An Exact Approach for the Balanced k-Way Partitioning Problem with Weight Constraints and its Application to Sports Team Realignment

TL;DR

This paper introduces an exact method for solving the balanced k-way partitioning problem with weight constraints, applied to sports team realignment to minimize travel distances, using integer programming and cutting-plane techniques.

Contribution

It develops two integer programming formulations and proves the validity of inequalities, enhancing solution methods for sports league realignment problems.

Findings

Optimal solution for Ecuadorian Football league realignment

Branch & Cut algorithm outperforms tabu search in efficiency

Method adaptable to other sports leagues' realignment

Abstract

In this work a balanced k-way partitioning problem with weight constraints is defined to model the sports team realignment. Sports teams must be partitioned into a fixed number of groups according to some regulations, where the total distance of the road trips that all teams must travel to play a Double Round Robin Tournament in each group is minimized. Two integer programming formulations for this problem are introduced, and the validity of three families of inequalities associated to the polytope of these formulations is proved. The performance of a tabu search procedure and a Branch & Cut algorithm, which uses the valid inequalities as cuts, is evaluated over simulated and real-world instances. In particular, an optimal solution for the realignment of the Ecuadorian Football league is reported and the methodology can be suitable adapted for the realignment of other sports leagues.