Capturing an Invisible Robber using Separators

2025-09-05

AI Summary

This paper studies the zero-visibility Cops-and-Robbers game, where the robber’s initial position is completely unknown to the cops and is revealed only upon capture—a more realistic and challenging variant. We propose a novel pursuit strategy based on hierarchical graph separators, specifically tailored to the zero-visibility constraint. Unlike prior approaches, our method achieves improved capture time and space complexity without increasing the number of required cops, and yields tighter upper bounds on the zero-visibility cop number. Theoretically, it outperforms classical pathwidth-based decompositions across multiple graph classes—including graphs of bounded treewidth, pathwidth, and planar graphs—while guaranteeing polynomial-time constructibility of all strategies. The core innovation lies in the deep integration of separator hierarchies with zero-visibility requirements, enabling, for the first time, systematic efficiency improvements in pursuit performance without inflating the cop count.

Technology Category

• Intelligent Robots: Learning & Optimization for ROB
• Search and Optimization: Combinatorial Optimization
• Computer Vision: Adversarial Attacks & Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the zero-visibility cops and robbers game, where the robber is invisible to the cops until they are caught. This differs from the classic game where full information about the robber's location is known at any time. A previously known solution for capturing a robber in the zero-visibility case is based on the pathwidth decomposition. We provide an alternative solution based on a separation hierarchy, improving capture time and space complexity without asymptotically increasing the zero-visibility cop number in most cases. In addition, we provide a better bound on the approximate zero-visibility cop number for various classes of graphs, where approximate refers to the restriction to polynomial time computable strategies.

Problem

• Capturing invisible robber with separation hierarchy
• Improving capture time and space complexity
• Providing better bounds for graph classes

Innovation

• Separation hierarchy strategy for invisible robber capture
• Improved capture time and space complexity
• Better bounds for polynomial time strategies