A Solution Merging Heuristic for the Steiner Problem in Graphs Using Tree Decompositions

Fixed parameter tractable algorithms for bounded treewidth are known to exist for a wide class of graph optimization problems. While most research in this area has been focused on exact algorithms, it is hard to find decompositions of treewidth sufficiently small to make these al- gorithms fast enough for practical use. Consequently, tree decomposition based algorithms have limited applicability to large scale optimization. However, by first reducing the input graph so that a small width tree decomposition can be found, we can harness the power of tree decomposi- tion based techniques in a heuristic algorithm, usable on graphs of much larger treewidth than would be tractable to solve exactly. We propose a solution merging heuristic to the Steiner Tree Problem that applies this idea. Standard local search heuristics provide a natural way to generate subgraphs with lower treewidth than the original instance, and subse- quently we extract an improved solution by solving the instance induced by this subgraph. As such the fixed parameter tractable algorithm be- comes an efficient tool for our solution merging heuristic. For a large class of sparse benchmark instances the algorithm is able to find small width tree decompositions on the union of generated solutions. Subsequently it can often improve on the generated solutions fast

The development and application of metaheuristics for problems in graph theory: A computational study

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.It is known that graph theoretic models have extensive application to real-life discrete optimization problems. Many of these models are NP-hard and, as a result, exact methods may be impractical for large scale problem instances. Consequently, there is a great interest in developing e±cient approximate methods that yield near-optimal solutions in acceptable computational times. A class of such methods, known as metaheuristics, have been proposed with success. This thesis considers some recently proposed NP-hard combinatorial optimization problems formulated on graphs. In particular, the min- imum labelling spanning tree problem, the minimum labelling Steiner tree problem, and the minimum quartet tree cost problem, are inves- tigated. Several metaheuristics are proposed for each problem, from classical approximation algorithms to novel approaches. A compre- hensive computational investigation in which the proposed methods are compared with other algorithms recommended in the literature is reported. The results show that the proposed metaheuristics outper- form the algorithms recommended in the literature, obtaining optimal or near-optimal solutions in short computational running times. In addition, a thorough analysis of the implementation of these methods provide insights for the implementation of metaheuristic strategies for other graph theoretic problems

A new effective mathematical programming model to design CDN topology

The Steiner Tree Problem is an umbrella of combinatorial optimization problems in graphs, most of them NP-Hard, within which, the Steiner Tree Problem in graphs (STP) is perhaps one of the most famous and widely studied. The STP consists in optimally interconnect a given set of terminal or mandatory nodes within a graph with edges of positive weights, eventually using other optional nodes. It has a wide range of applications from circuit layouts to network design, so plenty of models to find its exact solutions have been crafted. Traditionally, due to its intrinsic complexity, heuristic approaches have been used to find good quality solutions to the STP. Currently, the outstanding computing power resulting from combining developments in hardware and software capabilities makes it possible to rely upon exact formulations and generic algorithms to solve complex instances of the problem. This work introduces a flow-based mixed-integer problem formulation (MIP) for the STP using the SteinLib, a reference test-set repository. Later on, that MIP formulation is modified to solve the Quality of Service Multicast Tree problem (QoSTP). To the best of our knowledge, there is no previous MIP formulation. While existing approaches go all the way of approximation algorithms to find solutions, this MIP formulation shows promising experimental results. Optimal solutions are found for several instances, while low feasible-to-optimal gaps were obtained for most of the remaining ones

Un algoritmo multithreading para el problema del árbol de Steiner

Este artículo presenta una implementación paralela de la metaheurística SN [13] utilizando una técnica de programación multithreading y su aplicación a la resolución del problema del árbol de Steiner. Se describen las decisiones de diseño del algoritmo y se presentan experimentos realizados sobre un conjunto de problemas de prueba estándar, analizando la calidad de resultados obtenidos y la eficiencia computacional de la versión paralela del algoritmo.Eje: IV - Workshop de procesamiento distribuido y paraleloRed de Universidades con Carreras en Informática (RedUNCI

Un algoritmo multithreading para el problema del árbol de Steiner

Este artículo presenta una implementación paralela de la metaheurística SN [13] utilizando una técnica de programación multithreading y su aplicación a la resolución del problema del árbol de Steiner. Se describen las decisiones de diseño del algoritmo y se presentan experimentos realizados sobre un conjunto de problemas de prueba estándar, analizando la calidad de resultados obtenidos y la eficiencia computacional de la versión paralela del algoritmo.Eje: IV - Workshop de procesamiento distribuido y paraleloRed de Universidades con Carreras en Informática (RedUNCI

Randomized heuristics for the Capacitated Clustering Problem

In this paper, we investigate the adaptation of the Greedy Randomized Adaptive Search Procedure (GRASP) and Iterated Greedy methodologies to the Capacitated Clustering Problem (CCP). In particular, we focus on the effect of the balance between randomization and greediness on the performance of these multi-start heuristic search methods when solving this NP-hard problem. The former is a memory-less approach that constructs independent solutions, while the latter is a memory-based method that constructs linked solutions, obtained by partially rebuilding previous ones. Both are based on the combination of greediness and randomization in the constructive process, and coupled with a subsequent local search phase. We propose these two multi-start methods and their hybridization and compare their performance on the CCP. Additionally, we propose a heuristic based on the mathematical programming formulation of this problem, which constitutes a so-called matheuristic. We also implement a classical randomized method based on simulated annealing to complete the picture of randomized heuristics. Our extensive experimentation reveals that Iterated Greedy performs better than GRASP in this problem, and improved outcomes are obtained when both methods are hybridized and coupled with the matheuristic. In fact, the hybridization is able to outperform the best approaches previously published for the CCP. This study shows that memory-based construction is an effective mechanism within multi-start heuristic search techniques