[Paper Review] Distributed pursuit algorithms for probabilistic adversaries on connected graphs

Jesse Geneson|arXiv (Cornell University)|Oct 9, 2016

Crime, Illicit Activities, and Governance1 references3 citations

TL;DR

This paper studies distributed pursuit algorithms for tracking a probabilistic adversary (gambler) on connected graphs using k cops. It establishes that, for known, unknown, and known-changing gamblers, the expected capture time is Θ(n/k) when cops choose optimal initial positions, leveraging sector-based partitioning and depth-first search strategies to achieve tight bounds across all three adversary models.

ABSTRACT

A gambler moves between the vertices $1, \ldots, n$ of a graph using the probability distribution $p_{1}, \ldots, p_{n}$. Multiple cops pursue the gambler on the graph, only being able to move between adjacent vertices. We investigate the expected capture time for the gambler against $k$ cops as a function of $n$ and $k$ for three versions of the game: (1) known gambler: the cops know the gambler's distribution (2) unknown gambler: the cops do not know the gambler's distribution (3) known changing gambler: the gambler's distribution can change every turn, but the cops know all of the gambler's distributions from the beginning. We show for $n > k$ that if the cops are allowed to choose their initial positions before the game starts and before they know the gambler's distribution(s), and if both the gambler and the cops play optimally, then the expected capture time is $Θ(n/k)$ for the known gambler, the unknown gambler, and the known changing gambler.

Motivation & Objective

• To analyze the expected capture time for k cops pursuing a probabilistic gambler on connected n-vertex graphs under different information assumptions.
• To determine the optimal strategies for cops when they can choose initial positions before learning the gambler's distribution.
• To establish tight asymptotic bounds on expected capture time across three adversary models: known, unknown, and known-changing gamblers.
• To develop distributed algorithms using graph partitioning and depth-first search that achieve optimal performance under adversarial uncertainty.
• To resolve open questions on worst-case capture time bounds for various pursuit game variants on connected graphs.

Proposed method

• Partition the n-vertex connected graph into at most k connected sectors of size at most 2n/k + 1 using a spanning tree and Lemma 1 on vertex subset selection.
• Assign each cop to a sector and have them perform a depth-first search (DFS) on the sector's spanning subtree, with an extra turn at leaves to ensure coverage.
• Use random coin flips to determine DFS direction (forward or backward), reducing expected search time to at most 3n/k + 1 per cycle.
• Repeat DFS cycles until capture, with each round having an evasion probability ≤ e^(-2), leading to an expected number of rounds ≈ 1/(1 - e^(-2)).
• For the known changing gambler, use a two-phase round: first move to high-probability vertices, then patrol them; ensure coverage via shared vertex handling.
• Leverage the diameter d of the graph to derive tighter bounds for low-diameter graphs, using periodic cop repositioning every d turns.

Experimental results

Research questions

• RQ1 What is the maximum expected capture time for the known gambler versus k cops on a connected n-vertex graph?
• RQ2 What is the maximum expected capture time for the unknown gambler versus k cops on a connected n-vertex graph?
• RQ3 What is the maximum expected capture time for the known changing gambler versus k cops on a connected n-vertex graph?
• RQ4 What is the maximum expected capture time for the unknown changing gambler versus k cops on a connected n-vertex graph?
• RQ5 How does graph diameter affect the expected capture time in pursuit games with probabilistic adversaries?

Key findings

• For the known gambler, the expected capture time is exactly max(1, n/k) when cops choose optimal initial positions, regardless of the gambler's distribution.
• For the unknown gambler, the expected capture time is bounded above by 3.94n/k + 1.16, achieved through sector-based DFS with random direction selection.
• For the known changing gambler, the expected capture time is at most 6.33n/k + 3.17, using a two-phase round strategy that adapts to changing distributions.
• For graphs of diameter d, the known changing gambler has an expected capture time of at most 1 + dn/k, improving bounds for low-diameter graphs like stars.
• All three models (known, unknown, known-changing) achieve Θ(n/k) expected capture time when n > k, indicating tight asymptotic optimality.
• The lower bound of Ω(n/k) holds for the changing gambler, confirming that the upper bounds are asymptotically tight.