Heuristics for the mirrored traveling tournament problem
Introduction
Professional sports leagues are a major economic activity around the world. Teams and leagues do not want to waste their investments in players and structure in consequence of poor schedules of games. Game scheduling is a difficult task with multiple constraints and objectives (involving, e.g., logistic, organizational, economical, and fairness issues), as well as several decision makers (such as league officials, team managers, and TV executives).
Schedules with minimum traveling times and offering similar conditions to all teams are of major interest for teams, leagues, sponsors, fans, and the media. In the case of the Brazilian soccer national championship, a single trip from Porto Alegre to Belém takes almost a full journey and many stops, due to the absence of direct flights to cover a distance of approximately 4000 km. The total distance traveled becomes an important variable to be minimized, so as to reduce traveling costs and to give more time to the players for resting and training along a season that lasts for approximately six months. Another possible variable to be minimized is the maximum distance traveled over all teams.
Several authors in different contexts (see, e.g., [1], [2], [4], [24], [26], [27], [28], [29], [30]) tackled the problem of tournament scheduling in different leagues and sports such as soccer, basketball, hockey, baseball, rugby, cricket, and football, using different techniques such as integer programming, tabu search, genetic algorithms, and simulated annealing. Another promising technique for sports timetabling is constraint programming, which was already used by Henz [19] and lead to impressive improvements in terms of computation time with respect to [24]. The results in [19] were also improved by Zhang [31], using again constraint programming. We refer to Henz et al. [20] for recent advances in constraint programming for scheduling problems in sports. Easton et al. [10] reviewed scheduling problems in sports.
The traveling tournament problem is an intermural championship timetabling problem that abstracts certain characteristics of scheduling problems in sports [11]. It combines tight feasibility issues with a difficult optimization problem. The objective is to minimize the total distance traveled by the teams, subject to the constraint that no team can play on a row more than three games at home or away. Since the total distance traveled is a major issue for every team taking part in the tournament, solving a traveling tournament problem may be a starting point for the solution of real timetabling applications in sports.
In this paper, we propose new, effective heuristics for solving the mirrored version of the traveling tournament problem. The contribution of the paper is twofold: first, a new, fast constructive heuristic which provides good initial solutions; and second, a hybridization of the GRASP (Greedy Randomized Adaptive Search Procedure) and ILS (iterated local search) metaheuristics which computes high-quality solutions. The paper is organized as follows. In Section 2, the mirrored version of the traveling tournament problem is formulated in detail. In Section 3, we show how a solution of this problem may be represented as an oriented ordered 1-factorization of a complete graph. A constructive heuristic for building good feasible initial solutions is proposed in Section 4. Three simple neighborhoods and an ejection chain neighborhood are described in Section 5. A local search procedure based on these neighborhoods is described in Section 6. In Section 7, we present the new GRILS-mTTP heuristic for the mirrored traveling tournament problem, combining the main ideas of the GRASP and ILS metaheuristics. Computational results for 12 benchmark instances and one real-life problem are reported in Section 8. Concluding remarks are made in the last section.
Section snippets
The mirrored traveling tournament problem
We consider a tournament played by n teams, where n is an even number. In a simple round-robin (SRR) tournament, each team plays every other exactly once in n − 1 prescheduled rounds. The game between teams i and j is represented by the unordered pair {i, j}. There are n/2 games in each round. Each team plays exactly once in each round. In a double round-robin (DRR) tournament, each team plays every other twice, once at home and once away. A mirrored double round-robin (MDRR) tournament is a
Solutions as ordered oriented 1-factorizations
We use the same formulation proposed by de Werra [7], [8] for representing a tournament schedule using 1-factorizations of a graph. A factor of an undirected graph G = (V, E) is a graph G′ = (V, E′) with E′ ⊆ E. G′ is a 1-factor if all its nodes have a degree equal to one. A factorization of G is a set of edge-disjoint factors G1 = (V, E1), … , Gp = (V, Ep), with E1 ∪ ⋯ ∪ Ep = E. All factors in a 1-factorization of G are 1-factors. An oriented 1-factorization of G is a 1-factorization in which an orientation is set
A fast constructive heuristic for the mTTP
Good initial solutions can affect the performance of metaheuristic-based algorithms for timetabling problems. Elmohamed et al. [12] noticed that when a random initial configuration is used for timetabling problems, simulated annealing performs very poorly; however, there is a dramatic improvement in performance when a preprocessor is used to provide a good starting point for the annealing. Fast approximate algorithms are important to timetabling problems in sports, because the different and
Neighborhoods
Four neighborhood structures are defined and will be later used by the extended GRASP with ILS heuristic described in Section 7.
Local search
Four neighborhood structures were defined in the previous section. The first three and simpler neighborhoods HAS, TS, and PRS are explored by local search. The GR ejection chain neighborhood will be explored only as a diversification move performed less frequently by the heuristic described in the next section, due to the high computation costs associated with its investigation.
We use a first-improving strategy similar to the VND (variable neighborhood descent) procedure [18], [23] to implement
Extended GRASP with ILS
GRASP is a multi-start or iterative metaheuristic [13], [14], [25], in which each iteration consists of two phases: construction and local search. The construction phase builds a feasible solution, whose neighborhood is investigated until a local minimum is found during the local search phase. The best overall solution is kept as the result. An extensive survey of the literature is given in [15].
The ILS (iterated local search) metaheuristic [21], [22] starts from a locally optimal feasible
Computational results
The algorithms described in this paper were coded in C++ and compiled with version 2.96 of the gcc compiler with the optimization flag −03. The experiments were performed on a Pentium IV machine with a 2.0 GHz clock and 512 Mbytes of RAM memory.
Concluding remarks
In this work, we investigated the yet unexplored mirrored version of the traveling tournament problem (mTTP). This tournament structure is very common in Latin American tournaments.
The first contribution of this paper is a very fast constructive heuristic for the mirrored traveling tournament problem. It runs in approximately 1/1000 of a second on a Pentium IV machine with a 2.0 GHz clock for the largest benchmark instances. Computation times are smaller by several orders of magnitude with
References (31)
Geography, games and graphs
Discrete Applied Mathematics
(1980)Scheduling in sports
- et al.
A probabilistic heuristic for a computationally difficult set covering problem
Operations Research Letters
(1989) Ejection chains, reference structures and alternating path methods for traveling salesman problems
Discrete Applied Mathematics
(1996)- et al.
Global constraints for round robin tournament scheduling
European Journal of Operational Research
(2004) - et al.
Variable neighborhood search
Computers and Operations Research
(1997) Combinatorial aspects of construction of competition Dutch professional football leagues
Discrete Applied Mathematics
(1992)- A. Anagnostopoulos, L. Michel, P. Van Hentenryck, Y. Vergados, A simulated annealing approach to the traveling...
- et al.
Optimal schedules for even-numbered team athletic conferences
AIIE Transactions
(1997) - A. Cardemil, Optimización de fixtures deportivos: Estado del arte y un algoritmo tabu search para el traveling...