Sports tournaments, home-away assignments, and the break minimization problem

We consider the break minimization problem for fixing home-away assignments in round-robin sports tournaments. First, we show that for an opponent schedule with n teams and n− 1 rounds, there always exists a home-away assignment with at most 1 4n(n−2) breaks. Secondly, for infinitely many n, we construct opponent schedules for which at least 6n(n−1) breaks are necessary. Finally, we prove that break minimization for n teams and a partial opponent schedule with r rounds is an NP-hard problem for r ≥ 3. This is in strong contrast to the case of r = 2 rounds, which can be scheduled (in polynomial time) without any breaks.