Matheuristics: using mathematics for heuristic design

Matheuristics are heuristic algorithms based on mathematical tools such as the ones provided by mathematical programming, that are structurally general enough to be applied to different problems with little adaptations to their abstract structure. The result can be metaheuristic hybrids having components derived from the mathematical model of the problems of interest, but the mathematical techniques themselves can define general heuristic solution frameworks. In this paper, we focus our attention on mathematical programming and its contributions to developing effective heuristics. We briefly describe the mathematical tools available and then some matheuristic approaches, reporting some representative examples from the literature. We also take the opportunity to provide some ideas for possible future development

Robust Branch-Cut-and-Price for the Capacitated Minimum Spanning Tree Problem over a Large Extended Formulation

This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arbores- cence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms. Powerful new cuts expressed over a very large set of variables could be added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very signi¯cant improvements over previous algorithms. Several open instances could be solved to optimalityNo keywords;

Automatically Produced Algorithms for the Generalized Minimum Spanning Tree Problem

The generalized minimum spanning tree problem consists of finding a minimum cost spanning tree in an undirected graph for which the vertices are divided into clusters. Such spanning tree includes only one vertex from each cluster. Despite the diverse practical applications for this problem, the NP-hardness continues to be a computational challenge. Good quality solutions for some instances of the problem have been found by combining specific heuristics or by including them within a metaheuristic. However studied combinations correspond to a subset of all possible combinations. In this study a technique based on a genotype-phenotype genetic algorithm to automatically construct new algorithms for the problem, which contain combinations of heuristics, is presented. The produced algorithms are competitive in terms of the quality of the solution obtained. This emerges from the comparison of the performance with problem-specific heuristics and with metaheuristic approaches

Técnicas heurísticas para instâncias de grande porte do problema cabo-trincheira

Orientadores: Flávio Keidi Miyazawa, Eduardo Candido XavierDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O problema cabo trincheira foi apresentado em 2002 para modelar redes cabeadas. Esse problema pode ser visto como a união do problema de caminhos mínimos com o problema da árvore geradora mínima. Como entrada do problema temos um grafo $G=(V,E)$ com pesos nas arestas que indicam a distância entre os vértices incidentes na mesma. Há um vértice especial que representa uma instalação e demais vértices representam clientes. Uma solução para o problema é uma árvore geradora enraizada na instalação. O custo da solução é o custo da árvore geradora multiplicado por um fator de custo de trincheira mais os custos de cabos. Para cada cliente, o seu custo de cabo é dado pelo custo do caminho do cliente até a instalação multiplicado por um fator de custo de cabo. Esse problema modela cenários onde cada cliente deve ser conectado a uma instalação central através de um cabo dedicado. Cada cabo deve estar acomodado em uma trincheira e cada trincheira pode conter um número ilimitado de cabos. Sabendo que o custo dos cabos e trincheiras é proporcional a seu comprimento multiplicado por um fator de custo, o problema é encontrar uma rede com custo mínimo. Trabalhos anteriores  utilizaram o problema cabo trincheira para modelar problemas em telecomunicações, distribuição de energia, redes ferroviárias e até para reconstrução de vasos sanguíneos em exames de tomografia computadorizada. O trabalho foca na resolução do problema em instâncias de grande porte (superiores a 10 mil vértices). Foram desenvolvidas várias heurísticas para o problema. Na busca por simplificações de instâncias, foram demonstradas regras seguras, ou seja, que não comprometem nenhuma solução ótima, e heurísticas para a remoção de arestas eliminando aquelas que dificilmente estariam em ''boas soluções" de uma instância. Foi apresentado um algoritmo rápido para busca local capaz de ser executado mesmo em instâncias de grande porte. Foram desenvolvidos também algoritmos baseados em Greedy Randomized Adaptive Search Procedure (GRASP) e formulada uma heurística que contrai vértices. Com a contração de vértices, foram criadas instâncias do problema Cabo Trincheira com Demandas nos Vértices (CTDV). Essa versão com demandas tem um número menor de vértices que o problema original, o que viabiliza o uso de algoritmos baseados em programação linear para resolvê-lo. Foi demonstrado como é possível, ao resolver essa versão reduzida com demandas, remontar uma solução viável para o problema cabo trincheira original. Foram obtidos, com essas heurísticas, resultados melhores do que trabalhos anteriores encontrados na literatura do problema. Para além disso, foi demonstrado como essa técnica de contração de vértices tem o potencial para resolver instâncias de tamanhos ainda maior para o problema cabo trincheiraAbstract: The Cable Trench Problem (CTP) was presented in 2002 to model wired networks. This problem can be seen as the combination of the shortest path problem with the minimum spanning tree problem. An instance of the problem is composed by a graph $G=(V, E)$ with weigths, representing the distance between a pair of vertices. A special vertex represents a facility, and all others are clients. A solution to the problem is a spanning tree rooted in the facility. The solution's cost is given by the spanning tree cost multiplied by a trench cost factor, added by the cables cost reaching the root from each vertex in the graph. For each client, its cable cost is given by the path in the spanning tree, from the client to the root, multiplied by a cable cost factor. The CTP models a scenario where each client must be connected through a dedicated cable to a central facility. Each cable must be laying on a trench and a trench may hold an unlimited number of cables. Knowing that the cost of cables and trenches are proportional to its lengths multiplied by a cost factor, the problem is to find a network of minimum cost. Previous works in the literature used the CTP to model telecommunication problems, power distribution, rail networks, and even a blood vessel networks for computed tomography exams. In this research, we focused on large-scale instances of the problem (above 10 thousand vertices), achieving better results than previous works found in the literature. We developed a series of heuristics for the problem. Searching for a simplification for those instances, we present safe reductions, that do not affect any optimal solution, and heuristic reduction rules that are capable of removing edges unlikely to be part of ''good'' solutions in an instance. We present a fast local search algorithm, capable of improving even solutions for large-scale instances. We developed an algorithm based on a Greedy Randomized Adaptive Search Procedure (GRASP) and formulated a heuristic to cluster vertices. By clustering vertices, we represent a CTP instance as an instance of the Cable Trench Problem with Demands (CTPD). We represent the large-scale CTP instance into a vertex-wise smaller one adding demands to its vertices. Dealing with smaller instances, we enable a new range of techniques such as linear programming based algorithms to solve it. We demonstrate how this instances with demands can be used to build a viable solution for the original CTP instance. We also demonstrate how this vertex clustering technique has the potential to solve even larger scale instances for the CTPMestradoCiência da ComputaçãoMestre em Ciência da Computação133323/2018-8, 131175/2017-3CNP