  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 V_C 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 V_C 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. We show that the starting player may lose the
game, even if both players start from the same vertex. For bipartite graphs, we
show that the starting player always can avoid a loss. We also 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 V_C consists of leaves of T. For the
case where T is a star and V_C consists of n leaves of T, we give a simple and
fast strategy which is optimal for both players. If V_C consists not only of
leaves, the situation is more involved.
