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 ${1\over 4}n(n-2)$ breaks. Secondly, for infinitely many $n$, we construct opponent schedules for which at least ${1\over 6}n(n-1)$ 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 $r\ge 3$. This is in strong contrast to the case of $r=2$ rounds, which can be scheduled (in polynomial time) without any breaks.