On the Transit Obfuscation Problem

In applications such as sensitive cargo transportation or surveillance, it is sometimes necessary to route an agent from a start point to a goal while concealing the location of a transit point, which is either on the route or visible from the route, from adversaries. For example, in a cargo transport application, if a depot or drop location is located somewhere on the route, it is essential to prevent potential adversaries from deducing the transit point location to minimize the risk of theft or interception. In a surveillance application, it is important to be able to conceal which specific location a surveillance agent is targeting on its route.

Obfuscating an agent’s true intention has been previously studied in various fields, including path planning, robotics, and game theory Chakraborti et al. (2019). Previous work has primarily focused on the Goal Obfuscation Problem, which aims to prevent observers from deducing the agent’s actual goal, and has numerous applications, e.g., creating realistic non-player characters (NPCs) capable of deceiving humans Dias et al. (2013), or ensuring secure escorting of a VIP to a hidden location Keren et al. (2016).

Concealing non-goal locations is also important for customer privacy protection in real-world situations. For example, Enayati et al. (2022) considers a scenario where an UAV delivers packages to a private location and returns to the starting point. Chen et al. (2012) notes that some public transportation systems track the stations where each customer boards and leaves, and those systems potentially reveal the location of homes or workplaces, which are the transit points on the round-trip paths. However, Enayati et al. (2022) is limited to path planning in two-dimensional coordinates with the constraint that the start and goal locations are the same, and Chen et al. (2012) focuses on not path planning but anonymizing the collected sequential data.

In this paper, we study the Transit Obfuscation Problem, where given a graph, start location, end location, a target transit point, and a visibility function for an agent, the task is to generate a route from the start to the end location such that the agent’s visibility function covers the target, but the target location is concealed from adversaries. We assume a strong adversary that has full knowledge of the agent’s path, as well as full knowledge of the domain as well the internal decision-making process of the agent (i.e., the adversary has full access to the agent’s code).

We introduce the notion of $(k,\ell,m)$-Anonymity, which quantifies the level of concealment achieved by a path planner. If a path planner satisfies $(k,\ell,m)$-Anonymity, there exist at least $k$ transit points resulting in the same path up to the first $m$ steps, while the deviation among these candidate points is $\geq\ell$. Thus, $(k,\ell,\infty)$-Anonymity is a guarantee that even with full knowledge of the agent’s path and code, an adversary can not distinguish the true target transit point among $k$ possible candidates which are at least $\ell$ apart from each other. We analyze some theoretical properties of $(k,\ell,m)$-Anonymity and propose a graph partitioning-based approach to generating paths that guarantee $(k,\ell,m)$-Anonymity.

The rest of the paper is structured as follows. First, we define the Transit Obfuscation Problem (TOP) with respect to a path-planning problem with a transit point and visibility constraints (Section 2) Next, in Section 3, we define ($k,\ell,m)$-Anonymity for the TOP and analyze its theoretical properties. We also define some metrics for evaluating the tradeoffs between privacy and path costs for the TOP. Then, in Section 4, we propose a partitioning-based search algorithm for the TOP which guarantees anonymity even when when the adversary knows the complete path, i.e., $(k,\ell,\infty)$-Anonymity. Section 5 proposes algorithms which guarantees anonymity for up to $m<\infty$ steps. In Section 6, we experimentally evaluate our search algorithms on some standard benchmark game map instances. Section 7 discusses related work. Section 8 concludes with a discussion and directions for future work. Our code is available at: https://github.com/Koukyosyumei/TOP.

A path-planning domain with visibility constraints is denoted by a triple $\mathcal{D}=\langle\mathcal{N},\mathcal{E},\mathscr{T},c,v\rangle$, where

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  $\mathcal{N}$ is a non-empty set of nodes;

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  $\mathscr{T}\subseteq\mathcal{N}$ is a set of transit candidates;

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  $\mathcal{E}\subseteq\mathcal{N}\times\mathcal{N}$ is a set of edges between nodes;

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  $c:\mathcal{E}\rightarrow R^{+}_{0}$ is a function that returns the non-negative cost of an edge between two nodes.

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  $v:\mathcal{N}\rightarrow 2^{\mathcal{N}}$ is a visibility function that returns the set of visible nodes from a given node.

The cost of the shortest path (a.k.a minimum cost path) between node $a\in\mathcal{N}$ and node $b\in\mathcal{N}$ is denoted by $d(a,b)$. For simplicity, we assume that $E$ does not contain self-loop edges, i.e., $d(a,b)=\infty$ if $a=b$. A path $\pi$ in a path-planning domain $\mathcal{D}$ is a sequence of nodes $\pi=n_{1},n_{2},...,n_{|\pi|}$ such that $\forall{i\in\{1,...,|\pi|-1\}}\hskip 5.0pt(n_{i},n_{i+1})\in\mathcal{E}$, where $|\pi|$ represents the length of $\pi$. We also denote the subsequence of $\pi$ till $m$-th node as $\pi|_{m}$, i.e., $\pi|_{m}=n_{1},n_{2},...,n_{m}$. For convenience, we assume that $\pi|_{m}=\pi$ if $m>|\pi|$. The binary operator $\circ$ represents the concatenation of two paths. Specifically, when $\pi_{a}=a_{1},a_{2},...,a_{|\pi_{a}|}$, $\pi_{b}=b_{1},b_{2},...,b_{|\pi_{b}|}$, and $a_{|\pi_{a}|}=b_{1}$, we have that $\pi=\pi_{a}\circ\pi_{b}=a_{1},a_{2},...,a_{|\pi_{a}|},b_{2},...,b_{|\pi_{b}|}$. We also introduce $\operatorname*{\scalerel*{\|}{\sum}}$ notation, where $\operatorname*{\scalerel*{\|}{\sum}}^{x}_{i=1}\pi_{i}=\pi_{1}\circ\pi_{2}\circ...\circ\pi_{x}$. The cost of $\pi$ is the sum of the costs of each edge in $\pi$, given by $cost(\pi)=\sum^{|\pi|}_{i=2}c(\pi_{i-1},\pi_{i})$, where $\pi_{i}$ is the $i$-th node in $\pi$. We say that $\pi$ covers node $n\in\mathcal{N}$ if there exists an index $i$ such that $n\in v(\pi_{i})$.

A path-planning problem with visibility constraints and a transit point (PPVT) is represented by a tuple $\langle\mathcal{D},s,g,t\rangle$, where $\mathcal{D}=\langle\mathcal{N},\mathscr{T},\mathcal{E},c,v\rangle$ is a domain, $s\in\mathcal{N}$ is the start node, $g\in\mathcal{N}$ is the goal node, $t\in\mathscr{T}$ is the transit point that must be covered. The solution to a PPVT is a path $\pi$ such that $\pi_{1}=s$, $\pi_{|\pi|}=g$, and $\exists{i\in\{1,2,...,|\pi|\},\hskip 5.0ptt\in v(\pi_{i})}$.

A path planner $\mathcal{A}$ takes as input a PPVT and returns a feasible path $\pi$ for that problem, i.e., $\mathcal{A}(\langle\mathcal{D},s,g,t\rangle)=\pi$. $\mathcal{A}$ returns Failure when it cannot find a feasible path. For convenience, we define Failure such that it is not equal to itself, i.e., Failure $\neq$ Failure.

We assume that there is an adversary who seeks to deduce the actual transit point $t\in\mathscr{T}$ by observing the trajectory of the agent.

A Transit Obfuscation Problem (TOP) is a tuple $\langle\mathcal{D},s,g,\mathscr{O}\rangle$, where $\mathscr{O}$ describes what the adversary can observe.

We make the following assumptions about the abilities of the adversarial observer (similar to the set of assumptions by  Kulkarni et al. (2018)):

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  Complete Knowledge about the Domain and Transit Candidates: The adversary has complete knowledge about the domain $\mathcal{D}$.

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  Full Access to the Planner: The adversary has full access to and thoroughly understands the agent’s planning algorithm.

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  Independence of Inputs: The adversary can execute the agent’s planner with arbitrary input tuples at any time.

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  Observability of Path: The adversary can immediately observe the path executed by the agent so far.

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  Semi-Honest Adversary: The adversary is passive, and it does not disturb the action of the agent or gain any additional information beyond what has been specified above.

These are challenging assumptions when trying to conceal the transit point, as the adversary has full information about the mechanism of the path planning algorithm, as well as the ability to rerun/simulate the algorithm many times in order to gain information that might reveal the transit point. When $t=g$, and $v$ is the identity function (the only node visible from a node is itself), this special case of the TOP is similar to the Goal Obfuscation Problem Kulkarni et al. (2018, 2019). Unlike the Goal Obfuscation Problem, where the final node of the path always reveals the actual goal, in the TOP, when $t\neq g$, it is possible to have a path where an adversary cannot deduce the actual transit point even after observing the entire trajectory. For example, in Fig. 1, an agent travels from $s$ to $g$ while covering one of $\mathscr{T}=\{t_{1},t_{2},t_{3},t_{4}\}$, where black cells are obstacles. Then, if the agent can see nodes within a radius of one, any feasible path covers all of $\mathscr{T}$ so that an observer cannot infer the true transit point.

Next, we formally define the conditions when a path is anonymized for a transit point, what inputs are anonymizable, and what planners can achieve anonymization.

We now propose an algorithm for $(k,\ell,\infty)$-Anonymity. First, note that if we disregard path cost, then in principle, a relatively straightforward approach to achieve $(k,\ell,\infty)$-Anonymity would be to return some path which covers all transit candidates. However, a practical algorithm for Transit Obfuscation needs to effectively trade off the privacy objective vs. path costs, on average, overall $(s,g,t)$ tuples of interest. We first propose objectives that express this tradeoff and then propose a partitioning-based algorithm which seeks a solution which optimizes this objective. Proofs are in Supp. LABEL:supp:proof:sec4 Takahashi and Fukunaga (2024).

We now propose three planners that output $(k,\ell,m)$-Anonymized Paths with $m<\infty$, which means that the adversary cannot identify which node is the transit point until observing more than $m$ nodes. If $m$ is less than the length of the output path, the adversary might be able to identify the true transit point after obtaining the $m+1$st and later nodes. Proofs for this section can be found in Supp. LABEL:supp:proof:sec5 Takahashi and Fukunaga (2024).

This paper introduced the Transit Obfuscation Problem and proposed novel techniques to address this challenge. We introduced $(k,\ell,m)$-Anonymity as a measure of concealment achieved by a path planner. We proposed a Pbp, a partitioning based algorithm to achieve $(k,\ell,\infty)$-Anonymity, and evaluated its performance on 2D grid maps with obstacles. We showed that Pbp with a merge-based branch-and-bound strategy significantly outperforms baseline partitioning approaches.

Although we showed that Merge-BB with the APR and MAC objectives is a viable approach to partitioning, a complete branch-and-bound search to find and prove the optimality of a solution poses a scalability challenge. For example, although our current implementation of Pbp can find solutions for $|\mathscr{T}|=16$ (Tab. 1), it can not complete the search and prove optimality within the 300 sec. limit. Search algorithms finding good partitionings quickly without an optimality guarantee (e.g., local search/metaheuristics) are a direction for future work.

We also investigated algorithms for $m$-bounded anonymity ($m<\infty$). We showed that while $m$-Pbp yielded the best MAC scores, Cbp offers fairly good MAC scores but runs much faster. Future work will investigate additional approaches to trading off scalability vs. solution quality.

Finally, while this work focused on a single agent and single adversary in a static environment, an extension of our proposed techniques to more complex scenarios, such as multi-agent systems and dynamic environments, is another direction for future work.