Unweighted Layered Graph Traversal

Introduced by Papadimitriou and Yannakakis in 1989, layered graph traversal is an important problem in online algorithms and mobile computing that has been studied for several decades, and which now is essentially resolved in its original formulation. In this paper, we demonstrate that what appears to be an innocuous modification of the problem actually leads to a drastic (exponential) reduction of the competitive ratio. Specifically, we present an algorithm that is $O(\log^2 w)$-competitive for traversing unweighted layered graphs of width $w$. Our technique is based on a simple entropic regularizer, which evolves as the agent progresses in the layered graph. Our algorithm is randomized and simply maintains that at all layers, the probability distribution of the position of the mobile agent maximizes the entropic regularizer.