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 nn teams and n−1n−1 rounds, there always exists a home–away assignment with at most 14n(n−2) breaks. Secondly, for infinitely many nn, we construct opponent schedules for which at least 16n(n−1) breaks are necessary. Finally, we prove that break minimization for nn teams and a partial opponent schedule with rr rounds is an NP-hard problem for r≥3r≥3. This is in strong contrast to the case of r=2r=2 rounds, which can be scheduled (in polynomial time) without any breaks.