The influence of maximum (s,t)-cuts on the competitiveness of deterministic strategies for the Canadian Traveller Problem

Abstract

We study the k-Canadian Traveller Problem, where the objective is to lead a traveller in an undirected weighted graph G from a source s to a target t, knowing that at most k edges of the graph are blocked and cannot be traversed. The locations of blockages are unknown at the beginning of the walk but a blocked edge is revealed when the traveller visits one of its endpoints. There exist graphs for which the competitive ratio of any deterministic strategies cannot be smaller than 2k+1. Conversely, there exists a very simple strategy, REPOSITION, which achieves this ratio 2k+1. It consists in successively traversing shortest (s,t)-paths and coming back to s when the traveller is blocked. We refine this analysis by detecting families of graphs for which a smaller competitive ratio can be obtained. This paper produces a global analysis to understand the impact of the size of the maximum (s,t)-cuts of G on the competitiveness of deterministic strategies. We design deterministic strategies achieving a ratio ρk+O(λ), with ρ<2, for two different cut parameters λ. In particular, we propose a strategy called DETOUR with a competitive ratio 2k+O(μmaxE), where μmaxE is the size of the maximum edge (s,t)-cut. Another contribution is a strategy called BYPASS with a competitive ratio [Formula presented], where λmaxV is the size of the maximum vertex (s,t)-cut of all subgraphs of G. This produces an efficient algorithm for outerplanar graphs, which verify λmaxV≤2.