Integer Programming Models for Round Robin Tournaments

TL;DR

This paper introduces two new integer programming models for scheduling round robin tournaments, demonstrating that the matching formulation is stronger and more efficient, with practical algorithms and valid inequalities for improved solutions.

Contribution

The paper presents a novel matching formulation for round robin scheduling, compares its strength to traditional models, and develops a branch-and-price algorithm based on it.

Findings

Matching formulation is stronger than traditional models.

LP relaxation of the matching formulation is solvable in polynomial time.

The branch-and-price algorithm effectively finds optimal tournaments.

Abstract

Round robin tournaments are omnipresent in sport competitions and beyond. We propose two new integer programming formulations for scheduling a round robin tournament, one of which we call the matching formulation. We analytically compare their linear relaxations with the linear relaxation of a well-known traditional formulation. We find that the matching formulation is stronger than the other formulations, while its LP relaxation is still being solvable in polynomial time. In addition, we provide an exponentially sized class of valid inequalities for the matching formulation. Complementing our theoretical assessment of the strength of the different formulations, we also experimentally show that the matching formulation is superior on a broad set of instances. Finally, we describe a branch-and-price algorithm for finding round robin tournaments that is based on the matching formulation.