How Hard Is It to Compute Cover Time?

It is $\#\mathrm{P}\text{-hard}$. Even for connected simple undirected unweighted graphs, deciding whether an expected random-walk cover time is at most a given rational threshold is $\#\mathrm{P}\text{-hard}$ under polynomial-time Turing reductions, both from a specified starting vertex and when maximized over starting vertices. Both families have a universal vertex, called the hub, and diameter two. Deleting the hub leaves a planar bipartite graph, and every non-hub vertex has degree at most five. A shared construction encodes an Euler-tour count in rare covering walks. For specified starts, a number of fractional bits linear in the queried graph's vertex count suffices for recovery. For the maximum-start problem, a fixed cube attachment makes the maximizing starts known, and one interpolation of expected cover times recovers the count. Both reductions use polynomial-size graphs and polynomial-bit threshold queries. For general graphs, both comparison problems can be solved in polynomial time with access to an $\mathrm{NP}^{\mathrm{PP}}$ oracle.