Tight bounds for break minimization

We consider round-robin sports tournaments with n teams and n − 1 rounds. We construct an infinite family of opponent schedules for which every home-away assignment induces at least 1/4 n(n−2) breaks. This construction establishes a matching lower bound for a corresponding upper bound from the literature

Sports tournaments, home–away assignments, and the break minimization problem

We consider the break minimization problem for fixing home–away assignments in round-robin sports tournaments. First, we show that, for an opponent schedule with $n$ teams and $n−1$ rounds, there always exists a home–away assignment with at most ${1\over 4}n(n-2)$ breaks. Secondly, for infinitely many $n$, we construct opponent schedules for which at least ${1\over 6}n(n-1)$ breaks are necessary. Finally, we prove that break minimization for $n$ teams and a partial opponent schedule with $r$ rounds is an NP-hard problem for $r\ge 3$. This is in strong contrast to the case of $r=2$ rounds, which can be scheduled (in polynomial time) without any breaks

Calendarização dos jogos da primeira divisão distrital de futebol de Santarém

Trabalho de Projeto de Mestrado, Matemática Aplicada à Economia e Gestão, 2022, Universidade de Lisboa, Faculdade de CiênciasEste trabalho de projeto tem como principal objetivo o desenvolvimento de modelos matemáticos baseados em Programação Linear Inteira Binária para obter a calendarização dos jogos relativos à primeira divisão distrital de futebol de Santarém. Esta competição, na época 2021/2022, é composta por 16 equipas. As equipas jogam todas entre si duas vezes, uma em casa e outra fora. No fim da época a equipa que fica em primeiro lugar é a equipa campeã e apura-se para o campeonato nacional de Portugal que se realiza na época seguinte. As equipas que ficam nos dois últimos lugares descem para a segunda divisão distrital enquanto as equipas que ficam nos três primeiros lugares da segunda divisão distrital vão na época seguinte para a primeira divisão distrital. Atualmente o calendário dos jogos é feito com base num sorteio das equipas usando uma tabela fornecida pela Federação Portuguesa de Futebol. Apresenta-se também um modelo para a fase de Apuramento de Campeão no caso da primeira Divisão Nacional Feminina de futebol. Esta fase é composta pelas 4 primeiras equipas da série Norte e pelas 4 primeiras equipas da série Sul. Numa primeira fase serão descritas, em termos gerais, as caraterísticas do sistema de ligas de futebol em Portugal, tendo em especial consideração a primeira divisão distrital de Santarém, tais como o número de equipas, as regras que presidem ao agendamento dos jogos entre as equipas e o modo como é feito o calendário dos jogos. Faz-se uma revisão bibliográfica sobre a aplicação de técnicas de Investigação Operacional no desporto e nomeadamente no futebol. Adaptando algumas das restrições que regem a Primeira Liga de futebol em Portugal e outros campeonatos de futebol em outros países, à primeira divisão distrital de Santarém desenvolveu-se um modelo que foi implementado em linguagem Mosel. O modelo usado para o Apuramento de Campeão no caso da primeira Divisão Nacional Feminina de futebol é semelhante ao usado para a primeira divisão distrital de Santarém adicionando neste caso uma restrição que evite viagens longas e consecutivas. Nos modelos o objetivo considerado foi a minimização do número de jogos consecutivos em casa e fora para a mesma equipa. Apresentam-se os resultados obtidos e comparam-se com os atualmente usados na primeira divisão distrital de futebol de Santarém.This project work has as main objective the development of mathematical models to obtain the schedule of the games related to the first district division of football (soccer) of Santarém. This competition, in the season 2021/2022 , has 16 teams. All the teams play with each other twice, once at home and once away. At the end of the season the team that comes first is the champion and is qualified for the national championship of Portugal that takes place in the following season. Teams in the last two places fall to the second district division while teams in the top three of the second district division move the following season to the district first division. Currently the schedule of the games is made based on a draw of the teams using a table provided by the Portuguese Football Federation. It is also presented a model for the champion qualifying round of the first Women's National Division of football. This round consists of the first 4 teams of the North series and the first 4 teams of the South series. In a first phase, the characteristics of the football league system in Portugal will be described, considering the first district division of Santarém, such as the number of teams, the rules governing the scheduling of matches between the teams and how the schedule of the matches is made. A literature review is made on the application of Operational Research techniques in sport and in particular in football. A model was developed, for the first district division of Santarem, by adapting some of the restrictions of the First Football League in Portugal and other football championships in other countries. It was implemented in Mosel language. The model used for the Champion's round in the case of the first Women's National Football Division is similar to the one used for the first district division of Santarém with an additional constraint that avoids long consecutive trips. In the models the objective considered was to minimize the number of consecutive games at home and away for the same team. The results obtained are presented and compared with those currently used in the first district football division of Santarém

Time Relaxed Round Robin Tournament and the NBA Scheduling Problem

This dissertation study was inspired by the National Basketball Association regular reason scheduling problem. NBA uses the time-relaxed round robin tournament format, which has drawn less research attention compared to the other scheduling formats. Besides NBA, the National Hockey League and many amateur leagues use the time-relaxed round robin tournament as well. This dissertation study is the first ever to examine the properties of general time-relaxed round robin tournaments. Single round, double round and multiple round time-relaxed round robin tournaments are defined. The integer programming and constraint programming models for those tournaments scheduling are developed and presented. Because of the complexity of this problem, several decomposition methods are presented as well. Traveling distance is an important factor in the tournament scheduling. Traveling tournament problem defined in the time constrained conditions has been well studied. This dissertation defines the novel problem of time-relaxed traveling tournament problem. Three algorithms has been developed and compared to address this problem. In addition, this dissertation study presents all major constraints for the NBA regular season scheduling. These constraints are grouped into three categories: structural, external and fairness. Both integer programming and constraint programming are used to model these constraints and the computation studies are presente

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

Mathematical Modeling and Optimization Approaches for Scheduling the Regular-Season Games of the National Hockey League

RÉSUMÉ : La Ligue nationale de hockey (LNH) est une association sportive professionnelle de hockey sur glace regroupant des équipes du Canada et des États-Unis. Chaque année, la LNH dois compter sur un calendrier de haute qualité concernant des questions économiques et d'équité pour les 1230 matchs de sa saison régulière. Dans cette thèse, nous proposons le premier modèle de programmation linéaire en nombres entiers (PLNE) pour le problème de la planification de ces matchs. Basé sur la littérature scientifique en planification des horaires sportifs, et aussi sur un raisonnement pratique, nous identifions et soulignons des exigences essentielles et des préférences qui doivent être satisfaites par des calendriers de haute qualité pour la LNH. La construction de tels calendriers, tout comme la planification des horaires sportifs en général, s'avère une tâche très difficile qui doit prendre en compte des intérêts concurrents et, dans plusieurs cas, subjectifs. En particulier, les expérimentations numériques que nous décrivons dans cette étude fournissent des évidences solides suggérant qu'une approche basée sur la PLNE est actuellement incapable de résoudre des instances de taille réaliste pour le problème. Pour surmonter cet inconvénient, nous proposons ensuite un algorithme de recherche adaptative à voisinage large (ALNS) qui intègre à la fois des nouvelles stratégies et des heuristiques spécialisées provenant de la littérature scientifique. Afin de tester cette approche, nous générons plusieurs instances du problème. Toutes les instances sont basées sur les calendriers officiels de la LNH et, en particulier, utilisent les dates de matchs à domicile de chaque équipe comme des dates de disponibilité de son aréna. Dans les situations les plus difficiles, la disponibilité des arénas est rare ou est à son minimum. Dans tous les cas, en ce qui concerne les indicateurs de qualité soulevés, l'algorithme ALNS a été capable de générer des calendriers clairement meilleur que leur correspondants adoptés par la LNH. Les résultats obtenus suggèrent que notre approche pourrait certainement permettre aux gestionnaires de la LNH de trouver des calendriers de meilleur qualité par rapport à une variété de nouvelles préférences.----------ABSTRACT : The National Hockey League (NHL) is a major professional ice hockey league composed of 30 teams located throughout the United States and Canada. Every year, the NHL must rely on a high-quality schedule regarding both economic and fairness issues for the 1230 games of its regular season. In this thesis, we propose the first integer linear programming (IP) model for the problem of scheduling those games. Based both on the pertinent sports scheduling literature and on practical reasoning, we identify and point out essential requirements and preferences that should be satisfied by good NHL schedules. Finding such schedules, as many other sports scheduling problems, is a very difficult task that involves several stakeholders with many conflicting, and often subjective, interests. In fact, computational experiments that we describe in this study, provide compelling evidence that an IP approach is currently unable to solve instances of realistic size for the problem. To overcome such drawback, we propose then an Adaptive Large Neighborhood Search (ALNS) algorithm that integrates both novel strategies and specialized heuristics from the scientific literature. To test the approach, we generate instances based on past NHL schedules and on a given number of arena-available dates that are suitable for the home games of each team. In the most challenging instances, availability of arenas is scarce or at its minimum. In all cases, regarding the identified concerns, the ALNS algorithm was able to generate much better schedules than those implemented by the NHL. Results obtained suggest that our approach could certainly identify unnecessary weakness in NHL schedules, makes the NHL managers aware of better schedules with respect to different requirements, and even lead them to consider other desired features they might not have previously taken into account