On the complexity of the traveling umpire problem

The traveling umpire problem (TUP) consists of determining which games will be handled by each one of several umpire crews during a double round-robin tournament. The objective is to minimize the total distance traveled by the umpires, while respecting constraints that include visiting every team at home, and not seeing a team or venue too often. Even small instances of the TUP are very difficult to solve, and several exact and heuristic approaches for it have been proposed in the literature. To this date, however, no formal proof of the TUP’s computational complexity exists. We prove that the decision version of the TUP is NP-complete for certain values of its input parameters.