Improved Deterministic Strategy for the Canadian Traveller Problem Exploiting Small Max-(s, t)-Cuts

Abstract

The k-Canadian Traveller Problem consists in finding the optimal way from a source s to a target t on an undirected weighted graph G, knowing that at most k edges are blocked. The traveller, guided by a strategy, sees an edge is blocked when he visits one of its endpoints. A major result established by Westphal is that the competitive ratio of any deterministic strategy for this problem is at least $2k+1$. reposition and comparison strategies achieve this bound.

We refine this analysis by focusing on graphs with a maximum (s, t)-cut size $\mu _{\text {max}}$ less than k. A strategy called detour is proposed and its competitive ratio is $2\mu _{\text {max}}+ \sqrt{2}(k-\mu _{\text {max}}) + 1$ when $\mu _{\text {max}}< k$ which is strictly less than $2k+1$. Moreover, when $\mu _{\text {max}}\ge k$, the competitive ratio of detour is $2k+1$ and is optimal. Therefore, detour improves the competitiveness of the deterministic strategies known up to now.