Finding Densest $k$-Connected Subgraphs

The remainder of this paper is organized as follows. In Section 2, we review related work. In Section 3, we extend Mader’s theorem to edge-weighted graphs and design an algorithm for finding a Mader subgraph. In Sections 4 and 5, we present our bicriteria and ordinary approximation algorithms, respectively. We conclude with some open problems in Section 6.

Wu et al. [52] consider the problem of detecting a dense and connected subgraph in dual networks. A dual network is a pair of graphs $G=(V,E_{G})$ and $H=(V,E_{H})$ defined on the same vertex set $V$, which encode different types of connections using two edge sets $E_{G}$ and $E_{H}$. Wu et al. [52] introduced the following problem: given a dual network $(G,H)$, we are asked to find $S\subseteq V$ that maximizes $d(S)$ in $G$ under the constraint that $H[S]$ is connected (i.e., $1$-edge-connected). They proved that the problem is NP-hard and devised a scalable heuristic. Problem 2 with $k=1$, i.e., the densest $1$-edge-connected subgraph, on unweighted graphs, can be seen as a special case of their problem wherein two graphs $G$ and $H$ are the same, i.e., $E_{G}=E_{H}$. It is easy to see that unlike the general form of their problem, the densest $1$-edge-connected subgraph problem (on unweighted graphs) is polynomial-time solvable.

Two closely related papers are due to Tsourakakis [48] and Kawase and Miyauchi [30]. They aim to find a near-clique (which is robust to vertex/edge failure) by extending the densest subgraph problem. Tsourakakis [48] introduced the problem called the $k$-clique densest subgraph problem. In this problem, given an unweighted graph $G=(V,E)$, we are asked to find $S\subseteq V$ that maximizes the $k$-clique density $w_{k}(S)/|S|$, where $w_{k}(S)$ is the number of $k$-cliques (i.e., cliques with size $k$) in $G[S]$. Tsourakakis [48] showed that this problem (with constant $k$) remains polynomial-time solvable, and later, Mitzenmacher et al. [39] proposed a scalable algorithm that obtains a nearly-optimal solution. On the other hand, Kawase and Miyauchi [30] introduced the problem called the $f$-densest subgraph problem with convex $f$. In this problem, given an edge-weighted graph $G=(V,E,w)$, we are asked to find $S\subseteq V$ that maximizes $w(S)/f(|S|)$, where $f:\mathbb{Z}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$ is a monotonically non-decreasing function that satisfies $(f(x+2)-f(x+1))-(f(x+1)-f(x))\geq 0$ for any $x\in\mathbb{Z}_{\geq 0}$. This formulation generalizes the NP-hard optimal quasi-cliques problem due to Tsourakakis et al. [50, 49]. Kawase and Miyauchi [30] studied the hardness of the problem, and proposed a polynomial-time approximation algorithm. Although the above two problems contribute to computing a dense subgraph that is robust to vertex/edge failure, they cannot explicitly impose $k$-vertex/edge connectivity.

There are also some variants that take into account the robustness to the uncertainty of input graphs. Zou [53] studied the densest subgraph problem on uncertain graphs. Uncertain graphs are a generalization of graphs, which can model the uncertainty of the existence of edges. More formally, an uncertain graph consists of an unweighted graph $G=(V,E)$ and a function $p:E\rightarrow[0,1]$, where $e\in E$ is present with probability $p(e)$ whereas $e\in E$ is absent with probability $1-p(e)$. In the problem introduced by Zou [53], given an uncertain graph $G=(V,E)$ with $p$, we are asked to find $S\subseteq V$ that maximizes the expected value of the density. Zou [53] observed that this problem can be reduced to the original densest subgraph problem, and designed polynomial-time exact algorithm using the reduction. Very recently, Tsourakakis et al. [51] introduced the problem called the risk-averse DSD. In this problem, given an uncertain graph $G=(V,E)$ with $p$, we are asked to find $S\subseteq V$ that has a large expected density and at the same time has a small risk. The risk of $S\subseteq V$ is measured by the probability that $S$ is not dense on a given uncertain graph. They showed that the risk-averse DSD can be reduced to the densest subgraph problem with negative edge weights (which is NP-hard), and designed an efficient approximation algorithm based on the reduction.

Miyauchi and Takeda [41] considered the uncertainty of edge weights rather than the existence of edges. To model that, they assumed that they have an edge-weight space $W=\times_{e\in E}[l_{e},r_{e}]\subseteq\times_{e\in E}[0,\infty)$ that contains the unknown true edge weight $w$. To evaluate the performance of $S\subseteq V$ without any concrete edge weight, they employed a well-known measure in the field of robust optimization, called the robust ratio. In their scenario, the robust ratio of $S\subseteq V$ under $W$ is defined as the multiplicative gap between the density of $S$ in terms of edge weight $w^{\prime}$ and the density of $S^{*}_{w^{\prime}}$ in terms of edge weight $w^{\prime}$ under the worst-case edge weight $w^{\prime}\in W$, where $S^{*}_{w^{\prime}}$ is a densest subgraph of $G$ with $w^{\prime}$. Intuitively, $S\subseteq V$ with a large robust ratio has a density close to the optimal value even on $G$ with the edge weight selected adversarially from $W$. Using the robust ratio, they formulated the robust densest subgraph problem, where given an unweighted graph $G=(V,E)$ and an edge-weight space $W=\times_{e\in E}[l_{e},r_{e}]\subseteq\times_{e\in E}[0,\infty)$, we are asked to find $S\subseteq V$ that maximizes the robust ratio under $W$. Miyauchi and Takeda [41] designed an algorithm that returns $S\subseteq V$ with a robust ratio of at least $\frac{1}{\max_{e\in E}\frac{r_{e}}{l_{e}}}$ under some mild condition.

In addition to the above, there are many other problem variations. The most well-studied variants are size restricted ones [1, 6, 13, 31]. For example, in the densest $k$-subgraph problem [13], given an edge-weighted graph $G=(V,E,w)$ and a positive integer $k\in\mathbb{Z}_{>0}$, we are asked to find $S\subseteq V$ that maximizes $d(S)$ subject to the constraint $|S|=k$. It is known that such a restriction makes the problem much harder; indeed, the densest $k$-subgraph problem is NP-hard and the best known approximation ratio is $\Omega(1/n^{{1/4}+\epsilon})$ for any $\epsilon>0$ [6]. The densest subgraph problem has also been extended to more general computation models and graph structures. As for computation models, to cope with the dynamics of real-world graphs, some literature has considered dynamic settings [12, 26], and moreover, to model the limited computation resources in reality, some literature has considered streaming settings [2, 4, 7]. As for graph structures, the problem has been defined on hypergraphs [26, 40] and multilayer networks [17].

In the vertex connectivity problem, we are asked to compute $\kappa(G)$ for a given graph $G=(V,E)$. For this problem, Gabow [16] developed an $O(|V|(\kappa(G)^{2}\cdot\min\{|V|^{3/4},\kappa(G)^{3/2}\}+\kappa(G)|V|))$-time algorithm, which also computes a corresponding minimum vertex separator $S\subset V$. This is one of the current fastest deterministic algorithms for the problem, although there are various randomized algorithms (e.g., see [14, 24, 33, 43]). Note that there are linear-time algorithms that decide whether $G$ is 2-vertex-connected and 3-vertex-connected, respectively, due to Tarjan [47] and Hopcroft and Tarjan [25].

Another important problem related to vertex connectivity is to compute the family of maximal $k$-vertex-connected subgraphs, which will be solved in our bicriteria approximation algorithm for Problem 1. For $S\subseteq V$ and $k\in\mathbb{Z}_{>0}$, the induced subgraph $G[S]$ is called a maximal $k$-vertex-connected subgraph if $G[S]$ is $k$-vertex-connected and no superset of $S$ has this property. For this task, the first polynomial-time algorithm is given by Matula [37]. Note that maximal $k$-vertex-connected subgraphs may overlap each other; the design of the algorithm by Matula [37] is based on the fact that the maximum total number of maximal $k$-vertex-connected subgraphs is $O(|V|)$ [37]. Later, Makino [36] designed an $O(|V|\cdot T)$-time algorithm, where $T$ is the computation time required to find a vertex separator of size at most $k-1$. Combined with the above vertex connectivity algorithm by Gabow [16], the algorithm by Makino [36] yields the running time of $O(|V|^{2}(k^{2}\cdot\min\{|V|^{3/4},k^{3/2}\}+k|V|))$. For some special $k$, there are some existing algorithms that have better running time. For $k=2$ and 3, there are linear-time algorithms by Tarjan [47] and Hopcroft and Tarjan [25], respectively. For any constant $k$, Henzinger et al. [22] presented an $O(|V|^{3})$-time algorithm.

In the (global) minimum cut problem, given an edge-weighted graph $G=(V,E,w)$, we are asked to find the minimum weight cut $F\subseteq E$. For this problem, Nagamochi and Ibaraki [42] designed an $O(|V|(|E|+|V|\log|V|))$-time algorithm. Later, Stoer and Wagner [46] and Frank [15] independently presented a very simple algorithm that still has the same running time. For simple unweighted graphs, the seminal work by Karger [27] provides a randomized (Monte Carlo) algorithm that runs in nearly-linear, $O(|E|\log^{3}|V|)$, time. As this algorithm does not necessarily return the right answer, Karger [27] posed an open question to find a nearly-linear-time deterministic algorithm. In a recent breakthrough, Kawarabayashi and Thorup [28] answered the question; they developed a deterministic algorithm for simple unweighted graphs that runs in $O(|E|\log^{12}|V|)$ time. Very recently, Henzinger et al. [23] improved the running time to $O(|E|\log^{2}|V|\log\log^{2}|V|)$ time, which is better even than that of the randomized algorithm by Karger [27].

As in the vertex connectivity case, computing the family of maximal $k$-edge-connected subgraphs is also an important problem, which will be solved in our bicriteria approximation algorithm for Problem 2. For $S\subseteq V$ and $k\in\mathbb{R}_{>0}$, the induced subgraph $G[S]$ is called a maximal $k$-edge-connected subgraph if $G[S]$ is $k$-edge-connected and no superset of $S$ has this property. The problem can be solved using any minimum cut algorithm as follows: if the weight of the minimum cut of the graph is less than $k$, divide the graph into two subgraphs along with the cut and then repeat the procedure on the resulting subgraphs. For edge-weighted graphs, we can directly obtain an $O(|V|^{2}(|E|+|V|\log|V|))$-time algorithm using one of the above minimum cut algorithms by Nagamochi and Ibaraki [42], Stoer and Wagner [46], and Frank [15]. To the best of our knowledge, there is no existing algorithm that has a better running time. For simple unweighted graphs, we can again directly obtain an $O(|E||V|\log^{2}|V|\log\log^{2}|V|)$-time algorithm using the above minimum cut algorithm by Henzinger et al. [23]. Unlike the weighted case, for some special $k$, there are some existing algorithms that have a better running time. For $k=2$, there is a linear-time algorithm by Tarjan [47]. For any constant $k$, Henzinger et al. [22] presented an $O(|V|^{2}\log|V|)$-time algorithm, and more recently, Chechik et al. [9] provided an $O(\sqrt{|V|}(|E|+|V|\log|V|))$-time algorithm. The latter algorithm is efficient particularly for sparse graphs; indeed, the latter is better than the former when $|E|=o(|V|^{3/2}\log|V|)$. Very recently, for any $k\in\mathbb{Z}_{>0}$, Forster et al. [14] developed a randomized (Las Vegas) algorithm that has expected running time $O(k^{3}|V|^{3/2}\log|V|+k|E|\log^{2}|V|)$, which is faster than the algorithm by Chechik et al. [9] (for general $k\in\mathbb{Z}_{>0}$).