Integer Programming Models for Round Robin 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.

First-break-heuristically-schedule: Constructing highly-constrained sports timetables

“…Constraints (14) ensure that all games are scheduled. If the timetable needs to be phased, we add Constraints (15). Constraints ( 16) represent a set of application-specific constraints that involve the x i, j,s variables (see Section 3).…”

“…With regard to the quality of the opponent schedule, Van Bulck and Goossens [12] show that the generated lower bounds are as good as the Lagrangian relaxation relative to ( 12)-( 13) or relative to (14). Interestingly, once the HAP set is fixed, the LP-relaxation based on the x i, j,s variables is also relaxation-equivalent to an exponentially-sized model where there is one variable for every possible perfect matching in the complete graph K n (see van Doornmalen et al [15]). The proof for this result is trivial, and follows from the fact that the well-known odd-set inequalities are redundant in a complete bipartite graph.…”