Traveling Tournament Problem Research Articles

Practical Algorithms with Guaranteed Approximation Ratio for Traveling Tournament Problem with Maximum Tour Length 2

The traveling tournament problem (TTP) is a hard but interesting sports scheduling problem inspired by Major League Baseball, which is to design a double round-robin schedule such that each pair of teams plays one game in each other’s home venue, minimizing the total distance traveled by all n teams (n is even). In this paper, we consider TTP-2 (i.e., TTP under the constraint that at most two consecutive home games or away games are allowed for each team). In this paper, we propose practical algorithms for TTP-2 with improved approximation ratios. Because of the different structural properties of the problem, all known algorithms for TTP-2 are different for n/2 being odd and even, and our algorithms are also different for these two cases. For even n/2, our approximation ratio is [Formula: see text], improving the previous result of [Formula: see text]. For odd n/2, our approximation ratio is [Formula: see text], improving the previous result of [Formula: see text]. In practice, our algorithms are easy to implement. Experiments on well-known benchmark sets show that our algorithms beat previously known solutions for all instances with an average improvement of 5.66%. Funding: This work was supported by the National Natural Science Foundation of China [Grants 62372095 and 62172077] and the Sichuan Natural Science Foundation [Grant 2023NSFSC0059].

Approaching the Traveling Tournament Problem with Randomized Beam Search.

The traveling tournament problem is a well-known sports league scheduling problem famous for its practical hardness. Given an even number of teams with symmetric distances between their venues, a double round-robin tournament has to be scheduled minimizing the total travel distances over all teams. We consider the most common constrained variant without repeaters and a streak limit of three, for which we study a beam search approach based on a state-space formulation guided by heuristics derived from different lower bound variants. We solve the arising capacitated vehicle routing subproblems either exactly for small- to medium-sized instances up to 18 teams or heuristically also for larger instances up to 24 teams. In a randomized variant of the search, we employ random team ordering and add small amounts of Gaussian noise to the nodes' guidance for diversification when multiple runs are performed. This allows for a simple yet effective parallelization of the beam search. A final comparison is done on the NL, CIRC, NFL, and GALAXY benchmark instances with 12 to 24 teams, for which we report a mean gap difference to the best known feasible solutions of 1.2% and five new best feasible solutions.

Unconstrained traveling tournament problem is APX-complete

We show that the Unconstrained Traveling Tournament Problem (UTTP) is APX-complete by giving an L-reduction from the following version of (1,2)-TSP: Our analysis gives an inapproximability threshold of 1+((α−1)/200) for UTTP, where α denotes the inapproximability threshold for (1,2)-TSP.

The Linear Distance Traveling Tournament Problem Allows an EPTAS

The Linear Distance Traveling Tournament Problem Allows an EPTAS

Fair travel distances in tournament schedules: A cooperative game theory approach

The most studied problem in sports scheduling, so-called traveling tournament problem (TTP), aims at finding schedules minimizing the total distance traveled by the teams. While minimizing all the traveling between games is efficient from the overall perspective, it overlooks the distribution of the travel among the teams. Consequently, some teams may end up better than others with respect to their individual goals, an imbalance which may affect teams’ often-limited resources or preparedness for the games. This article adopts a cooperative game theory framework to obtain tournament schedules where the distances traveled by the teams are allocated according to fairness criteria. The approach consists of three steps. First, the scheduling problem is reformulated as a transferable utility game. Second, by means of well-established allocation methods, an ideal distance distribution among the teams is determined. Third, we introduce fairness measures to produce a schedule which approximately resembles the ideal distribution. We also discuss the case of not pursuing fairness, but rather a compromise between fairness and minimum total distance. We illustrate the approach by a numerical example in one of the classic TTP data instances.

A polyhedral study for the cubic formulation of the unconstrained traveling tournament problem

We consider the unconstrained traveling tournament problem, a sports timetabling problem that minimizes traveling of teams. Since its introduction about 20 years ago, most research was devoted to modeling and reformulation approaches. In this paper we carry out a polyhedral study for the cubic integer programming formulation by establishing the dimension of the integer hull as well as of faces induced by model inequalities. Moreover, we introduce a new class of inequalities and show that they are facet-defining. Finally, we evaluate the impact of these inequalities on the linear programming bounds.

A Comparative Analysis of Application of Genetic Algorithm and Particle Swarm Optimization in Solving Traveling Tournament Problem (TTP)

Traveling Tournament Problem (TTP) has been a major area of research due to its huge application in developing smooth and healthy match schedules in a tournament. The primary objective of a similar problem is to minimize the travel distance for the participating teams. This would incur better quality of the tournament as the players would experience least travel; hence restore better energy level. Besides, there would be a great benefit to the tournament organizers from the economic point of view as well. A well constructed schedule, comprising of diverse combinations of the home and away matches in a round robin tournament would keep the fans more attracted, resulting in turnouts in a large number in the stadiums and a considerable amount of revenue generated from the match tickets. Hence, an optimal solution to the problem is necessary from all respects; although it becomes progressively harder to identify the optimal solution with increasing number of teams. In this work, we have described how to solve the problem using Genetic algorithm and particle swarm optimization.

Particle Swarm Algorithm for Improved Handling of the Mirrored Traveling Tournament Problem

Particle Swarm Algorithm for Improved Handling of the Mirrored Traveling Tournament Problem

The Linear Distance Traveling Tournament Problem

We introduce a linear distance relaxation of the n-team Traveling Tournament Problem (TTP), a simple yet powerful heuristic that temporarily "assumes"' the n teams are located on a straight line, thereby reducing the n(n–1)/2 pairwise distance parameters to just n–1 variables. The modified problem then becomes easier to analyze, from which we determine an approximate solution for the actual instance on n teams. We present combinatorial techniques to solve the Linear Distance TTP (LD-TTP) for n = 4 and n = 6, without any use of computing, generating the complete set of optimal distances regardless of where the n teams are located. We show that there are only 295 non-isomorphic schedules that can be a solution to the 6-team LD-TTP, and demonstrate that in all previously-solved benchmark TTP instances on 6 teams, the distance-optimal schedule appears in this list of 295, even when the six teams are arranged in a circle or located in three-dimensional space. We then extend the LD-TTP to multiple rounds, and apply our theory to produce a nearly-optimal regular-season schedule for the Nippon Pro Baseball league in Japan. We conclude the paper by generalizing our theory to the n-team LD-TTP, producing a feasible schedule whose total distance is guaranteed to be no worse than 4/3 times the optimal solution.

Hybrid Quantum - Classical AI based approach to solve the Traveling Tournament Problem

Hybrid Quantum - Classical AI based approach to solve the Traveling Tournament Problem

A new approach based on graph matching and evolutionary approach for sport scheduling problem

The Traveling Tournament Problem (TTP) is concerned with finding a double round-robin tournament schedule that minimizes the total distances traveled by the teams. It has attracted significant interest recently since a favorable TTP schedule can result in significant savings for the league. This paper proposes an original evolutionary algorithm for TTP. We first propose a quick and effective constructive algorithm to construct a Double Round Robin Tournament (DRRT) schedule with low travel cost. We then describe an enhanced genetic algorithm with a new crossover operator to improve the travel cost of the generated schedules. A new heuristic for ordering efficiently the scheduled rounds is also proposed. The latter leads to significant enhancement in the quality of the schedules. The overall method is evaluated on publicly available standard benchmarks and compared with other techniques for TTP and UTTP (Unconstrained Traveling Tournament Problem). The computational experiment shows that the proposed approach could build very good solutions comparable to other state-of-the-art approaches or better than the current best solutions on UTTP. Further, our method provides new valuable solutions to some unsolved UTTP instances and outperforms prior methods for all US National League (NL) instances.

A Physics Based Novel Approach for Travelling Tournament Problem: Optics Inspired Optimization

Computational intelligence search and optimization algorithms have been efficiently adopted and used for many types of complex problems. Optics Inspired Optimization (OIO) is one of the most recent physics inspired computational intelligence methods which treats the search space of the problem to be optimized as a wavy mirror in which each peak is assumed to reflect as a convex mirror and each valley to reflect as a concave one. Each candidate solution is treated as an artificial light point that its glittered ray is reflected back by the search space of the problem and the artificial image is formed based on mirror equations adopted from Optics, as a new candidate solution. In this study, OIO for the first time has been designed as solution search strategy for travelling tournament problem which is one of the current sports problems and aids to minimize transportation and total movement of teams. Furthermore, this problem has been firstly solved by League Championship Algorithm and obtained results from both synthetic and real datasets have been compared in this study for the first time. Obtained results show the superiority of OIO which is a novel algorithm and seems to efficiently solve many complex problems.

Scheduling Argentina’s professional basketball leagues: A variation on the Travelling Tournament Problem

Operations research methods are applied to design the season schedules of Argentina’s professional basketball leagues using a format adopted in 2014 by the top two divisions. Following the setup used by the National Basketball Association (NBA) in North America, games are played any day of the week and away games are scheduled in road trips of one to four consecutive games. The main scheduling objective is to reduce the teams’ total travel distance compared to previous season formats through the use of predetermined trips submitted by the teams. The mathematical form of the problem is a variation on the well-known Travelling Tournament Problem. The modelling is divided into two successive stages, the first one defining the sequences in which each team plays the other teams and the second one assigning the days on which each game is played. Both stages use integer programming models that incorporate a series of constraints reflecting criteria requested by the Argentine Basketball Club Association. Implementation of the models has generated average travel distance reductions of more than 30% per away game, with consequential benefits in lower travel costs and less player fatigue.

A Cooperative Local Search Method for Solving the Traveling Tournament Problem

Constrained optimization is the process of optimizing a certain objective function subject to a set of constraints. The goal is not necessarily to find the global optimum. We try to explore the search space more efficiently in order to find a good approximate solution. The obtained solution should verify the hard constraints that are required to be satisfied. In this paper, we propose a cooperative search method that handles optimality and feasibility separately. We take the traveling tournament problem (TTP) as a case study to show the applicability of the proposed idea. TTP is the problem of scheduling a double round-robin tournament that satisfies a set of related constraints and minimizes the total distance traveled by the teams. The proposed method for TTP consists of two main steps. In the first step, we ignore the optimization criterion. We reduce the search only to feasible solutions satisfying the problem's constraints. For this purpose, we use constraints programming model to ensure the feasibility of solutions. In the second step, we propose a stochastic local search method to handle the optimization criterion and find a good approximate solution that verifies the hard constraints. The overall method is evaluated on benchmarks and compared with other well-known techniques for TTP. The computational results are promising and show the effectiveness of the proposed idea for TTP.

Complexity of the Unconstrained Traveling Tournament Problem

The Traveling Tournament problem is a problem of scheduling round robin leagues which minimizes the total travel distance maintaining some constraints on consecutive home and away matches. The problem was proven NP-hard when the upper bound on any consecutive home or away stint is 3. In this paper, we prove that even without the constraints on the consecutive home or away matches, the problem remains NP-Hard.

A new neighborhood structure for round robin scheduling problems

In recent years, several works proposed local search heuristics for round robin sport scheduling problems. It is known that the neighborhood structures used in those works do not fully connect the solution space. The aim of this paper is to present a novel neighborhood structure for single round robin sport scheduling problems. The neighborhood structure is described in graph theory terms and its correctness is proven. We show that the new neighborhood structure increases the connectivity of the solution space when compared to previous neighborhood structures. We evaluate its performance using the Traveling Tournament Problem with Predefined Venues and the Weighted Carry-Over Effects Value Minimization Problem as case studies.

A combined approximation for the traveling tournament problem and the traveling umpire problem

AbstractWe consider the traveling tournament problem (TTP) and the traveling umpire problem (TUP). In TTP, the task is to design a double round-robin schedule, where no two teams play against each other in two consecutive rounds, and the total travel distance is minimized. In TUP, the task is to find an assignment of umpires for a given tournament such that every umpire handles at least one game at every team’s home venue and an umpire neither visits a venue nor sees a team (home or away) too often. The task is to minimize the total distance traveled by the umpires. We present a combined approximation for this problem, when the number of umpires is odd. We therefore first design an approximation algorithm for TTP and then show how to define an umpire assignment for this tournament such that a constant-factor approximation for TUP is guaranteed.

A Mathematical Model for the Traveling Tournament Problem

A Mathematical Model for the Traveling Tournament Problem

Solving scheduling tournament problems using a new version of CLONALG

The travelling tournament problem (TTP) is an important and well-known problem within the collective sports research community. The problem is NP-hard which makes difficult finding quality solution in short amount of time. Recently a new kind of TTP has been proposed ‘The Relaxed Travelling Tournament Problem’. This version of the problem allows teams to have some days off during the tournament. In this paper, we propose an immune algorithm that is able to solve both problem versions. The algorithm uses moves which are based on the team home/away patterns. One of these moves has been specially designed for the relaxed travel tournament instances. We have tested the algorithm using well-known problem benchmarks and the results obtained are very encouraging.

Solving the Traveling Tournament Problem by Packing Three-Vertex Paths

The Traveling Tournament Problem (TTP) is a complex problem in sports scheduling whose solution is a schedule of home and away games meeting specific feasibility requirements, while minimizing the total distance traveled by all the teams. A recently-developed "hybrid" algorithm, combining local search and integer programming, has resulted in best-known solutions for many TTP instances. In this paper, we tackle the TTP from a graph-theoretic perspective, by generating a new "canonical" schedule in which each team's three-game road trips match up with the underlying graph's minimum-weight P_3-packing. By using this new schedule as the initial input for the hybrid algorithm, we develop tournament schedules for five benchmark TTP instances that beat all previously-known solutions.