Characterizing Feasible Pattern Sets with a Minimum Number of Breaks

In sports timetabling, creating an appropriate timetable for a round-robin tournament with home–away assignment is a significant problem. To solve this problem, we need to construct home–away assignment that can be completed into a timetable; such assignment is called a feasible pattern set. Although finding feasible pattern sets is at the heart of many timetabling algorithms, good characterization of feasible pattern sets is not known yet. In this paper, we consider the feasibility of pattern sets, and propose a new necessary condition for feasible pattern sets. In the case of a pattern set with a minimum number of breaks, we prove a theorem leading a polynomial-time algorithm to check whether a given pattern set satisfies the necessary condition. Computational experiment shows that, when the number of teams is less than or equal to 26, the proposed condition characterizes feasible pattern sets with a minimum number of breaks.