Traveling salesmen in the presence of competition

We propose the “Competing Salesmen Problem” (CSP), a 2-player competitive version of the classical Traveling Salesman Problem. This problem arises when considering two competing salesmen instead of just one. The concern for a shortest tour is replaced by the necessity to reach any of the customers before the opponent does. In particular, we consider the situation where players take turns, moving along one edge at a time within a graph G = (V,E). The set of customers is given by a subset VC ⊆ V of the vertices. At any given time, both players know of their opponent’s position. A player wins if he is able to reach a majority of the vertices in VC before the opponent does. We prove that the CSP is PSPACE-complete, even if the graph is bipartite, and both players start at distance 2 from each other. Furthermore, we show that the starting player may not be able to avoid losing the game, even if both players start from the same vertex. However, for the case of bipartite graphs, we show that the starting player always can avoid a loss. On the other hand, we show that the second player can avoid to lose by more than one customer, when play takes place on a graph that is a tree T , and VC consists of leaves of T . It is unclear whether a polynomial strategy exists for any of the two players to force this outcome. For the case where T is a star (i. e., a tree with only one vertex of degree higher than two) and VC consists of n leaves of T , we give a simple and fast strategy which is optimal for both players. If VC consists not only of leaves, we point out that the situation is more involved.