Traveling Tournament Problem Research Articles

A combined local search and integer programming approach to the traveling tournament problem

The traveling tournament problem is a well-known combinatorial optimization problem with direct applications to sport leagues scheduling, that sparked intensive algorithmic research over the last decade. With the Challenge Traveling Tournament Instances as an established benchmark, the most successful approaches to the problem use meta-heuristics like tabu search or simulated annealing, partially heavily parallelized. Integer programming based methods on the other hand are hardly able to tackle larger benchmark instances. In this work we present a hybrid approach that draws on the power of commercial integer programming solvers as well as the speed of local search heuristics. Our proposed method feeds the solution of one algorithm phase to the other one, until no further improvements can be made. The applicability of this method is demonstrated experimentally on the galaxy instance set, resulting in currently best known solutions for most of the considered instances.

An Approximation Algorithm for the Bipartite Traveling Tournament Problem

The bipartite traveling tournament problem (BTTP) is an NP-complete scheduling problem whose solution is a double round-robin inter-league tournament with minimum total travel distance. The 2n-team BTTP is a variant of the well-known traveling salesman problem (TSP), albeit much harder as it involves the simultaneous coordination of 2n teams playing a sequence of home and away games under fixed constraints, rather than a single entity passing through the locations corresponding to the teams' home venues. As the BTTP requires a distance-optimal schedule linking venues in close proximity, we provide an approximation algorithm for the BTTP based on an approximate solution to the corresponding TSP. We prove that our polynomial-time algorithm generates a 2n-team inter-league tournament schedule whose total distance is at most 1 + 2c/3 + (3 − c)/(3n) times the total distance of the optimal BTTP solution, where c is the approximation factor of the TSP. In practice, the actual approximation factor is far better; we provide a specific example by generating a nearly-optimal inter-league tournament for the 30-team National Basketball Association, with total travel distance just 1.06 times the trivial theoretical lower bound.

Balancing the Traveling Tournament Problem for Weekday and Weekend Games

The Traveling Tournament Problem (TTP) is a well-known NP-complete problem in sports scheduling that was inspired by the application of optimizing schedules for Major League Baseball to reduce total team travel. The techniques and heuristics from the n-team TTP can be extended to optimize the scheduling of other sports leagues, such as the Nippon Professional Baseball (NPB) league in Japan. In this paper, we describe the additional scheduling constraints required by the NPB league, such as the requirement that each team play the same number of weekend home games, weekday home games, weekend road games, and weekday road games. We fully solve this TTP-variant for the case n = 6, and conclude the paper by presenting the official 2013 NPB Central League Schedule, where we helped this Japanese baseball league reduce total team travel by over six thousand kilometres.

Solving the Traveling Tournament Problem with Iterative-Deepening A*

We give an overview of our journal paper on applying iterative-deepening A* to the traveling tournament problem, a combinatorial optimization problem from the sports scheduling literature. This approach involved combining past ideas and creating new ideas to help reduce node expansion. This resulted in a state-of-the-art approach for optimally solving instances of the traveling tournament problem. It was the first approach to solve the classic NL10 and CIRC10 instances, which had not been solved since the problem’s introduction.

A complete search method for the relaxed traveling tournament problem

The traveling tournament problem is a sports scheduling problem that includes two major issues in creating timetables: home/away pattern feasibility and travel distance. In this problem, the schedule must be compact: every team plays in every time slot. However, there are some sports leagues that have both home/away pattern restrictions and distance limits, but do not require a compact schedule. In such schedules, one or more teams can have a bye in any time slot. This leads us to a variant of the problem: the relaxed traveling tournament problem. We present a complete search method to solve this problem based on branch-and-bound, metaheuristics and dynamic programming.

Generating Approximate Solutions to the TTP using a Linear Distance Relaxation

In some domestic professional sports leagues, the home stadiums are located in cities connected by a common train line running in one direction. For these instances, we can incorporate this geographical information to determine optimal or nearly-optimal solutions to the n-team Traveling Tournament Problem (TTP), an NP-hard sports scheduling problem whose solution is a double round-robin tournament schedule that minimizes the sum total of distances traveled by all n teams. We introduce the Linear Distance Traveling Tournament Problem (LD-TTP), and solve it for n=4 and n=6, generating the complete set of possible solutions through elementary combinatorial techniques. For larger n, we propose a novel "expander construction" that generates an approximate solution to the LD-TTP. For n congruent to 4 modulo 6, we show that our expander construction produces a feasible double round-robin tournament schedule whose total distance is guaranteed to be no worse than 4/3 times the optimal solution, regardless of where the n teams are located. This 4/3-approximation for the LD-TTP is stronger than the currently best-known ratio of 5/3 + epsilon for the general TTP. We conclude the paper by applying this linear distance relaxation to general (non-linear) n-team TTP instances, where we develop fast approximate solutions by simply "assuming" the n teams lie on a straight line and solving the modified problem. We show that this technique surprisingly generates the distance-optimal tournament on all benchmark sets on 6 teams, as well as close-to-optimal schedules for larger n, even when the teams are located around a circle or positioned in three-dimensional space.

New models for the Mirrored Traveling Tournament Problem

The Mirrored Traveling Tournament Problem (mTTP) is a challenging combinatorial optimization problem which consists in generating a timetable for sports tournaments with two half series, what is equivalent to a double round-robin timetable problem. The distance traveled by the teams should be minimized in the final timetable, and a new objective is to minimize the longest distance traveled, named MinMaxTTP. It is proposed an integer programming formulation to the mTTP and two models with dynamic constraints to its solution. Both models are based on the detection of independent sets on conflict graphs, whose use has not been reported in the literature about the problem. Real data benchmarks from a baseball tournament are used in the experiments carried out.

Traveling baseball players’ problem in Korea

We study the so-called the traveling tournament problem (TTP), to find an optimal tournament schedule. Differently from the original TTP, in which the total travel distance of all the participants is the objective function to minimize, we instead seek to maximize the fairness of the round robin tournament schedule of the Korean Baseball League. The standard deviation of the travel distances of teams is defined as the energy function, and the Metropolis Monte-Carlo method combined with the simulated annealing technique is applied to find the ground state configuration. The resulting tournament schedule is found to satisfy all the constraint rules set by the Korean Baseball Organization, but with drastically increased fairness in traveling distances.

A 2.75-approximation algorithm for the unconstrained traveling tournament problem

A 2.75-approximation algorithm is proposed for the unconstrained traveling tournament problem, which is a variant of the traveling tournament problem. For the unconstrained traveling tournament problem, this is the first proposal of an approximation algorithm with a constant approximation ratio. In addition, the proposed algorithm yields a solution that meets both the no-repeater and mirrored constraints. Computational experiments show that the algorithm generates solutions of good quality.

An Application of the Traveling Tournament Problem: The Argentine Volleyball League

This article describes an optimization process used to schedule the First Division of Argentina's professional volleyball league. The teams in the league are grouped into couples, and matches are held on Thursdays and Saturdays. In each pair of consecutive Thursday–Saturday matches, the two teams in each couple play against two teams from another couple. Minimization of travel distances is critical because the teams' home locations are scattered throughout the country and teams do not return to their home sites between consecutive away matches, making this problem a variation of the well-known traveling tournament problem. The coupled format gives rise to two key decisions: (1) how to couple the teams, and (2) how to schedule the matches. We apply integer programming techniques and a tabu search heuristic to solve these questions. The league successfully used the resulting schedules in its 2007–2008, 2008–2009, 2009–2010, and 2010–2011 seasons, reducing the total travel distance while meeting all of the teams' requirements. This is the first application of the traveling tournament problem to a real-world sports league reported in the optimization literature.

Approximation algorithms for TTP(2)

We consider the traveling tournament problem, which is a well-known benchmark problem in tournament timetabling. It consists of designing a schedule for a sports league of n teams such that the total traveling costs of the teams are minimized. The most important variant of the traveling tournament problem imposes restrictions on the number of consecutive home games or away games a team may have. We consider the case where at most two consecutive home games or away games are allowed. We show that the well-known independent lower bound for this case cannot be reached and present two approximation algorithms for the problem. The first algorithm has an approximation ratio of \({3/2+\frac{6}{n-4}}\) in the case that n/2 is odd, and of \({3/2+\frac{5}{n-1}}\) in the case that n/2 is even. Furthermore, we show that this algorithm is applicable to real world problems as it yields close to optimal tournaments for many standard benchmark instances. The second algorithm we propose is only suitable for the case that n/2 is even and n ≥ 12, and achieves an approximation ratio of 1 + 16/n in this case, which makes it the first \({1+\mathcal{O}(1/n)}\) -approximation for the problem.

AISTTP: AN ARTIFICIAL IMMUNE ALGORITHM TO SOLVE TRAVELING TOURNAMENT PROBLEMS

The traveling tournament problem is a well-known and important problem within the collective sports communities. This is a NP-hard problem which makes difficult finding quality solutions in a short amount of time. In this paper, we include new components to the Clonal Selection Algorithm to tackle the Traveling Tournament Problem. We introduce a new selection procedure focused on improving diversity of the selected cells set. The algorithm also uses a new move which is based on the team home/away patterns. We have tested the algorithm with well-known benchmarks of the problem, and the results obtained are very encouraging. A statistical analysis is also reported. The results obtained show a new application of an artificial immune algorithm based on the CLONALG framework.

A 5.875-approximation for the Traveling Tournament Problem

In this paper we propose an approximation for the Traveling Tournament Problem which is the problem of designing a schedule for a sports league consisting of a set of teams T such that the total traveling costs of the teams are minimized. It is not allowed for any team to have more than k home-games or k away-games in a row. We propose an algorithm which approximates the optimal solution by a factor of 2+2k/n+k/(n−1)+3/n+3/(2⋅k) which is not more than 5.875 for any choice of k≥4 and n≥6. This is the first constant factor approximation for k>3. We furthermore show that this algorithm is also applicable to real-world problems as it produces solutions of high quality in a very short amount of time. It was able to find solutions for a number of well known benchmark instances which are even better than the previously known ones.

A hybrid PSO-SA algorithm for the travelling tournament problem

Sports scheduling has become an important area of applied operations research in recent years, since satisfying fans and teams’ requests and revenues of a sports league and TV networks may be affected by quality of the league schedule. While this type of scheduling problem can be solved by mathematical methods and exact solutions are accessible, it computationally leads to hard problems. The travelling tournament problem (TTP) is defined as minimising total travelling distance for all teams in a league. In this study, a new mathematical model for the TTP with the no-repeater constraint is presented. In addition, a very fast hybrid metaheuristic algorithm is proposed, which combines particle swarm optimisation (PSO) and simulated annealing (SA). Our computational experiments on standard instances show that the hybrid approach results in comparable to or even better than current best known solutions, specifically in computational time.

An Efficient Simulated Annealing Approach to the Travelling Tournament Problem

Scheduling sports leagues has drawn significant attention to itself in recent years, as it involves considerable revenue as well as challenging combinatorial optimization problems. A particular class of these problems is the Traveling Tournament Problem (TTP) which focuses on minimizing the total traveling distance for teams. In this paper, an efficient simulated annealing approach is presented for TTP which applies two simultaneous and disparate models for the problem in order to search the solutions space more effectively. Also, a computationally efficient modified greedy scheme is proposed for constructing a favorable initial solution for the simulated annealing algorithm. Our computational experiments, carried out on standard instances, demonstrate that this approach competes with previous offered methods in quality of found solutions and their computational time.

An Improved Approximation Algorithm for the Traveling Tournament Problem

This paper describes the traveling tournament problem, a well-known benchmark problem in the field of tournament timetabling. We propose an approximation algorithm for the traveling tournament problem with the constraints such that both the number of consecutive away games and that of consecutive home games are at most k. When k≤5, the approximation ratio of the proposed algorithm is bounded by (2k−1)/k+O(k/n) where n denotes the number of teams; when k>5, the ratio is bounded by (5k−7)/(2k)+O(k/n). For k=3, the most investigated case of the traveling tournament problem to date, the approximation ratio of the proposed algorithm is 5/3+O(1/n); this is better than the previous approximation algorithm proposed for k=3, whose approximation ratio is 2+O(1/n).

A distributed and hierarchical strategy for autonomic grid-enabled cooperative metaheuristics with applications

The adoption of the same cluster-based programming strategies for grid applications, although requiring minimal effort from a programmer's point of view, does not always take advantage of the available computational resources to their fullest extent. This paper investigates the impact of a distributed and hierarchical autonomic strategy on the performance of parallel metaheuristics to solve hard combinatorial optimization problems on grids. Two problems, the mirrored traveling tournament problem and the bounded diameter minimum spanning tree problem, for which high quality sequential heuristics based on the paradigms of the GRASP and Iterated Local Search metaheuristics already exist, are employed as case-studies. The computational results obtained on a grid by the novel autonomic strategy show that outstanding performance improvements over the traditional master–worker parallelization approach can be achieved.

A multi-round generalization of the traveling tournament problem and its application to Japanese baseball

In a double round-robin tournament involving n teams, every team plays 2( n − 1) games, with one home game and one away game against each of the other n − 1 teams. Given a symmetric n by n matrix representing the distances between each pair of home cities, the traveling tournament problem (TTP) seeks to construct an optimal schedule that minimizes the sum total of distances traveled by the n teams as they move from city to city, subject to several natural constraints to ensure balance and fairness. In the TTP, the number of rounds is set at r = 2. In this paper, we generalize the TTP to multiple rounds ( r = 2 k, for any k ⩾ 1) and present an algorithm that converts the problem to finding the shortest path in a directed graph, enabling us to apply Dijkstra’s Algorithm to generate the optimal multi-round schedule. We apply our shortest-path algorithm to optimize the league schedules for Nippon Professional Baseball (NPB) in Japan, where two leagues of n = 6 teams play 40 sets of three intra-league games over r = 8 rounds. Our optimal schedules for the Pacific and Central Leagues achieve a 25% reduction in total traveling distance compared to the 2010 NPB schedule, implying the potential for considerable savings in terms of time, money, and greenhouse gas emissions.

Solving the traveling tournament problem with iterative-deepening A∗

This work presents an iterative-deepening A∗ (IDA∗) based approach to the traveling tournament problem (TTP). The TTP is a combinatorial optimization problem which abstracts the Major League Baseball schedule. IDA∗ is able to find optimal solutions to this problem, with performance improvements coming from the incorporation of various past concepts including disjoint pattern databases, symmetry breaking, and parallelization along with new ideas of subtree skipping, forced deepening, and elite paths to help to reduce the search space. The results of this work show that an IDA∗ based approach can find known optimal solutions to most TTP instances much faster than past approaches. More importantly, it has been able to optimally solve two larger instances that have been unsolved since the problem’s introduction in 2001. In addition, a new problem set called GALAXY is introduced, using a 3D space to create a challenging problem set.

The Multi-Round Balanced Traveling Tournament Problem

Given an n-team sports league, the Traveling Tournament Problem (TTP) seeks to determine an optimal double round-robin schedule minimizing the sum total of distances traveled by the n teams as they move from city to city. In the TTP, the number of "rounds" is fixed at r = 2. In this paper, we propose the Multi-Round Balanced Traveling Tournament Problem (mb-TTP), inspired by the actual league structure of Japanese professional baseball, where n = 6 teams play 120 intra-league games over r = 8 rounds, subject to various constraints that ensure competitive balance. These additional balancing constraints enable us to reformulate the 2k-round mb-TTP as a shortest path problem on a directed graph, for all k >= 1. We apply our theoretical algorithm to the 6-team Nippon (Japanese) Professional Baseball Central League, creating a distance-optimal schedule with 57836 kilometres of total travel, a 26.8% reduction compared to the 79067 kilometres traveled by these six teams during the 2010 regular season.