## 1. Introduction

Sports have become a big business in a global economy. Tournaments are followed by millions of people across the world. Teams make big investments in new players. Broadcast rights amount to hundreds of millions of dollars in some competitions. Countries and cities fight for the right to organize worldwide events such as the Olympics and the Football World Cup.

Professional sport leagues involve millions of fans and significant investments in players, broadcast rights, merchandising, and advertising, facing challenging optimization problems. On the other side, amateur leagues involve less investments, but also require coordination and logistical efforts due to the large number of tournaments and competitors.

The main problem in sports scheduling consists in determining the date and the venue in which each game of a tournament will be played. Applications are found in the scheduling of tournaments of sports such as football, baseball, basketball, cricket, and hockey. These problems have been solved by different exact and approximate approaches, including integer programming, constraint programming, metaheuristics, and hybrid methods.

There are many relevant aspects to be considered in the determination of the best schedule for a tournament. In some situations, one seeks for a schedule minimizing the total traveled distance, as in the case of the traveling tournament (Easton et al., 2001) and in that of its mirrored variant (Ribeiro and Urrutia, 2007b), which is common to many tournaments in South America (Durán et al., 2007b). Other problems attempt to minimize the total number of breaks, i.e., the number of pairs of consecutive home games or consecutive away games played by the same team. The minimization of the carry-over effects value (Russell, 1980) is another fairness criterion leading to an even distribution of the sequence of games along the schedule. Some problems in sports scheduling have a multi-criteria nature. Ribeiro and Urrutia (2007a, 2009) tackled the scheduling of the yearly Brazilian football tournament, preliminarily formulated as a bicriteria optimization problem in which one of the objectives consisted in maximizing the number of games that could be broadcast by open TV channels (to increase the revenues from broadcast rights) and the other consisted in finding a balanced schedule with a minimum number of home breaks and away breaks (for sake of fairness). A multi-criteria version of a referee assignment problem arising in amateur leagues (Duarte et al., 2007a, 2007b) was tackled by Duarte and Ribeiro (2008).

This paper provides an introductory review to the main problems in sports scheduling, also covering the principal practical applications. Although being more focused on problems and applications, it also addresses the main solution methods and innovative algorithmic approaches applied in their solution. It should be considered as a starting point for newcomers and research in the area. The interested reader is referred to Rasmussen and Trick (2008) for a comprehensive survey of the literature on round robin tournament scheduling and to Kendall et al. (2010) for a rather complete bibliography of scheduling problems in sports.

The remaining of this paper is organized as follows. Section 'Definitions' reviews the main definitions and basic issues. Section 'Some fundamental problems' presents an overview of the main problems arising in sports scheduling and their formulations: breaks minimization, distance minimization and the traveling tournament problem (TTP), and carry-over effects minimization. Problem reformulation techniques are investigated in Section 'Reformulations', which explores a variant of the TTP where the venues are known beforehand. Section 'Applications' surveys applications in different sport disciplines like football, baseball, basketball, cricket, and hockey. A case study of a recent application of integer programming in the scheduling of the yearly first-division football tournament in Brazil is reported in Section 'Application: scheduling the Brazilian football tournament'. Concluding remarks and references to other scheduling problems in sports are given in the last section.