A Beam Search Approach to the Traveling Tournament Problem

Abstract

The well-known traveling tournament problem is a hard optimization problem in which a double round robin sports league schedule has to be constructed while minimizing the total travel distance over all teams. The teams start and end their tours at their home venues, are only allowed to play a certain maximum number of games in a row at home or away, and must not play against each other in two consecutive rounds. The latter aspects introduce also a difficult feasibility aspect. In this work, we study a beam search approach based on a recursive state space formulation. We compare different state ordering heuristics for the beam search based on lower bounds derived by means of decision diagrams. Furthermore, we introduce a randomized beam search variant that adds Gaussian noise to the heuristic value of a node for diversifying the search in order to enable a simple yet effective parallelization. In our computational study, we use randomly generated instances to compare and tune algorithmic parameters and present final results on the classical National League and circular benchmark instances. Results show that this purely construction-based method provides mostly better solutions than existing ant-colony optimization and tabu search algorithms and it comes close to the leading simulated annealing based approaches without using any local search. For two circular benchmark instances we found new best solutions for which the last improvement was twelve years ago. The presented state space formulation and lower bound techniques could also be beneficial for exact methods like A$^*$ or DFS$^*$ and may be used to guide the randomized construction in ACO or GRASP approaches.