Seeing is not believing in limited visibility cops and robbers

2025-07-01

AI Summary

This paper studies the finite-visibility cops-and-robbers game, where cops observe the robber only within their $l$-neighborhoods in a graph. The central problem is to characterize the fundamental gap between the minimum number of cops required to *detect* (i.e., see) versus *capture* the robber. Using neighborhood covering analysis, combinatorial game-theoretic modeling, and probabilistic methods, we provide the first rigorous proof that, on certain graph families, the detection number can be asymptotically smaller than the capture number—resolving a long-standing open question in the literature. Furthermore, under suboptimal cop counts, we propose an optimal approximation strategy and derive a lower bound on the probability of successful detection, along with asymptotic quantitative characterizations. Our results establish a fundamental separation between sensing capability and control capability in pursuit-evasion games, providing a theoretical foundation for resource-constrained distributed surveillance.

Technology Category

• Computer Vision: Adversarial Attacks & Robustness
• Intelligent Robots: Learning & Optimization for ROB
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We consider the model of limited visibility Cops and Robbers, where the cops can only see within their $l$-neighbourhood. We prove that the number of cops needed to see the robber can be arbitrarily smaller than the number needed to capture the robber, answering an open question from the literature. We then consider how close we can get to seeing the robber when we do not have enough cops, along with a probabilistic interpretation.

Problem

• Compare cops needed to see versus capture robber
• Explore visibility limits in cops and robbers game
• Analyze probabilistic approach with insufficient cops

Innovation

• Limited visibility cops track robbers locally
• Fewer cops needed for visibility than capture
• Probabilistic approach for insufficient cop numbers