A Timecop's Chase Around the Table

TL;DR

This paper studies a cops and robber game on time-varying graphs with periodic edges, proving NP-hardness of winning strategies, establishing bounds related to cycle lengths and period LCMs, and improving complexity bounds to PSPACE.

Contribution

It introduces the complexity of the cops and robber game on periodic TVGs, proves NP-hardness even on simple graphs, and refines the complexity class from EXPTIME to PSPACE.

Findings

NP-hardness for simple cycle graphs

Matching bounds between cycle length and period LCM

Improved upper bound to PSPACE for the problem

Abstract

We consider the cops and robber game variant consisting of one cop and one robber on time-varying graphs (TVG). The considered TVGs are edge periodic graphs, i.e., for each edge, a binary string $s_e$ determines in which time step the edge is present, namely the edge $e$ is present in time step $t$ if and only if the string $s_e$ contains a $1$ at position $t \mod |s_e|$. This periodicity allows for a compact representation of the infinite TVG. We proof that even for very simple underlying graphs, i.e., directed and undirected cycles the problem whether a cop-winning strategy exists is NP-hard and W[1]-hard parameterized by the number of vertices. Our second main result are matching lower bounds for the ratio between the length of the underlying cycle and the least common multiple (LCM) of the lengths of binary strings describing edge-periodicies over which the graph is robber-winning. Our third main result improves the previously known EXPTIME upper bound for Periodic Cop and Robber on general edge periodic graphs to PSPACE-membership.