A tight lower bound for the capture time of the Cops and Robbers game

Sebastian Brandt,Jara Uitto,Roger Wattenhofer,Yuval Emek

doi:10.1016/j.tcs.2020.06.004

Sebastian Brandt, Jara Uitto + Show 2 more

Open Access

https://doi.org/10.1016/j.tcs.2020.06.004

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Abstract

For the game of Cops and Robbers, it is known that in 1-cop-win graphs, the cop can capture the robber in O(n) time, and that there exist graphs in which this capture time is tight. When k≥2, a simple counting argument shows that in k-cop-win graphs, the capture time is at most O(nk+1), however, no non-trivial lower bounds were previously known; indeed, in their 2011 book, Bonato and Nowakowski ask whether this upper bound can be improved. In this paper, the question of Bonato and Nowakowski is answered on the negative, proving that the O(nk+1) bound is asymptotically tight for any constant k≥2. This yields a surprising gap in the capture time complexities between the 1-cop and the 2-cop cases.

Highlights

The game of Cops and Robbers is a perfect information two-player zero-sum game played on an undirected n-vertex graph G = (V, E), where the first player is identified with k ≥ 1 cops, indexed by the integers 0, . . . , k − 1, and the second player is identified with a single robber
The goal of the cop player is to ensure that r(t − 1) ∈ {c0(t), . . . , ck−1(t)} for some finite round t, referred to as capturing the robber
The capture time of graph G is defined to be the minimum capture time of any cop strategy over G

Summary

Introduction

Bonato et al [7] studied the capture time in single cop games and proved that every 1-copwin graph admits a cop strategy that captures the robber in O(n) rounds. Notice that for constant k ≥ 2, this theorem provides an (existential) Ω(nk+1) lower bound on the capture time in k-cop-win graphs. The main feature in the latter argument is an exit component X ; if the robber manages to reach X , she can force the game to shift backwards to the beginning, i.e., to σ(0) This threat is the key ingredient in the analysis of the cop strategy: we construct the cops’ components so that they must strictly follow their canonical paths in order to cover all exits in X. We say that a node is covered, resp. uncovered, if at least one cop covers it, resp. if no cop covers it

Overview

The Graph Construction

Observations

The Robber’s Strategy

The Cops’ Strategy

A Lower Bound for the Robber’s Strategy

Findings

An Upper Bound for the Cops’ Strategy