Evasive Random Walks and the Clairvoyant Demon

TL;DR

This paper investigates the conditions under which two random walks on a graph can be scheduled to avoid collisions, introducing the concept of evasive walks and analyzing their existence and properties.

Contribution

It introduces the notion of evasive walks, characterizes graphs that admit them, and provides algorithms for successful scheduling in certain graph classes.

Findings

Graphs with evasive walks are explicitly characterized.

On cycles, tokens tend to collide quickly with high probability.

Algorithms are developed for successful scheduling under specific graph conditions.

Abstract

A pair of random walks $(R,S)$ on the vertices of a graph $G$ is {\it successful} if two tokens can be scheduled (moving only one token at a time) to travel along $R$ and $S$ without colliding. We consider questions related to P. Winkler's {\it clairvoyant demon problem}, which asks whether for random walks $R$ and $S$ on $G$, $Pr[\ (R,S) \mbox{ is successful }] >0$. We introduce the notion of an {\it evasive} walk on $G$: a walk $S$ so that for a random walk $R$ on $G$, $Pr[\ (R,S) \mbox{ is successful }]>0$. We characterize graphs $G$ having evasive walks, giving explicit constructions on such $G$. On a cycle, we show that with high probability the tokens must collide quickly. Finally we consider two variants of the problem for which, under certain assumptions on the graph $G$, we provide algorithms that schedule $(R,S)$ successfully with positive probability.