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Minimizing breaks by minimizing odd cycle transversals

Koichi Fujii, Tomomi Matsui

Abstract

Constructing a suitable schedule for sports competitions is a crucial issue in sports scheduling. The round-robin tournament is a competition adopted in many professional sports. For most round-robin tournaments, it is considered undesirable that a team plays consecutive away or home matches; such an occurrence is called a break. Accordingly, it is preferable to reduce the number of breaks in a tournament. A common approach is first to construct a schedule and then determine a home-away assignment based on the given schedule to minimize the number of breaks (first-schedule-then-break). In this study, we concentrate on the problem that arises in the second stage of the first-schedule-then-break approach, namely, the break minimization problem(BMP). We prove that this problem can be reduced to an odd cycle transversal problem, the well-studied graph problem. These results lead to a new approximation algorithm for the BMP.

Minimizing breaks by minimizing odd cycle transversals

Abstract

Constructing a suitable schedule for sports competitions is a crucial issue in sports scheduling. The round-robin tournament is a competition adopted in many professional sports. For most round-robin tournaments, it is considered undesirable that a team plays consecutive away or home matches; such an occurrence is called a break. Accordingly, it is preferable to reduce the number of breaks in a tournament. A common approach is first to construct a schedule and then determine a home-away assignment based on the given schedule to minimize the number of breaks (first-schedule-then-break). In this study, we concentrate on the problem that arises in the second stage of the first-schedule-then-break approach, namely, the break minimization problem(BMP). We prove that this problem can be reduced to an odd cycle transversal problem, the well-studied graph problem. These results lead to a new approximation algorithm for the BMP.

Paper Structure

This paper contains 6 sections, 5 theorems, 12 equations, 2 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Break Minimization Problem
  4. Odd Cycle Transversal
  5. Main theorem
  6. Example

Key Result

Theorem 3.1

Let $\tau$ be a given timetable and $B_{\min} (\tau)$ the optimal value of the BMP defined by $\tau.$ Then, $B_{\min} (\tau)$ is equal to the size of the minimum OCT of the auxiliary graph $G(\tau).$

Figures (2)

  • Figure 1: Auxiliary graph
  • Figure 2: Rectangular cycles connected to $C(t_1^*, t_2^*)$.

Theorems & Definitions (9)

  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Corollary 3.5