Random walks which prefer unvisited edges. Exploring high girth even degree expanders in linear time

We consider a modified random walk which uses unvisited edges whenever possible, and makes a simple random walk otherwise. We call such a walk an edge-process. We assume there is a rule A, which tells the walk which unvisited edge to use whenever there is a choice. In the simplest case, A is a uniform random choice over unvisited edges incident with the current walk position. However we do not exclude arbitrary choices of rule A. For example, the rule could be determined on-line by an adversary, or could vary from vertex to vertex. For even degree expander graphs, of bounded maximum degree, we have the following result. Let G be an n vertex even degree expander graph, for which every vertex is in at least one vertex induced cycle of length L. Any edge-process on G has cover time (n+ (n log n)/L). This result is independent of the rule A used to select the order of the unvisited edges, which can be chosen on-line by an adversary. As an example, With high probability, random r-regular graphs, (r at least 4, even), are expanders for which L = Omega(log n). Thus, for almost all such graphs, the vertex cover time of the edge-process is Theta(n). This improves the vertex cover time of such graphs by a factor of log n, compared to the Omega(n log n) cover time of any weighted random walk.

💡 Research Summary

The paper introduces a variant of the classical random walk called the “edge‑process.” In this process a walker, when situated at a vertex, first checks whether there are any incident edges that have never been traversed. If such edges exist, the walker must select one of them and move along it; only when all incident edges have already been used does the walker revert to a simple, unbiased random step. The rule that resolves the choice among several unvisited edges is denoted by A. The authors deliberately keep A completely general: it may be a uniform random selection, a deterministic rule that varies from vertex to vertex, or even an adversarial online algorithm that decides the next edge based on the entire history of the walk.

The main setting of the analysis is a family of bounded‑degree, even‑degree expander graphs. Formally, let G be an n‑vertex graph in which every vertex has even degree and the maximum degree Δ is a constant. Moreover, G is assumed to be an expander with a spectral gap bounded away from zero, and every vertex belongs to at least one vertex‑induced cycle of length L (the “girth” condition). Under these structural hypotheses the authors prove a universal upper bound on the vertex cover time of any edge‑process, regardless of the choice of A:

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