Time Dependent Biased Random Walks

TL;DR

This paper extends the study of biased random walks with a controller to a time-dependent setting, providing new bounds on cover and hitting times, and analyzing the complexity of optimal strategies.

Contribution

It introduces a time-dependent model of biased random walks, refines conjectures about stationary probabilities, and proves PSPACE-completeness for computing optimal strategies.

Findings

New bounds on cover and hitting times for the model.

Confirmation of a refined conjecture for a broad class of graphs.

Proving PSPACE-completeness for directed graphs in computing optimal strategies.

Abstract

We study the biased random walk where at each step of a random walk a "controller" can, with a certain small probability, move the walk to an arbitrary neighbour. This model was introduced by Azar et al. [STOC'1992]; we extend their work to the time dependent setting and consider cover times of this walk. We obtain new bounds on the cover and hitting times. Azar et al. conjectured that the controller can increase the stationary probability of a vertex from $p$ to $p^{1-\epsilon}$; while this conjecture is not true in full generality, we propose a best-possible amended version of this conjecture and confirm it for a broad class of graphs. We also consider the problem of computing an optimal strategy for the controller to minimise the cover time and show that for directed graphs determining the cover time is PSPACE-complete.