Multiple Canadians on the road: minimizing the distance competitive ratio

Abstract

The online k-Canadian Traveller Problem ($k$-CTP), known to be PSPACE-complete, asks for the best strategy a traveller has to follow in order to traverse with minimum distance a graph from s to t where at most k edges are blocked. A blocked edge is revealed when the traveller visits one of its endpoints. It is proven that for any deterministic strategy, the competitive ratio is larger than $2k+1$. Indeed, the distance traversed by the traveller is potentially greater than $2k+1$ times the optimal journey. For randomized strategies, this ratio becomes $k+1$. We complement the work of Zhang et al. on $k$-CTP with multiple travellers by evaluating the distance competitive ratio of deterministic and randomized strategies for complete and partial communication. We compare these ratios with two other communication levels: when travellers do not communicate at all and when they communicate only before beginning to move. Eventually, we provide a wide picture of the distance competitiveness reachable for the $k$-CTP in function of the number of travellers and communication levels.