Branch-and-Price and Improved Bounds to the Traveling Umpire Problem

The present paper proposes a branch-and-price approach to the Traveling Umpire Problem (TUP). In this hard combinatorial optimization problem, umpires (or referees) have to be assigned to games in a double round robin tournament. The objective is to obtain a solution with minimum total travel distance over all umpires, while respecting several hard constraints. Dantzig-Wolfe decomposition is applied to an existing Integer Programming formulation to be used in a branch-and-price framework. The pricing problems are solved using a specialized branch-and-bound algorithm, which applies multiple pruning techniques. Two branching strategies (best-first and depth-first) were employed and result in many improved lower bounds compared to the previous best known. In addition, five new best solutions were found and four instances with 16 teams were proven to be infeasible.