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The Home-Away Assignment Problems and Break Minimization/Maximization Problems in Sports Scheduling

Suppose that we have a timetable of a round-robin tournament with a number of teams, and distances between their homes. The home-away assignment problem is to find a home-away assignment that minimizes the total traveling distance of the teams; the break minimization/maximization problem is to find a home-away assignment that minimizes/maximizes the number of breaks, i.e., the number of occurrences of consecutive matches held either both at away or both at home for a team. The aim of this paper is to give a unified view to the three problems. We see that optimal solutions of the break minimization/maximization problems are obtained by solving the home-away assignment problem. For these problems, we propose formulations and approximation preserving reductions, and report known approximation algorithms. For the home-away assignment problem, we give a formulation as an integer program and some rounding algorithms. We also provide a technique to transform the home-away assignment problem to MIN RES CUT and apply Goemans and Williamson’s algorithm for MAX RES CUT, which is based on a positive semidefinite programming relaxation, to the obtained MIN RES CUT instances. Our computational experiments show that the proposed approaches quickly generate solutions of good approximation ratios.