Time dependent biased random walks

Journal article

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 p1-ε; 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.

Authors: Haslegrave, J; Sauerwald, T; Sylvester, J

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Association for Computing Machinery
• Journal: ACM Transactions on Algorithms
• Volume: 18
• Issue: 2
• Article number: 12
• Publication date: 2022-03-15
• Acceptance date: 2021-11-01
• DOI: 10.1145/3498848
• EISSN: 1549-6333
• ISSN: 1549-6325
• Language: English
• Keywords: Markov chain; random walk; cover time; FFR; Markov decision process; PSPACE