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

An approximate solution method based on tabu search for k-minimum spanning tree problems

This paper considers k-minimum spanning tree problems. An existing solution algorithm based on tabu search, which was proposed by Katagiri et al., includes an iterative solving procedure of minimum spanning tree (MST) problems for subgraphs to obtain a local optimal solution of k-minimum spanning tree problems. This article provides a new tabu-searchbased approximate solution method that does not iteratively solve minimum spanning tree problems. Results of numerical experiments show that the proposed method provides a good performance in terms of accuracy over those of existing methods for relatively high cardinality k

Automated Reconstruction of Dendritic and Axonal Trees by Global Optimization with Geometric Priors

We present a novel probabilistic approach to fully automated delineation of tree structures in noisy 2D images and 3D image stacks. Unlike earlier methods that rely mostly on local evidence, ours builds a set of candidate trees over many different subsets of points likely to belong to the optimal tree and then chooses the best one according to a global objective function that combines image evidence with geometric priors. Since the best tree does not necessarily span all the points, the algorithm is able to eliminate false detections while retaining the correct tree topology. Manually annotated brightfield micrographs, retinal scans and the DIADEM challenge datasets are used to evaluate the performance of our method. We used the DIADEM metric to quantitatively evaluate the topological accuracy of the reconstructions and showed that the use of the geometric regularization yields a substantial improvemen

The multiple pheromone Ant clustering algorithm

Ant Colony Optimisation algorithms mimic the way ants use pheromones for marking paths to important locations. Pheromone traces are followed and reinforced by other ants, but also evaporate over time. As a consequence, optimal paths attract more pheromone, whilst the less useful paths fade away. In the Multiple Pheromone Ant Clustering Algorithm (MPACA), ants detect features of objects represented as nodes within graph space. Each node has one or more ants assigned to each feature. Ants attempt to locate nodes with matching feature values, depositing pheromone traces on the way. This use of multiple pheromone values is a key innovation. Ants record other ant encounters, keeping a record of the features and colony membership of ants. The recorded values determine when ants should combine their features to look for conjunctions and whether they should merge into colonies. This ability to detect and deposit pheromone representative of feature combinations, and the resulting colony formation, renders the algorithm a powerful clustering tool. The MPACA operates as follows: (i) initially each node has ants assigned to each feature; (ii) ants roam the graph space searching for nodes with matching features; (iii) when departing matching nodes, ants deposit pheromones to inform other ants that the path goes to a node with the associated feature values; (iv) ant feature encounters are counted each time an ant arrives at a node; (v) if the feature encounters exceed a threshold value, feature combination occurs; (vi) a similar mechanism is used for colony merging. The model varies from traditional ACO in that: (i) a modified pheromone-driven movement mechanism is used; (ii) ants learn feature combinations and deposit multiple pheromone scents accordingly; (iii) ants merge into colonies, the basis of cluster formation. The MPACA is evaluated over synthetic and real-world datasets and its performance compares favourably with alternative approaches

Solution of minimum spanning forest problems with reliability constraints

We propose the reliability constrained k-rooted minimum spanning forest, a relevant optimization problem whose aim is to find a k-rooted minimum cost forest that connects given customers to a number of supply vertices, in such a way that a minimum required reliability on each path between a customer and a supply vertex is satisfied and the cost is a minimum. The reliability of an edge is the probability that no failure occurs on that edge, whereas the reliability of a path is the product of the reliabilities of the edges in such path. The problem has relevant applications in the design of networks, in fields such as telecommunications, electricity and transports. For its solution, we propose a mixed integer linear programming model, and an adaptive large neighborhood search metaheuristic which invokes several shaking and local search operators. Extensive computational tests prove that the metaheuristic can provide good quality solutions in very short computing times

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

The Sampling-and-Learning Framework: A Statistical View of Evolutionary Algorithms

Evolutionary algorithms (EAs), a large class of general purpose optimization algorithms inspired from the natural phenomena, are widely used in various industrial optimizations and often show excellent performance. This paper presents an attempt towards revealing their general power from a statistical view of EAs. By summarizing a large range of EAs into the sampling-and-learning framework, we show that the framework directly admits a general analysis on the probable-absolute-approximate (PAA) query complexity. We particularly focus on the framework with the learning subroutine being restricted as a binary classification, which results in the sampling-and-classification (SAC) algorithms. With the help of the learning theory, we obtain a general upper bound on the PAA query complexity of SAC algorithms. We further compare SAC algorithms with the uniform search in different situations. Under the error-target independence condition, we show that SAC algorithms can achieve polynomial speedup to the uniform search, but not super-polynomial speedup. Under the one-side-error condition, we show that super-polynomial speedup can be achieved. This work only touches the surface of the framework. Its power under other conditions is still open