Lower bounds for large traveling umpire instances: New valid inequalities and a branch-and-cut algorithm

Abstract

Given a double round-robin tournament, the Traveling Umpire Problem (TUP) seeks to assign umpires to the games of the tournament while minimizing the total distance traveled by the umpires. The assignment must satisfy constraints that prevent umpires from seeing teams and venues too often, while making sure all games have umpires in every round, and all umpires visit all venues. We propose a new integer programming model for the TUP that generalizes the two best existing models, introduce new families of strong valid inequalities, and implement a branch-and-cut algorithm to solve instances from the TUP benchmark. When compared against published state-of-the-art methods, our algorithm significantly improves all best known lower bounds for large TUP instances (with 20 or more teams).