Online graph exploration on a restricted graph class: Optimal solutions for tadpole graphs

Abstract

We study the problem of online graph exploration on undirected graphs, where a searcher has to visit every vertex and return to the origin. Once a new vertex is visited, the searcher learns of all neighboring vertices and the connecting edge weights. The goal of such an exploration is to minimize its total cost, where each edge traversal incurs a cost of the corresponding edge weight.We investigate the problem on tadpole graphs (also known as dragons, kites), which consist of a cycle with an attached path. The construction by Miyazaki et al. (The online graph exploration problem on restricted graphs, IEICE Transactions 92-D (9), 2009) can be extended to show that every online algorithm on these graphs must have a competitive ratio of 2−ε, but the authors did not investigate non-unit edge weights. We show via amortized analysis that a greedy approach yields a matching competitive ratio of 2 on tadpole graphs, for arbitrary non-negative edge weights. Moreover, we also briefly discuss the topic of advice complexity on cycle and tadpole graphs.