Better and faster solutions for the maximum diversity problem

The aim of the Maximum Diversity Problem (MDP) is to extract a subset M of given cardinality from a set of elements N, in such a way that the sum of the pairwise distances between the elements of M is maximum. This problem, introduced by Glover [7], has been deeply studied using GRASP methodologies [6, 1, 17, 2, 16]. Usually, effective algorithms owe their success more to the careful exploitation of problem-specific features than to the application of general-purpose methods. A solution for MDP has a very simple structure which can not be exploited for sophisticated neighborhood search. This paper explores the performance of three alternative solution approaches, that is Tabu Search, Variable Neighborhood Search and Scatter Search, comparing them with those of best GRASP algorithms in literature. We also focus our attention on the comparison of these three methods applied in their pure form

GRASP and BRKGA metaheuristics applied to the maximum diversity problem

Orientadores: Marcia Aparecida Gomes Ruggiero, Kelly Cristina PoldiDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Neste trabalho estudamos o Problema da Diversidade Máxima (PDM), que consiste em selecionar entre um conjunto de elementos, um subconjunto que seja o mais diverso possível. O problema é classificado como NP-difícil. Apresentamos as formulações quadrática e a linear inteira mista, aplicações e exemplo numérico. Resolvemos problemas pequenos de maneira exata e notamos que para problemas maiores é necessário o uso de heurísticas ou meta-heurísticas em sua resolução. Sendo assim, escolhemos as meta-heurísticas Greedy Randomized Adaptive Search Procedure (GRASP) e Biased Random-Key Genetic Algorithm (BRKGA) para serem aplicadas ao PDM. No GRASP, adotamos uma construção de solução selecionando o elemento de acordo com a correspondente contribuição do mesmo ao valor da função objetivo. Após a construção de cada solução, aplicamos uma busca local e em seguida acionamos a técnica path relinking. No procedimento de busca local, empregamos uma busca do tipo exaustiva, realizando a melhor troca entre um elemento pertencente a solução e outro não pertencente. Geradas as soluções, aplicamos o path relinking na expectativa que entre cada par de soluções, exista uma solução com melhor valor para a função objetivo. Já no BRKGA, implementamos uma função fitness e um decodificador de soluções adaptados ao Problema da Diversidade Máxima. A função fitness adotada é a soma das diversidades entre os elementos selecionados e a decodificação de soluções é baseada na ordenação das chaves aleatórias. O objetivo deste trabalho é analisar e comparar os resultados obtidos pelas meta-heurísticas GRASP e BRKGA, tendo como referência os melhores resultados da literatura. Os problemas analisados nos testes computacionais foram extraídos da biblioteca MDPLIB. Observamos que as duas meta-heurísticas apresentam bons resultados na resolução do PDM, sendo que para problemas de pequeno e médio porte BRKGA obteve melhor desempenho que GRASP, enquanto que para problemas de grande porte, GRASP supera o desempenho do BRKGAAbstract: In this work we study the Maximum Diversity Problem (MDP), which consists of selecting among a set of elements, a subset as diverse as possible. The problem is classified as NP-hard. We present the quadratic and mixed integer linear formulations, applications and numerical example. We solve small problems exactly and we note that for larger problems it is necessary to use heuristics or metaheuristics on its resolution. Therefore, we chose Greedy Randomized Adaptive Search Procedure (GRASP) and Biased Random-Key Genetic Algorithm (BRKGA) metaheuristics to be applied to the MDP. In GRASP, we adopted a solution construction by selecting the element according to its corresponding contribution to the objective function value. After the construction of each solution, we apply a local search and then we activate the path relinking technique. In the local search procedure, we used a exhaustive search, making the best exchange between an element which belongs to solution and another element that does not belong. Once the solutions were generated, we apply path relinking in the expectation that between each pair of solutions, there is a solution with better objective function value. In BRKGA, we implemented a fitness function and a solution decoder adapted to the Maximum Diversity Problem. The fitness function adopted is the sum of diversities between selected elements e the decoding of solutions is based on sorting random-keys. The objective of this work is to analyze and compare the results obtained by GRASP and BRKGA metaheuristics, having as reference the best results in the literature. The problems analyzed in the computational tests were extracted from the MDPLIB library. We observed that the two metaheuristics showed good results on MDP's resolution, moreover for small and medium sized problems BRKGA obtained better performance than GRASP, while for large problems, GRASP outperforms BRKGAMestradoMatematica AplicadaMestre em Matemática AplicadaCAPE

Un problema de dispersión con capacidades

Los problemas de dispersión y diversidad surgen de la inquietud de encontrar las mejores ubicaciones para instalaciones no deseadas, administración de personal y en el contexto de las redes sociales, entre otros. Maximizar la diversidad se ocupa, en términos generales, de seleccionar un subconjunto de elementos de un conjunto dado de tal manera que se maximice la distancia entre los elementos seleccionados. En este trabajo presentamos una variante del problema de la Dispersión con restricciones de capacidad y costos dado por Rosenkrantz, Tayi y Ravi (1999), que muestra una heurística con garantía de desempeño de 2, lo que significa que, en todos los casos, el valor de la solución óptima dividido por el valor de su solución es inferior o igual a 2. Para realizar esta variante, investigamos la adaptación de las metodologías de búsqueda adaptativa aleatoria codificada (GRASP) y la búsqueda de entornos variables (VNS) al problema de dispersión con capacidades (CDP), con el objetivo de proponer una hibridación entre GRASP y VNS que se implementará mediante un procedimiento de oscilación estratégica, que nos ayudará a encontrar un método competitivo en la búsqueda de soluciones de alta calidad al problema (CDP). Para evaluar el rendimiento de nuestra propuesta, realizamos una extensa experimentación para establecer primero los parámetros clave de búsqueda de la heurística y luego compararlos con el método anterior. Además, proponemos un modelo matemático para obtener soluciones óptimas para instancias de pequeño tamaño y comparar nuestras soluciones con el conocido software Local-Solver

Approximation algorithms for geometric dispersion

The most basic form of the max-sum dispersion problem (MSD) is as follows: given n points in R^q and an integer k, select a set of k points such that the sum of the pairwise distances within the set is maximal. This is a prominent diversity problem, with wide applications in web search and information retrieval, where one needs to find a small and diverse representative subset of a large dataset. The problem has recently received a great deal of attention in the computational geometry and operations research communities; and since it is NP-hard, research has focused on efficient heuristics and approximation algorithms. Several classes of distance functions have been considered in the literature. Many of the most common distances used in applications are induced by a norm in a real vector space. The focus of this thesis is on MSD over these geometric instances. We provide for it simple and fast polynomial-time approximation schemes (PTASs), as well as improved constant-factor approximation algorithms. We pay special attention to the class of negative-type distances, a class that includes Euclidean and Manhattan distances, among many others. In order to exploit the properties of this class, we apply several techniques and results from the theory of isometric embeddings. We explore the following variations of the MSD problem: matroid and matroid-intersection constraints, knapsack constraints, and the mixed-objective problem that maximizes a combination of the sum of pairwise distances with a submodular monotone function. In addition to approximation algorithms, we present a core-set for geometric instances of low dimension, and we discuss the efficient implementation of some of our algorithms for massive datasets, using the streaming and distributed models of computation

Proceedings /5th International Symposium on Industrial Engineering – SIE2012, June 14-15, 2012., Belgrade

editors Dragan D. Milanović, Vesna Spasojević-Brkić, Mirjana Misit

Proceedings /5th International Symposium on Industrial Engineering – SIE2012, June 14-15, 2012., Belgrade

editors Dragan D. Milanović, Vesna Spasojević-Brkić, Mirjana Misit