The break minimization problem is solvable in polynomial time when the optimal value is less than the number of teams

The break minimization problem is to find a home-away assignment that minimizes the number of breaks for a given timetable of a round-robin tournament. In a recent paper, Elf, Jünger and Rinaldi conjectured that the break minimization problem is solvable in polynomial time when the optimal value is less than the number of teams. This paper proves their conjecture affirmatively by showing that a problem related to the break minimization problem is solvable in polynomial time. Our approach is to transform an instance of the related problem into instances of the 2-satisfiability problem. In this paper, we prove a previously proposed conjecture about the break minimization problem in sports timetabling. We consider a round-robin tournament with the following properties: • the number of teams is 2n (n ∈ IN), and the number of slots, i.e., the days when matches are held, is 2n− 1; • each team plays one match in each slot; • each team plays every other team once; • each team has its home, and each match is held at the home of one of the corresponding two teams. Figure 1 is a timetable of a round-robin tournament satisfying these properties. In the figure, each match with ‘@’ means that the match is held at the home of the opponent; without ‘@’ is held at the home of the team corresponding to the row. For example, in slot 5 team 2 plays team 3 at the home of team 2. In other words, team 3 plays at away in slot 5, whereas team 2 plays at home. If a team plays either both at home or both at away in slots s− 1 and s, it is said that the team has a break at slot s. In Fig. 1, team 3 plays at home in slots 1 and 2, and thus team 3 has a break at slot 2. In total, the timetable has six breaks, each of which is represented as a line under the corresponding entry. Given a timetable not assigned where to play, one should decide a homeaway assignment to complete a timetable (Fig. 2). In most of practical sports ? Supported by Superrobust Computation Project of the 21st Century COE Program “Information Science and Technology Strategic Core,” Japan.