}n)$ recursive depth with high probability. $\Box$

We can now prove Lemma 5.1. The argument is again similar to [Li21] except we have to incorporate the approximation errors.

Proof of Lemma 5.1.

By Lemma 5.3, the algorithm does not abort with high probability.

Let $\Delta=O(\frac{n\lg^{6}n}{\varepsilon})$ be such that with high probability $\lambda_{G_{v}}(p,q)\leq\lambda_{G}(p,q)+\Delta$ for $p,q\in U_{v}$ and similarly $\lambda_{G_{large}}(p,q)\leq\lambda_{G}(p,q)+\Delta$ for $p,q\in U_{large}$ . The existence of $\Delta$ is guaranteed by Lemma 5.2. We prove by induction on $i=0,\dots,t_{\max}$ , that the output to the instances at level $t_{\max}-i$ of the recursion are $2i\Delta$ -approximate Gomory-Hu Steiner trees. This holds trivially for $i=0$ as the instances on that level have $|U|=1$ and the tree is the trivial one-vertex tree approximating no cuts at all. Let $i\geq 1$ and assume inductively that the result holds for smaller $i$ . In particular, if $(T,f)$ is the output of an instance at recursion level $i$ , then the trees $(T_{v},f_{v})$ and $(T_{\text{large}},f_{\text{large}})$ are $2(i-1)\Delta$ -approximate Gomory-Hu Steiner trees of their respective $G_{v}$ or $G_{\text{large}}$ graphs.

Consider any internal edge $(a,b)\in T_{\text{large}}$ (without loss of generality, what follows also holds for $(a,b)\in T_{v})$ . Let $U^{\prime}$ and $U^{\prime}_{\text{large}}$ be the connected component containing $a$ after removing $(a,b)$ from $T$ and $T_{\text{large}}$ , respectively. By design of Algorithm 5, $f^{-1}_{\text{large}}(U^{\prime}_{\text{large}})$ and $f^{-1}(U^{\prime})$ are the same except each contracted vertex $y_{v}\in f^{-1}_{\text{large}}(U^{\prime}_{\text{large}})$ appears as ${\hat{S}}_{v}\subseteq f^{-1}(U^{\prime})$ . It follows that $w_{G_{\text{large}}}(f_{\text{large}}^{-1}(U^{\prime}_{\text{large}}))=w_{G}(f^{-1}(U^{\prime}))$ . By the inductive hypothesis, $(T_{\text{large}},f_{\text{large}})$ is an approximate Gomory-Hu Steiner tree, so $w_{G_{\text{large}}}(f_{\text{large}}^{-1}(U^{\prime}_{\text{large}}))=w_{T_{\text{large}}}(a,b)$ . Therefore setting $w_{T}(a,b)=w_{T_{\text{large}}}(a,b)=w_{G}(f^{-1}(U^{\prime}))$ has the correct cost for $T$ according to the definition of an approximate Gomory-Hu Steiner tree.

Furthermore, on the new edges $(f_{v}(x_{v}),f_{\text{large}}(y_{v}))$ , the weight $w({\hat{S}}_{v})$ is the correct weight for that edge in $T$ as ${\hat{S}}_{v}$ is the $f_{v}(x_{v})$ side of the connected component after removing that edge. Finally, by adding these new edges, the resulting tree is a spanning tree. So, the structure of the tree is correct, and it remains to argue that the cuts induced by the tree (via minimum edges on shortest paths) are approximate Min- $s$ - $t$ -Cuts.

Consider any $p,q\in U$ . Let $(a,b)$ be the minimum edge on the shortest path in $T$ . Note that it is always the case that $w_{T}(a,b)\geq\lambda(a,b)$ as $w_{T}(a,b)$ corresponds to the value of a cut in $G$ separating $a$ and $b$ . We will proceed by cases.

If $(a,b)\in T_{\text{large}}$ , then by induction, $w_{T}(a,b)=w_{T_{\text{large}}}(a,b)\leq\lambda_{G_{\text{large}}}(a,b)+(2i-2)\Delta$ . By Lemma 5.2, it follows that $w_{T}(a,b)\leq\lambda_{G}(a,b)+(2i-1)\Delta$ . The exact same argument applies if $(a,b)\in T_{v}$ for some $v\in R$ .

The case that remains is if $(a,b)$ is a new edge with $a=f_{v}^{-1}(x_{v})\in U_{v}$ and $b=f_{\text{large}}^{-1}(y_{v})\in U_{\text{large}}$ . Then, $w_{T}(a,b)=w_{G}({\hat{S}}_{v})$ . By Lemma 4.1, ${\hat{S}}_{v}$ is an approximate Min- $v$ - $s$ -Cut and $w_{T}(a,b)\leq\lambda_{G}(v,s)+\Delta$ . To connect this value to $\lambda_{G}(a,b)$ , note that over the choices of where $v$ and $s$ lie on the Min- $a$ - $b$ -Cut,

\lambda_{G}(a,b)\geq\min\MT@delim@Auto\brace@star{\lambda_{G}(a,v),\lambda_{G}(v,s),\lambda_{G}(s,b)}.

Let $S^{\prime}_{a}$ be the $a$ side of the $(a,v)$ cut induced by the approximate Gomory-Hu Steiner tree $(T_{v},f_{v})$ . As $a=f_{v}^{-1}(x_{v})$ and $s\in x_{v}$ , $S^{\prime}_{a}$ is also the $s$ side of a $(v,s)$ cut. Therefore, $w_{G}(S^{\prime}_{a})\geq\lambda_{G}(v,s)$ . On the other hand, by our inductive hypothesis and Lemma 5.2, this is an approximate Min- $a$ - $v$ -Cut: $w_{G}(S^{\prime}_{a})\leq\lambda_{G}(a,v)+(2i-1)\Delta$ . The analogous argument holds to show $\lambda_{G}(v,s)\leq\lambda_{G}(s,b)+(2i-1)\Delta$ . Therefore,

\lambda_{G}(v,s)\leq\min\MT@delim@Auto\brace@star{\lambda_{G}(a,v),\lambda_{G}(s,b)}+(2i-1)\Delta

and

w_{T}(a,b)\leq\lambda_{G}(v,s)+\Delta\leq\lambda_{G}(a,b)+2i\Delta.

In all cases, $w_{T}(a,b)\leq\lambda_{G}(a,b)+2i\Delta$ . As the cut corresponding to the edge $(a,b)$ is on the path from $p$ to $q$ , it is also a $(p,q)$ -cut, so $\lambda_{G}(p,q)\leq w_{G}(a,b)$ . Furthermore, it must the case that there is an edge $(a^{\prime},b^{\prime})$ along the path between $p$ to $q$ such that $a^{\prime}$ and $b^{\prime}$ are in different sides of the true Min- $p$ - $q$ -Cut. Otherwise, $p$ and $q$ would be on the same side of the cut. Therefore, $\lambda_{G}(p,q)\geq\lambda_{G}(a^{\prime},b^{\prime})$ . As we chose $(a,b)$ to be the minimum weight edge,

w_{T}(a,b)\leq w_{T}(a^{\prime},b^{\prime})\leq\lambda_{G}(a^{\prime},b^{\prime})+2i\Delta\leq\lambda_{G}(p,q)+2i\Delta.

This completes the induction. It follows that the call to $\text{PrivateGHSteinerTree}{}(G,V,\varepsilon,0)$ outputs a $2t_{\max}\Delta$ -approximate Gomory-Hu Steiner tree $T$ . Substituting in the values $t_{\max}=O(\lg^{2}n)$ and $\Delta=O(\frac{n\lg^{6}n}{\varepsilon})$ gives the approximation guarantee. $\Box$

We now state our main result on the approximation guarantee of Algorithm 3.

Theorem 5.1.

Let $T=(V_{T},E_{T},w_{T})$ be the weighted tree output by PrivateGHTree $(G=(V,E,w),\varepsilon)$ on a weighted graph $G$ . For each edge $e\in E_{T}$ , define $S_{e}$ to be the set of vertices of one of the connected components of $T\setminus\{e\}$ . Let $u,v\in V$ be distinct vertices and let $e_{\min}$ be an edge on the unique $u$ - $v$ path in $T$ such that $w_{T}(e_{\min})$ is minimal. With high probability, $S_{e_{\min}}$ is an $O(\frac{n\lg^{8}n}{\varepsilon})$ -approximate Min- $u$ - $v$ -Cut and moreover, $|\lambda_{G}(u,v)-w_{T}(e_{\min})|=O(\frac{n\lg^{8}n}{\varepsilon})$ .

Proof.

Note that for each edge $e$ , $w_{T}(e)$ is obtained by adding noise $\text{Lap}\MT@delim@Auto\paren@star{\frac{2(n-1)}{\varepsilon}}$ to the cut value $w(S_{e})$ . Thus, $|w(S_{e})-w_{T}(e)|=O(\frac{n\lg n}{\varepsilon})$ with high probability for all $e\in T$ . Now let $e_{0}$ be an edge on the unique $u$ - $v$ path in $T$ such that $w(S_{e_{0}})$ is minimal. Then, by Lemma 5.1, $w(S_{e_{0}})\leq\lambda_{G}(u,v)+O(\frac{n\lg^{8}n}{\varepsilon})$ with high probability. As $e_{\min}$ was chosen as an edge on the $u$ - $v$ path in $T$ of minimal weight, $w_{T}(e_{\min})\leq w_{T}(e_{0})$ , and so

	$\displaystyle w(S_{e_{\min}})$	$\displaystyle\leq w_{T}(e_{\min})+O\MT@delim@Auto\paren@star{\frac{n\lg n}{\varepsilon}}\leq w_{T}(e_{0})+O\MT@delim@Auto\paren@star{\frac{n\lg n}{\varepsilon}}$
		$\displaystyle\leq w(S_{e_{0}})+O\MT@delim@Auto\paren@star{\frac{n\lg n}{\varepsilon}}\leq\lambda_{G}(u,v)+O\MT@delim@Auto\paren@star{\frac{n\lg^{8}n}{\varepsilon}}.$

On the the other hand, $S$ defines a $(u,v)$ -cut, so $\lambda_{G}(u,v)\leq w(S)$ . This proves the first statement. Moreover, the string of inequalities above combined with $\lambda_{G}(u,v)\leq w(S)$ in particular entails that

\lambda_{G}(u,v)\leq w_{T}(e_{\min})+O\MT@delim@Auto\paren@star{\frac{n\lg n}{\varepsilon}}\leq\lambda_{G}(u,v)+O\MT@delim@Auto\paren@star{\frac{n\lg^{8}n}{\varepsilon}},

from which the final statement follows. $\Box$

5.2 Privacy

We will refer to any subroutine or algorithm as being $\varepsilon$ -DP $(G,G^{\prime})$ if it satisfies the $\varepsilon$ -DP condition for a fixed pair of neighboring graphs $G,G^{\prime}$ . We will prove privacy by proving that $\varepsilon$ -DP $(G,G^{\prime})$ holds for any $G,G^{\prime}$ .

Theorem 5.2.

PrivateGHTree $(G,U,\varepsilon)$ is $\varepsilon$ -DP.

Proof.

Let $\sigma_{GH}(t)$ be the privacy of the topology of the tree released by PrivateGHSteinerTree on a graph with at most $n$ vertices, privacy parameter $\varepsilon$ , and at recursive depth $t$ . Note that the edge weights of the tree are not private but no computation depends on these values. Until the end of the analysis, we will ignore the edge weights on the tree in terms of privacy. We want to show $\sigma_{GH}(0)\leq\varepsilon$ .

Consider two neighboring graphs $G$ and $G^{\prime}$ which differ by $1$ on the weight on a single edge $e=(a,b)$ . Consider a call to PrivateGHSteinerTree with privacy parameter $\varepsilon$ and recursion depth $t$ . Let $z_{a},z_{b}$ be the vertices containing $a,b$ (possibly after contractions). If $z_{a}=z_{b}$ , the output of PrivateGHSteinerTree is $0$ -DP $(G,G^{\prime})$ as $e$ has been contracted and will have no effect on the output of the algorithm. By Lemma 4.3, the call to PrivateGHTreeStep is $\frac{\varepsilon}{4t_{\max}}$ -DP. Manipulation of the resulting vertex sets ${\hat{S}}_{v}$ does not hurt privacy via post-processing and as they do not depend on edge structure or weights. We will proceed by cases on where $z_{a},z_{b}$ belong.

Case 1: $z_{a},z_{b}\in{\hat{S}}_{v}$

In this setting, $z_{a},z_{b}$ belong to the same approximate min isolating cut. From the for loop on 7, releasing any $(T_{v^{\prime}},f_{v^{\prime}})$ for $v^{\prime}\neq v$ is $0$ -DP $(G,G^{\prime})$ as $z_{a},z_{b}$ will be contracted to the same vertex $x_{v^{\prime}}$ . On the other hand, $z_{a},z_{b}$ will exist as single vertices in $G_{v}$ and releasing $(T_{v^{\prime}},f_{v^{\prime}})$ is $\sigma_{GH}(t+1)$ -DP $(G,G^{\prime})$ . Finally, releasing $(T_{\text{large}},f_{\text{large}})$ is $0$ -DP $(G,G^{\prime})$ as $z_{a},z_{b}$ will be both contracted to $y_{v}$ . The Combine algorithm is only doing post-processing and does not hurt privacy. By basic composition (Theorem 2.1), the overall privacy in this case is

\frac{\varepsilon}{4t_{\max}}+\sigma_{GH}(t+1).

Case 2: $z_{a}\in{\hat{S}}_{v},z_{b}\in{\hat{S}}_{v^{\prime}}$

Here, $z_{a},z_{b}$ belong to separate approximate min isolating cuts ${\hat{S}}_{v},{\hat{S}}_{v^{\prime}}$ . For any $v^{\prime\prime}\notin\{v,v^{\prime}\}$ , releasing $(T_{v^{\prime\prime}},f_{v^{\prime\prime}})$ is $0$ -DP $(G,G^{\prime})$ . Consider releasing $(T_{v},f_{v})$ where $z_{a}$ will be included in $G_{v}$ and $z_{b}$ will be contracted to $x_{v}$ . On 9, we add $\text{Lap}\MT@delim@Auto\paren@star{\frac{8t_{\max}}{\varepsilon}}$ to each edge between $x_{v}$ and all vertices in ${\hat{S}}_{v}$ , including to $e$ , and as a post-processing step, we truncate the edge weights to zero. This ensures that releasing $(T_{v},f_{v})$ is $\frac{\varepsilon}{8t_{\max}}$ -DP $(G,G^{\prime})$ via the Laplace mechanism (Theorem 2.2). Any computation done in the resulting recursive branch does not hurt privacy via post-processing. The same argument applies without loss of generality to releasing $(T_{v^{\prime}},f_{v^{\prime}})$ . Releasing $(T_{\text{large}},f_{\text{large}})$ involves a recursive call to a graph where $z_{a},z_{b}$ are contracted to separate vertices $y_{v},y_{v^{\prime}}$ , and is $\sigma_{GH}(t+1)$ private. The overall privacy in this case is

\frac{\varepsilon}{4t_{\max}}+2\frac{\varepsilon}{8t_{\max}}+\sigma_{GH}(t+1)=\frac{\varepsilon}{2t_{\max}}+\sigma_{GH}(t+1).

Case 3: $z_{a}\in{\hat{S}}_{v},z_{b}\in V\setminus\bigcup_{v\in R}{\hat{S}}_{v}$

Here, $z_{a}$ belongs to an approximate min isolating cut and $z_{b}$ does not. Once again, for any $v^{\prime}\neq v$ , releasing $(T_{v^{\prime}},f_{v^{\prime}})$ is $0$ -DP $(G,G^{\prime})$ . By the same argument above via the Laplace mechanism, releasing $(T_{v},f_{v})$ is $\frac{\varepsilon}{8t_{\max}}$ -DP $(G,G^{\prime})$ . Finally, releasing $(T_{\text{large}},f_{\text{large}})$ is $\sigma_{GH}(t+1)$ private. The overall privacy, in this case, is

\frac{\varepsilon}{4t_{\max}}+\frac{\varepsilon}{8t_{\max}}+\sigma_{GH}(t+1)=\frac{3\varepsilon}{8t_{\max}}+\sigma_{GH}(t+1)

Case 4: $z_{a},z_{b}\in V\setminus\bigcup_{v\in R}{\hat{S}}_{v}$

In this case, neither $z_{a}$ nor $z_{b}$ belong to an approximate min isolating cut. Therefore, releasing all $(T_{v^{\prime}},f_{v^{\prime}})$ is $0$ -DP $(G,G^{\prime})$ . Releasing $(T_{\text{large}},f_{\text{large}})$ is $\sigma_{GH}(t+1)$ private. The overall privacy, in this case, is

\frac{\varepsilon}{4t_{\max}}+\sigma_{GH}(t+1)

Across all cases, the maximum privacy cost is incurred in Case 2. By basic composition, the privacy of the algorithm is bounded by the recurrence

\sigma_{GH}(t)\leq\frac{\varepsilon}{2t_{\max}}+\sigma_{GH}(t+1).

As the recurrence ends at $t^{\prime}<t_{\max}$ ,

\sigma_{GH}(0)\leq t_{\max}\MT@delim@Auto\paren@star{\frac{\varepsilon}{2t_{\max}}}=\frac{\varepsilon}{2}.

Therefore, releasing the unweighted tree from PrivateGHTree is $\frac{\varepsilon}{2}$ -DP. Each edge weight in the tree comes from calculating the weight of a given cut, which can change by at most $1$ between two neighboring graphs. As the tree contains $n-1$ edges, all tree weights can be released via the Laplace mechanism by adding $\text{Lap}\MT@delim@Auto\paren@star{\frac{2(n-1)}{\varepsilon}}$ noise to each edge weight, resulting in $\frac{\varepsilon}{2}$ -DP. Then, releasing the weighted tree is $\varepsilon$ -DP, as required. $\Box$

5.3 Runtime

While runtime is not our main focus, as a final note, our algorithm can be implemented to run in near-quadratic time in the number of vertices of the graph. The runtime is inherited directly from prior work of [AKL⁺22], which utilizes the same recursive algorithm introduced in [Li21]. The overall structure of their main algorithm and subroutines remains in our work with changes of the form (a) altering runtime-independent conditions in if statements or (b) adding noise to cut values or edges in the graph. While left unspecified here, computation of single source Min- $s$ - $v$ -Cuts in Algorithm 2 should be done via the runtime-optimized algorithm of prior work to achieve the best bound. Then, via Theorem 1.3 of [AKL⁺22], Algorithm 3 runs in time $\tilde{O}(n^{2})$ .

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Appendix A Proof of Corollary 1.3

We recall the statement.

See 1.3

Proof.

We follow the proof of [SV95]⁵⁵5more precisely the lecture notes in https://courses.grainger.illinois.edu/cs598csc/sp2009/lectures/lecture_7.pdf, but replace the ‘looking at’ the exact GH-tree with our approximate version. The algorithm is simple: we cut the edges corresponding to the union of cuts given by the smallest $k-1$ edges of our approximate GH-tree $T$ of Theorem 1.1. If this produces more than $k$ pieces, arbitrarily add back cut edges until we reach a $k$ -cut.

For the analysis, consider the optimal $k$ -cut with partitions $V_{1},\ldots,V_{k}$ and let $w(V_{1})\leq\ldots\leq w(V_{k})$ denote the weight of the edges leaving each partition without loss of generality. Since every edge in the optimum is adjacent to exactly two pieces of the partition, it follows that $\sum_{i}w(V_{i})$ is twice the cost of the optimal $k$ -cut. We will now demonstrate $k-1$ different edges in $T$ which have cost at most $\sum_{i}w(V_{i})$ , up to additive error $O(k\Delta)=\tilde{O}(nk/\varepsilon)$ , where $\Delta=\tilde{O}(n/\varepsilon)$ is the additive error from Theorem 1.1.

As in the proof in [SV95], contract the vertices in $V_{i}$ in $T$ for all $i$ . This may create parallel edges, but the resulting graph is connected since $T$ was connected to begin with. Make this graph into a spanning tree by removing parallel edges arbitrarily, root this graph at $V_{k}$ , and orient all edges towards $V_{k}$ .

Consider an arbitrary $V_{i}$ where $i\neq k$ . The ‘supernode’ for $V_{i}$ has a unique edge leaving it, which corresponds to a cut between some vertex $v\in V_{i}$ with $w\not\in V_{i}$ . Since $T$ is an approximate-GH tree, the weight of this edge must be upper bounded by $w(V_{i})$ (which is also a valid cut separating $v$ and $w$ ), up to additive error $\Delta$ . The proof now follows by summing across $V_{i}$ . $\Box$

	$\displaystyle\sum_{r\in R}\lambda_{H_{r}}(r,t_{r})$	$\displaystyle\leq\sum_{r\in R}w_{H_{r}}(W_{r}\cap S_{r})$
		$\displaystyle=\sum_{r\in R}\left(w_{G}(W_{r}\cap S_{r})+\|W_{r}\cap S_{r}\cap U\|\cdot O((n+\lg(1/\beta))\lg^{2}(\|R\|)/(\varepsilon\|U\|))\right)$
		$\displaystyle\leq\sum_{r\in R}w_{G}(W_{r}\cap S_{r})+O((n+\lg(1/\beta))\lg^{2}(\|R\|)/\varepsilon)$
		$\displaystyle\leq\sum_{r\in R}w_{G}(S_{r})+O((n+\lg(1/\beta))\lg^{2}(\|R\|)/\varepsilon),$

	$\displaystyle w({\mathcal{C}}\cap W_{r})+(B_{\text{H}}{}\cdot(n+\lg(1/\beta))\lg^{2}(\|R\|)/(\varepsilon\|U\|))\|{\mathcal{C}}\cap W_{r}\cap U\|$
	$\displaystyle\leq w(S_{r}\cap W_{r})+(B_{\text{H}}{}\cdot(n+\lg(1/\beta))\lg^{2}(\|R\|)/(\varepsilon\|U\|))\|S_{r}\cap W_{r}\cap U\|+O((n+\lg(1/\beta))(\lg(\|R\|))/\varepsilon)$
	$\displaystyle\leq w(S_{r})+(B_{\text{H}}{}\cdot(n+\lg(1/\beta))\lg^{2}(\|R\|)/(\varepsilon\|U\|))\|S_{r}\cap W_{r}\cap U\|+O((n+\lg(1/\beta))(\lg^{2}(\|R\|))/\varepsilon).$		(By 2)

$\displaystyle\|{\mathcal{C}}\cap W_{r}\cap U\|$	$\displaystyle\leq\frac{O((n+\lg(1/\beta))(\lg^{2}(\|R\|))/\varepsilon)}{B_{\text{H}}{}\cdot(n+\lg(1/\beta))\lg^{2}(\|R\|)/(\varepsilon\|U\|)}+\|S_{r}\cap W_{r}\cap U\|$
	$\displaystyle\leq 0.4\|U\|+\|S_{r}\cap W_{r}\cap U\|$	(By setting $B_{\text{H}}{}$ large enough)
	$\displaystyle\leq 0.4\|U\|+\|S_{r}\cap U\|$
	$\displaystyle\leq 0.9\|U\|,$

	$\displaystyle{\hat{w}}({\hat{S}}_{v}^{i^{\prime}})$	$\displaystyle\leq w(\tilde{S}_{v}^{i^{\prime}})+\|w(\tilde{S}_{v}^{i^{\prime}})-{\hat{w}}({\hat{S}}_{v}^{i^{\prime}})\|$
		$\displaystyle\leq w(\tilde{S}_{v}^{i^{\prime}})+\|w(\tilde{S}_{v}^{i^{\prime}})-w({\hat{S}}_{v}^{i^{\prime}})\|+\|w({\hat{S}}_{v}^{i^{\prime}})-{\hat{w}}({\hat{S}}_{v}^{i^{\prime}})\|$
		$\displaystyle\leq w(\tilde{S}_{v}^{i^{\prime}})+{\Gamma_{\text{iso}}}$
		$\displaystyle\leq\hat{\lambda}(s,v)+\MT@delim@Auto\paren@star{2(\MT@delim@Auto\floor@star{\lg(\|U\|)}-i^{\prime})+1}{\Gamma_{\text{iso}}}+{\Gamma_{\text{values}}}.$

	$\displaystyle w({\hat{S}}_{v}^{i})$	$\displaystyle\leq{\Gamma_{\text{iso}}}+{\hat{w}}({\hat{S}}_{v}^{i})$
		$\displaystyle\leq{\Gamma_{\text{iso}}}+\hat{\lambda}(s,v)+\MT@delim@Auto\paren@star{2(\MT@delim@Auto\floor@star{\lg\|U\|}-i)+1}{\Gamma_{\text{iso}}}+{\Gamma_{\text{values}}}$
		$\displaystyle\leq{\Gamma_{\text{iso}}}+(\lambda(s,v)+{\Gamma_{\text{values}}})+\MT@delim@Auto\paren@star{2(\MT@delim@Auto\floor@star{\lg\|U\|}-i)+1}{\Gamma_{\text{iso}}}+{\Gamma_{\text{values}}}$
		$\displaystyle\leq\lambda(s,v)+2(\MT@delim@Auto\floor@star{\lg\|U\|}+1){\Gamma_{\text{iso}}}+2{\Gamma_{\text{values}}}.$

Differentially Private Gomory-Hu Trees

Abstract

1 Introduction

1.1 Our Contribution

Theorem 1.1.

Corollary 1.1.

Corollary 1.2.

Corollary 1.3.

1.2 Technical Overview

1.2.1 Obstacles in privatizing Gomory-Hu Trees

1.2.2 DP Approximate Min Isolating Cuts (Section 3)

Definition 1.1 (Min Isolating Cuts [LP20, AKT21]).

Theorem 1.2.

1.2.3 DP Gomory-Hu Tree (Sections 4 and 5)

Approximately good sets.

The privacy guarantee.

1.3 Organization

1.4 Open Problems

2 Preliminaries

2.1 Notation

2.2 Graph Cuts

Definition 2.1 (Vertex contractions).

Lemma 2.1 (Submodularity of Cuts [Cun85]).

Fact 2.1 ([LP20]).

Definition 2.2 (Gomory-Hu Steiner tree [Li21]).

2.3 Differential Privacy

Definition 2.3 (Edge-Neighboring Graphs).

Definition 2.4 (Differential Privacy [Dwo06]).

Theorem 2.1 (Basic composition [DMNS06, DL09]).

Theorem 2.2 (Laplace mechanism [DR14]).

Theorem 2.3 (Private Min-ss-tt-Cut [DMN23]).

Corollary 2.1 (Private Min-SS-TT-Cut).

2.4 Concentration Inequalities

Theorem 2.4 (Sums of Exponential Random Variables (Theorem 5.1 of [Jan17])).

3 Private Min Isolating Cuts

Lemma 3.1.

Proof.

Privacy analysis.

Error analysis.

Claim 1.

Proof.

Claim 2.

Claim 3.

Claim 4.

Proof.

Additional guarantee.

4 Core Recursive Step

4.1 Correctness

Lemma 4.1.

Lemma 4.2.

Proof.

Proof of Lemma 4.1.

4.2 Privacy

Lemma 4.3.

Proof.

5 Final Algorithm

Definition 5.1.

5.1 Correctness

Lemma 5.1.

Lemma 5.2.

Proof.

Lemma 5.3.

Proof.

Proof of Lemma 5.1.

Theorem 5.1.

Proof.

5.2 Privacy

Theorem 5.2.

Proof.

Case 1: za,zb∈S^vz_{a},z_{b}\in{\hat{S}}_{v}

Case 2: za∈S^v,zb∈S^v′z_{a}\in{\hat{S}}_{v},z_{b}\in{\hat{S}}_{v^{\prime}}

Case 3: za∈S^v,zb∈V∖⋃v∈RS^vz_{a}\in{\hat{S}}_{v},z_{b}\in V\setminus\bigcup_{v\in R}{\hat{S}}_{v}

Case 4: za,zb∈V∖⋃v∈RS^vz_{a},z_{b}\in V\setminus\bigcup_{v\in R}{\hat{S}}_{v}

5.3 Runtime

References

Appendix A Proof of Corollary 1.3

Proof.

Theorem 2.3 (Private Min- $s$ - $t$ -Cut [DMN23]).

Corollary 2.1 (Private Min- $S$ - $T$ -Cut).

Case 1: $z_{a},z_{b}\in{\hat{S}}_{v}$

Case 2: $z_{a}\in{\hat{S}}_{v},z_{b}\in{\hat{S}}_{v^{\prime}}$

Case 3: $z_{a}\in{\hat{S}}_{v},z_{b}\in V\setminus\bigcup_{v\in R}{\hat{S}}_{v}$

Case 4: $z_{a},z_{b}\in V\setminus\bigcup_{v\in R}{\hat{S}}_{v}$