On the Competitiveness of Memoryless Strategies for the k-Canadian Traveller Problem
The k-Canadian Traveller Problem (\(k\)-CTP), proven PSPACE-complete by Papadimitriou and Yannakakis, is a generalization of the Shortest Path Problem which admits blocked edges. Its objective is to determine the strategy that makes the traveller traverse graph G between two given nodes s and t with the minimal distance, knowing that at most k edges are blocked. The traveller discovers that an edge is blocked when arriving at one of its endpoints.
We study the competitiveness of randomized memoryless strategies to solve the \(k\)-CTP. Memoryless strategies are attractive in practice as a decision made by the strategy for a traveller in node v of G does not depend on his anterior moves. We establish that the competitive ratio of any randomized memoryless strategy cannot be better than \(2k + O\left( 1\right) \). This means that randomized memoryless strategies are asymptotically as competitive as deterministic strategies which achieve a ratio \(2k+1\) at best.