Finding optimal realignments in sports leagues using a branch-and-cut-and-price approach

The sports team realignment problem can be modelled as k-way equipartition: given a complete graph Kn = (V, E), with edge weight ce on each edge, partition the vertices V into k divisions that have exactly S vertices, so as to minimise the total weight of the edges that have both endpoints in the same division. In this paper, we discuss solving k-way equipartition problem using branch-and-price scheme. We demonstrated the necessity of cutting planes for this problem and suggested an effective way of adding cutting planes in the branch-and-price framework. We solved the pricing subproblem as an integer programming problem. Using this method, we found the optimal realignment solution for three major professional sports leagues in North America (basketball, hockey, football). We also present computational results on some larger randomly generated microaggregation problems.