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.

翻译：我们研究有偏向的随机行走,在随机行走的每一步,“控制者”都可以以某种小的概率将行走移动到任意的邻居。这个模型是由Azar等人[STOC'1992] 推出的;我们把他们的工作扩大到根据时间而定的设置,并考虑行走的覆盖时间。我们在封面和打击时间上获得了新的界限。Azar 等人推测,控制者可以增加一个脊椎的固定概率,从美元到1美元到1美元;虽然这一推测并不完全属实,但我们提出一个最有可能修改的这一推测的版本,并为广泛的图表类别加以确认。我们还考虑为控制者计算一个最佳策略以尽量缩短覆盖时间的问题,并表明确定覆盖时间的定向图形是PSPACE完成的。