Celso C. Ribeiro

• Professor (Full) at Fluminense Federal University

Polarization arises when the underlying network connecting the members of a community or society becomes characterized by highly connected groups with weak intergroup connectivity. The increasing polarization, the strengthening of echo chambers, and the isolation caused by information filters in social networks are increasingly attracting the atten...

Let \(G=(V, E)\) be a graph with vertex set V and edge set E, and consider \(\gamma \in [0,1)\) to be a real constant. A \(\gamma \)-clique (or quasi-clique) is a subset \(V'\subseteq V\) inducing a subgraph of G with edge density at least \(\gamma \). In this paper, we tackle the minimum quasi-clique partitioning problem (MQCPP), which consists of...

The scheduling of a sports tournament involves some fundamental issues, such as the type of league, the tournament characteristics, the timing of the scheduling process, and the goals and constraints of each specific problem.

Combinatorial optimization, which includes sports scheduling problems as one of its areas of application, consists in finding optimal solutions to problems defined over a discrete set of feasible solutions. This chapter overviews approximate optimization methods, from greedy and semi-greedy construction algorithms to local search and from local sea...

Mathematical models based on theoretical graph concepts help to formulate and solve fundamental scheduling problems. In this chapter, we consider some elementary sports scheduling problems and concentrate on the combinatorial structures involved in their models and solution approaches.

This chapter reports in detail three applications of sports tournament scheduling. They involve the use of approximate and exact methods described in the previous chapters: the minimization of carry-over effects; the formulation, implementation, and practical use of the optimization software developed for scheduling the first division of the annual...

Integer programming has been widely used to solve round-robin tournament scheduling problems, either as standalone formulations or embedded in decomposition strategies. Methods applied to sports scheduling problems include branch-and-bound, branch-and-cut, Benders decomposition, and column generation. They are used to solve scheduling problems in r...

Brazilian electronic invoices (Nota Fiscal eletrônica, NFe) are digital documents that register the purchase and sale operations of goods and services. In this work, we developed and used clustering methods and outlier detection algorithms to investigate the presence of two main risk patterns in the electronic invoices data of public entities, such...

Many sports leagues organize their competitions as round‐robin tournaments. This tournament design has a rich mathematical structure that has been studied in the literature over the years. We review some of the main properties and fundamental scheduling methods of round‐robin tournaments. Special attention is given to extra‐mural tournaments in whi...

Given a graph and a subset of its vertices , the subgraph induced by in G is that with vertex set and edge set formed by all the edges in E linking two vertices in . Mixed integer programming (MIP) approaches are among the most successful techniques for solving induced graph optimization problems, that is, those related to obtaining maximum or mini...

Polarization arises when the underlying network connecting the members of a community or society becomes characterized by highly connected groups with weak inter-group connectivity. The increasing polarization, the strengthening of echo chambers, and the isolation caused by information filters in social networks are increasingly attracting the atte...

Given a simple graph G=(V,E) and a real constant γ∈(0,1], a γ-clique (or γ-quasi-clique) is a subset V′⊆V inducing a subgraph with edge density at least γ. The minimum quasi-clique (or γ-clique) partitioning problem (MQCPP) consists in partitioning the vertices of the graph in γ-cliques to minimize the number of elements in the partition. In this p...

Data mining in networks studies properties of complex systems where the entities are represented by vertices and the relationships between them are represented by edges. The analysis of networks has relevant applications, for example in research related to the spread of pandemics such as Covid-19, in technological collaborations, in carbon capture...

Given an undirected graph G=(V,E), the longest induced path problem (LIPP) consists of obtaining a maximum cardinality subset W⊆V such that W induces a simple path in G. In this paper, we propose two new formulations with an exponential number of constraints for the problem, together with effective branch-and-cut procedures for its solution. While...

Given a graph with a weight associated with each vertex , the maximum weighted induced forest problem (MWIF) consists of encountering a maximum weighted subset of the vertices such that induces a forest. This NP‐hard problem is known to be equivalent to the minimum weighted feedback vertex set problem, which has applicability in a variety of domain...

Real‐world networks are often extremely polarized because the communication between different groups of vertices can be weak and, most of the time, only vertices within the same group or sharing the same beliefs communicate to each other. In this work, we introduce the minimum‐cardinality edge addition problem (MinCEAP) as a strategy for reducing p...

Given an undirected graph $G=(V,E)$, the longest induced path problem (LIPP) consists of obtaining a maximum cardinality subset $W\subseteq V$ such that $W$ induces a simple path in $G$. In this paper, we propose two new formulations with an exponential number of constraints for the problem, together with effective branch-and-cut procedures for its...

In a biased random-key genetic algorithm, a deterministic decoder algorithm takes a solution represented by a vector of real numbers (random-keys) and builds a feasible solution for the problem at hand. Selection is said to be biased not only because one parent is always a high-quality solution, but also because it has a higher probability of passi...

Given a graph $G=(V,E)$ with a weight $w_v$ associated with each vertex $v\in V$, the maximum weighted induced forest problem consists of encountering a maximum weighted subset $V'\subseteq V$ of the vertices such that $V'$ induces a forest. This NP-hard problem is known to be equivalent to the minimum weighted feedback vertex set problem, which ha...

The longest induced path problem consists in finding a maximum subset of vertices of a graph such that it induces a simple path. We propose a new exact enumerative algorithm that solves problems with up to 138 vertices and 493 edges and a heuristic for larger problems. Detailed computational experiments compare the results obtained by the new algor...

Over the past few years, investigators in Brazil have been uncovering numerous corruption and money laundering schemes at all levels of government and in the country's largest corporations. It is estimated that between 2% and 5% of the global GDP is lost annually because of such practices, not only directly impacting public services and private sec...

The problem of routing and wavelength assignment in optical networks consists in minimizing the number of wavelengths that are needed to route a set of demands, such that demands routed using lightpaths that share common links are assigned to different wavelengths. We present a biased random-key genetic algorithm for approximately solving the probl...

Given a weighted graph G=(V,E), the minimum weighted feedback vertex set problem consists in obtaining a minimum weight subset F⊆V of the vertex set whose removal makes the graph acyclic. Differently from other approaches in the literature, in this work we tackle this problem via the maximum weighted induced forest problem. First, we propose two ne...

Real-world networks are often extremely polarized, because the communication between groups of vertices can be weak and, most of the time, only vertices in the same groups or sharing the same beliefs communicate to each other. We formulate the Minimum-Cardinality Balanced Edge Addition Problem as a strategy for reducing polarization in real-world n...

Sistemas de recomendação na internet nos levam a crer ainda mais naquilo em que já acreditamos

Given a graph $G=(V,E)$ and a threshold $\gamma \in (0,1]$, the maximum cardinality quasi-clique problem consists in finding a maximum cardinality subset $C^*$ of the vertices in $V$ such that the density of the graph induced in $G$ by $C^*$ is greater than or equal to the threshold $\gamma$. This problem has a number of applications in data mining...

Given a graph with its vertex set partitioned into a set of groups, nonnegative costs associated to its edges, and nonnegative prizes associated to its vertices, the prize‐collecting generalized minimum spanning tree problem consists in finding a subtree of this graph that spans exactly one vertex of each group and minimizes the sum of the costs of...

The graph coloring problem consists in coloring the vertices of a graph $G=(V,E)$ with a minimum number of colors, such as that any two adjacent vertices receive different colors. The minimum cost chromatic partition problem (MCCPP) is an extension of the graph coloring problem in which there are costs associated with the colors and one seeks a ver...

Given a graph and a threshold , the maximum quasi‐clique problem amounts to finding a maximum cardinality subset of the vertices in V such that the edge density of the graph induced in G by is greater than or equal to the threshold. This problem is NP‐hard and has a number of applications in data mining, for example, in social networks or phone cal...

In this paper, we investigate a multicast routing problem with quality of service constraints on ad hoc vehicular networks. An integer programming formulation for the problem is proposed that forms the basis of a relax‐and‐fix heuristic designed with the goal of producing feasible solutions of good quality. In addition, preprocessing procedures rel...

This paper proposes and explores a new quantitative characterization of the polarization phenomenon in networks. New tools for evaluating the polarization of a network are presented. We first characterize the homophily of each node individually. We depart from the definition of a new measure of the homophily of the nodes of a network and we conside...

Given a graph G=(V,E) and a threshold γ ∈ (0, 1], the maximum cardinality quasi-clique problem consists in finding a maximum cardinality subset C* of the vertices in V such that the density of the graph induced in G by C* is greater than or equal to the threshold γ. This problem is NP-hard, since it admits the maximum clique problem as a special ca...

Time-to-target plots (ttt-plots) are a useful tool to characterize, evaluate, and compare the behavior of randomized heuristics for a given problem instance of some combinatorial optimization problem. Multiple time-to-target plots (mttt-plots) are their natural extension to sets of multiple instances. We show how to build an mttt-plot from the indi...

The traveling salesman problem (TSP) is one of the most studied problems in combinatorial optimization. Given a set of nodes and the distances between them, it consists in finding the shortest route that visits each node exactly once and returns to the first. Nevertheless, more flexible and applicable formulations of this problem exist and can be c...

A divisible load is an amount W of computational work that can be arbitrarily divided into independent chunks of load. In many divisible load applications, the load can be parallelized in a master–worker fashion, where the master distributes the load among a set P of worker processors to be processed in parallel. The master can only send load to on...

Tumor growth is a complex process that requires mathematical modeling approaches for studying real life cancer behavior. The use of Cellular Automata (CA) to represent tumor growth in its avascular stage is explained in this work and Stochastic CA describing tumor growth is obtained, based on a differential equations system in the range of continuu...

Finding an implicit polynomial that fits a set of observations X is the goal of many researches in recent years. However, most existing algorithms assume the knowledge of the degree of the implicit polynomial that best represents the points. This paper presents two main contributions. First, a new distance measure between X and the implicit polynom...

Path-relinking is a search intensification strategy. As a major enhancement to heuristic search methods for solving combinatorial optimization problems, its hybridization with other metaheuristics has led to significant improvements in both solution quality and running times of hybrid heuristics. In this chapter, we review the fundamentals of path-...

This chapter presents the basic structure of a greedy randomized adaptive search procedure (or, more simply, GRASP). We first introduce random and semi-greedy multistart procedures and show how solutions produced by both procedures differ. The hybridization of a semi-greedy procedure with a local search method constitutes a GRASP heuristic. The cha...

In this final chapter of the book, we consider four case studies to illustrate the application and implementation of GRASP heuristics. These heuristics are for 2-path network design, graph planarization, unsplittable multicommodity flows, and maximum cut in a graph. The key point here is not to show numerical results or compare these GRASP heuristi...

In Chapter 3, we considered cardinality-based and quality-based adaptive greedy algorithms as a generalization of greedy algorithms. Next, we presented semi-greedy algorithms that are obtained by randomizing adaptive greedy algorithms and constitute the main foundation for developing the construction phase of GRASP heuristics. In this chapter, we c...

This chapter addresses the construction of feasible solutions. We begin by considering greedy algorithms and show their relationship with matroids. We then consider adaptive greedy algorithms, a generalization of greedy algorithms. Next, we present semi-greedy algorithms, obtained by randomizing adaptive greedy algorithms. The chapter concludes wit...

Local search methods start from any feasible solution and visit other (feasible or infeasible) solutions, until a feasible solution that cannot be further improved is found. Local improvements are evaluated with respect to neighboring solutions that can be obtained by slight modifications applied to a solution being visited. We introduce in this ch...

Parallel computers and parallel algorithms have increasingly found their way into metaheuristics. Most parallel implementations of GRASP found in the literature consist in either partitioning the search space or the GRASP iterations and assigning each partition to a processor. GRASP is applied to each partition in parallel. These implementations ca...

Runtime distributions or time-to-target plots display on the ordinate axis the probability that an algorithm will find a solution at least as good as a given target value within a given running time, shown on the abscissa axis. They provide a very useful tool to characterize the running times of stochastic algorithms for combinatorial optimization...

This chapter introduces combinatorial optimization problems and their computational complexity. We first formulate some fundamental problems already introduced in the previous chapter and then consider basic concepts of the theory of computational complexity, with special emphasis on decision problems, polynomial-time algorithms, and NP-complete pr...

Path-relinking is a major enhancement to GRASP, adding a long-term memory mechanism to GRASP heuristics. GRASP with path-relinking implements long-term memory using an elite set of diverse high-quality solutions found during the search. In its most basic implementation, at each iteration the path-relinking operator is applied between the solution f...

Continuous GRASP, or C-GRASP, extends GRASP to the domain of continuous box-constrained global optimization. The algorithm searches the solution space over a dynamic grid. Each iteration of C-GRASP consists of two phases. In the construction (or diversification) phase, a greedy randomized solution is constructed. In the local search (or intensifica...

RESUMO Dado um conjunto de requisições de caminhos óticos, o problema de roteamento e atri-buição de comprimentos de onda em redes óticas WDM consiste em rotear um subconjunto destas requisições e atribuir um comprimento de onda para cada um deles, de modo que dois caminhos óticos cujas rotas compartilham alguma fibra ótica usem comprimentos de ond...

We consider the multi-item uncapacitated lot-sizing problem with inventory bounds, in which a production plan for multiple items has to be determined considering that they share a storage capacity. We present (a) a shortest path formulation and (b) a formulation based on the a priori addition of valid inequalities, which are compared with a facilit...

Given a set of lightpath requests, the problem of routing and wavelength (RWA) assignment in wavelength division multiplexing (WDM) optical networks consists in routing a subset of these requests and assigning a wavelength to each of them, such that two lightpaths that share a common link are assigned to different wavelengths. There are many varian...

Feature subset selection (FSS) is an important preprocessing step for the classification task, especially in the case of datasets with high dimensionality, i.e., thousands of potentially predictive attributes. There is an extensive literature on methods for performing FSS, but most of them do not apply to datasets with high dimensionality because o...

A biased random-key genetic algorithm (BRKGA) is a general search procedure for finding optimal or near-optimal solutions to hard combinatorial optimization problems. It is derived from the random-key genetic algorithm of Bean (1994), differing in the way solutions are combined to produce offspring. BRKGAs have three key features that specialize ge...

Energy consumption is one of the most critical issues in wireless ad hoc and sensor networks. A considerable amount of energy is dissipated due to radio transmission power and interference (message collisions). A typical topology control technique aims at reducing energy consumption while ensuring specific desired properties to the established wire...

Structural approaches for pattern recognition frequently make use of graphs to represent objects. The concept of object similarity is of great importance in pattern recognition. The graph edit distance is often used to measure the similarity between two graphs. It basically consists in the amount of distortion needed to transform one graph into the...

In this work, we consider some basic sports scheduling problems and introduce the notions of graph theory which are needed to build adequate models. We show, in particular, how edge coloring can be used to construct schedules for sports leagues. Due to the emergence of various practical requirements, one cannot be restricted to classical schedules...

This is the first book to cover GRASP (Greedy Randomized Adaptive Search Procedures), a metaheuristic that has enjoyed wide success in practice with a broad range of applications to real-world combinatorial optimization problems. The state-of-the-art coverage and carefully crafted pedagogical style lends this book highly accessible as an introducto...

In this first chapter, we introduce general optimization problems and the class of combinatorial optimization problems. As a motivation, we present a number of fundamental combinatorial optimization problems that will be revisited along the next chapters of this book. We also contrast exact and approximate solution methods and trace a brief history...

Feature subset selection is an important preprocessing step for the classification task, specially in the case of datasets with high dimensionality, i.e., thousands of potentially predictive attributes. There is an extensive literature on methods for performing FSS, but most of them do not apply to datasets with high dimensionality because of the p...

A divisible load is an amount W ≥ 0 of computational work that can be arbitrarily divided into chunks and distributed among a set P of worker processors to be processed in parallel. The Divisible Load Scheduling Problem consists in (a) selecting a subset of active workers, (b) defining the order in which the chunks will be transmitted to each of th...

Given a graph G = (V, E), the maximum cardinality quasi-clique problem amounts to finding a maximum cardinality subset C * of the nodes in V such that the density of the graph induced in G by C * is greater than or equal to a given threshold. This problem has a number of applications in data mining, e.g., in social networks or phone call graphs. We...

Given a set of lightpath requests, the problem of routing and wavelength assignment (RWA) in optical networks consists in routing a subset of these requests and assigning a wavelength to each of them, such that two lightpaths that share a common link are as signed to different wavelengths. There are many variants of this problem in the literature....

A divisible load is an amount W of computational work that can be arbitrarily divided into chunks and distributed among a set P of worker processors to be processed in parallel. Divisible load applications occur in many fields of science and engineering. They can be parallelized in a master-worker fashion, but they pose several scheduling challenge...

We consider the freight consolidation and containerization problem, which consists of loading items into containers and then shipping these containers to different warehouses from where they are delivered to their final destinations. We show through computational experiments that very good solutions can be obtained by heuristically aggregating the...

This work addresses characteristics of software environments for mathematical modeling and proposes a system for developing and managing models of linear and integer programming (IP) problems. The main features of this modeling environment are: version control of models and data; client-server architecture, which allows the interaction among modele...

Run time distributions or time-to-target plots display on the ordinate axis the probability that an algorithm will find a solution at least as good as a given target value within a given running time, shown on the abscissa axis. Given a pair of different randomized algorithms \(A_1\) and \(A_2\) , we describe a numerical method that gives the proba...

GRASP is a multi-start metaheuristic for combinatorial optimization problems, in whicheach iteration consists basically of two phases: Construction and local search. The construction phase builds a feasible solution, whose neighborhood is investigated until a local minimum isfound during the local search phase. The best overall solution is kept as...

The set k-covering problem is an extension of the set covering pro- blem, in which each object has to be covered at least k times. We describe a GRASP with path-relinking heuristic for its solution, as well as the template of a family of Lagrangean heuristics. The hybrid GRASP Lagrangean heuristic employs the GRASP with path-relinking heuristic usi...

The main drawback of most metaheuristics is the absence of effective stopping criteria. Most implementations of such algorithms stop after performing a given maximum number of iterations or a given maximum number of consecutive iterations without improvement in the best-known solution value, or after the stabilization of the set of elite solutions...

Multi-start methods strategically sample the solution space of an optimization problem. The most successful of these methods have two phases that are alternated for a certain number of global iterations. The first phase generates a solution and the second seeks to improve the outcome. Each global iteration produces a solution that is typically a lo...

Sports, with their massive investments in players and structures, have become a big business. Professional and amateur leagues face challenging problems, including logistics, revenue maximization, broadcast rights, fairness issues, game attractiveness, and security. The annual Brazilian soccer tournament is a compact, mirrored double round-robin to...

The Traveling Tournament Problem with Predefined Venues (TTPPV) is a single round robin variant of the Traveling Tournament Problem, in which the venue of each game to be played is known beforehand. We propose an Iterated Local Search (ILS) heuristic for solving real-size instances of the TTPPV, based on two types of moves. Initial solutions are de...

Path-relinking is major enhancement to heuristic search methods for solving combinatorial optimization problems, leading to significant improvements in both solution quality and running times. We review its fundamentals and implementation strategies, as well as advanced hybridizations with more elaborate metaheuristic schemes such as tabu search, G...

Sports scheduling problems mainly consist in determining the date and the venue in which each game of a tournament will be played. Integer programming, constraint programming, metaheuristics, and hybrid methods have been successfully applied to the solution of different variants of this problem. This paper provides an introductory review of fundame...

The carry-over effects value is one of the various measures one can consider to assess the quality of a round robin tournament schedule. We introduce and discuss a new, weighted variant of the minimum carry-over effects value problem. The problem is formulated by integer programming and an algorithm based on the hybridization of the Iterated Local...

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 perf...

Run time distributions or time-to-target plots are very useful tools to characterize the running times of stochastic algorithms for combinatorial optimization. We further explore run time distributions and describe a new tool to compare two algorithms based on stochastic local search. For the case where the running times of both algorithms fit expo...

GRASP with path-relinking is a hybrid metaheuristic, or stochastic local search (Monte Carlo) method, for combinatorial optimization. A restart strategy in GRASP with path-relinking heuristics is a set of iterations {i 1, i 2, …} on which the heuristic is restarted from scratch using a new seed for the random number generator. Restart strategies ha...

The set k-covering problem (SCk P) is a variant of the classical set covering problem, in which each object is required to be covered at least k times. We describe a hybrid Lagrangean heuristic, named LAGRASP, which combines subgradient optimization and GRASP with path-relinking to solve the SCk P. Computational experiments carried out on 135 test...

The problem of routing and wavelength assignment in wavelength division multiplexing optical networks consists in routing a set of lightpaths and assigning a wavelength to each of them, such that lightpaths whose routes share a common fiber are assigned different wavelengths. This problem was shown to be NP-hard when the objective is to minimize th...

This paper focuses on the use of different memory strategies to improve multistart methods. A network design problem in which the costs are given by discrete stepwise increasing cost functions of the capacities installed in the edges is used to illustrate the contributions of adaptive memory and vocabulary building strategies. Heuristics based on s...

The main drawback of most metaheuristics is the absence of effective stopping criteria. Most implementations stop after performing a given maximum number of iterations or a given maximum number of consecutive iterations without improvement in the best known solution value, or after the stabilization of the set of elite solutions found along the sea...

We consider the problem of assigning transmission powers to the nodes of an ad hoc wireless network, so as that the total power consumed is minimized and the resulting network is biconnected, i.e., there are at least two node-disjoint paths between any pair of nodes. Biconnected communication graphs are important to ensure fault tolerance, since ad...

An equitable k-coloring of a graph is defined by a partition of its vertices into k disjoint stable subsets, such that the difference between the cardinalities of any two subsets is at most one. The equitable coloring problem consists of finding the minimum value of k such that a given graph can be equitably k-colored. We present two new integer pr...

Scatter search is an evolutionary metaheuristic that explores solution spaces by evolving a set of reference points, operating on a small set of solutions while making only limited use of randomization. We give a comprehensive description of the elements and methods that make up its template, including the most recent elements incorporated in succe...

GRASP is a multi-start metaheuristic for combinatorial optimization problems, in which each iteration consists basically 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 a...

The diameter-constrained minimum spanning tree problem consists in finding a minimum spanning tree of a given graph, subject to the constraint that the maximum number of edges between any two vertices in the tree is bounded from above by a given constant. This problem typically models network design applications where all vertices communicate with...