Intelligent variable neighbourhood search for the minimum labelling spanning tree problem

Given a connected, undirected graph whose edges are labelled (or coloured), the minimum labelling spanning tree (MLST) problem seeks a spanning tree whose edges have the smallest number of distinct labels (or colours). In recent work, the MLST problem has been shown to be NP-hard and some effective heuristics have been proposed and analyzed. In a currently ongoing project, we investigate an intelligent optimization algorithm to solve the problem. It is obtained by the basic Variable Neighbourhood Search heuristic with the integration of other complements from machine learning, statistics and experimental algorithmics, in order to produce high-quality performance and to completely automate the resulting optimization strategy. Computational experiments show that the proposed metaheuristic has high-quality performance for the MLST problem and it is able to obtain optimal or near-optimal solutions in short computational running time.     

Greedy Randomized Adaptive Search and Variable Neighbourhood Search for the minimum labelling spanning tree problem

This paper studies heuristics for the minimum labelling spanning tree (MLST) problem. The purpose is to find a spanning tree using edges that are as similar as possible. Given an undirected labelled connected graph, the minimum labelling spanning tree problem seeks a spanning tree whose edges have the smallest number of distinct labels. This problem has been shown to be NP-hard. A Greedy Randomized Adaptive Search Procedure (GRASP) and a Variable Neighbourhood Search (VNS) are proposed in this paper. They are compared with other algorithms recommended in the literature: the Modified Genetic Algorithm and the Pilot Method. Nonparametric statistical tests show that the heuristics based on GRASP and VNS outperform the other algorithms tested. Furthermore, a comparison with the results provided by an exact approach shows that we may quickly obtain optimal or near-optimal solutions with the proposed heuristics.     

Variable neighbourhood search heuristics for the probabilistic multi-source Weber problem

The Multi-source Weber Problem (MWP) is concerned with locating m facilities in the Euclidean plane, and allocating these facilities to n customers at minimum total cost. The deterministic version of the problem, which assumes that customer locations and demands are known with certainty, is a non-convex optimization problem and difficult to solve. In this work, we focus on a probabilistic extension and consider the situation where customer locations are randomly distributed according to a bivariate distribution. We first present a mathematical programming formulation for the probabilistic MWP called the PMWP. For its solution, we propose two heuristics based on variable neighbourhood search (VNS). Computational results obtained on a number of test instances show that the VNS heuristics improve the performance of a probabilistic alternate location-allocation heuristic referred to as PALA. In its original form, the applicability of the new heuristics depends on the existence of a closed-form expression for the expected distances between facilities and customers. Unfortunately, such an expression exists only for a few distance function and probability distribution combinations. We therefore use two approximation methods for the expected distances, which make the VNS heuristics applicable for any distance function and bivariate distribution of customer locations.     

Local search for the Steiner tree problem in the Euclidean plane

Most heuristics for the Steiner tree problem in the Euclidean plane perform a series of iterative improvements using the minimum spanning tree as an initial solution. We may therefore characterize them as local search heuristics. In this paper, we first give a survey of existing heuristic approaches from a local search perspective, by setting up solution spaces and neighbourhood structures. Secondly, we present a new general local search approach which is based on a list of full Steiner trees constructed in a preprocessing phase. This list defines a solution space on which three neighbourhood structures are proposed and evaluated. Computational results show that this new approach is very competitive from a cost–benefit point of view. Furthermore, it has the advantage of being easy to apply to the Steiner tree problem in other metric spaces and to obstacle avoiding variants.     

系列平行图上带时间约束的Steiner最小树问题

对一类特殊系列平行图上带有时间约束的Steiner最小树问题,证明了其复杂性为NPC,并给出了一个完全多项式时间近似方案.     

Variable neighbourhood search for bandwidth reduction

The problem of reducing the bandwidth of a matrix consists of finding a permutation of rows and columns of a given matrix which keeps the non-zero elements in a band as close as possible to the main diagonal. This NP-complete problem can also be formulated as a vertex labelling problem on a graph, where each edge represents a non-zero element of the matrix. We propose a variable neighbourhood search based heuristic for reducing the bandwidth of a matrix which successfully combines several recent ideas from the literature. Empirical results for an often used collection of 113 benchmark instances indicate that the proposed heuristic compares favourably to all previous methods. Moreover, with our approach, we improve best solutions in 50% of instances of large benchmark tests.     

Variable neighborhood search for the vertex weighted k-cardinality tree problem

This paper presents some new heuristics based on variable neighborhood search to solve the vertex weighted k-cardinality tree problem. An efficient local search procedure is also developed for use within these heuristics. Our computational results demonstrate that the new heuristics substantially outperform the state-of-the-art methodologies, including a tabu search and genetic algorithm recently proposed in the literature. We also show that a decomposition approach is best for larger problem sizes than previously investigated. Thus, our findings advance in a significant way the capacity to solve this important class of problems.     

A tabu search heuristic for the generalized minimum spanning tree problem

This paper describes an attribute based tabu search heuristic for the generalized minimum spanning tree problem (GMSTP) known to be NP-hard. Given a graph whose vertex set is partitioned into clusters, the GMSTP consists of designing a minimum cost tree spanning all clusters. An attribute based tabu search heuristic employing new neighborhoods is proposed. An extended set of TSPLIB test instances for the GMSTP is generated and the heuristic is compared with recently proposed genetic algorithms. The proposed heuristic yields the best results for all instances. Moreover, an adaptation of the tabu search algorithm is proposed for a variation of the GMSTP in which each cluster must be spanned at least once.     

Variable neighbourhood search for colour image quantization

^** Email: nenad.mladenovic{at}brunel.ac.uk Colour image quantization is a data compression technique thatreduces the total set of colours in a digital image to a representativesubset. This problem is first expressed as a large M-medianone. The advantages of this model over the usual minimum sum-of-squaresmodel are discussed first and then, the heuristic based on variableneighbourhood search metaheuristic is applied to solve it. Computationalexperience proves that this approach compares favourably withtwo other recent state-of-the-art heuristics, based on geneticand particle swarm searches.     

Variable neighbourhood search for redundancy allocation problems

^** Email: ycliang{at}saturn.yzu.edu.tw^*** Email: s929512{at}mail.yzu.edu.tw^**** Email: s927522{at}mail.yzu.edu.tw A variable neighbourhood search (VNS) algorithm has been developedto solve the redundancy allocation problem (RAP). The VNS methodis perfectly suited to those combinatorial problems with potentialneighbourhood structures, as in the case of the RAP. The moststudied configuration of the RAP is a series system of s-independentk-out-of-n:G subsystems the so-called series–parallelsystem. The RAP is to select the optimal combination and redundancylevels of components to meet system-level constraints. Two typesof objectives are considered in this study—system reliabilitymaximization and system cost minimization. The VNS algorithmis tested on sets of benchmark problems and compared to thebest heuristics in the literature such as tabu search, multipleweighted objective heuristic, ant colony optimization and geneticalgorithm. Computational results show the advantages and benefitsof VNS for solving both types of RAP while considering bothsolution quality and computational efficiency.     

Breakout local search for the Steiner tree problem with revenue,budget and hop constraints

The Steiner tree problem (STP) is one of the most popular combinatorial optimization problems with various practical applications. In this paper, we propose a Breakout Local Search (BLS) algorithm for an important generalization of the STP: the Steiner tree problem with revenue, budget and hop constraints (STPRBH), which consists of determining a subtree of a given undirected graph which maximizes the collected revenues, subject to both budget and hop constraints. Starting from a probabilistically constructed initial solution, BLS uses a Neighborhood Search (NS) procedure based on several specifically designed move operators for local optimization, and employs an adaptive diversification strategy to escape from local optima. The diversification mechanism is implemented by adaptive perturbations, guided by dedicated information of discovered high-quality solutions. Computational results based on 240 benchmarks show that BLS produces competitive results with respect to several previous approaches. For the 56 most challenging instances with unknown optimal results, BLS succeeds in improving 49 and matching one best known results within reasonable time. For the 184 instances which have been solved to optimality, BLS can also match 167 optimal results.     

The multi-weighted Steiner tree problem

We formulate and investigate the Multi-Weighted Steiner Problem (MWS), a generalization of the Steiner problem in graphs, involving more than one weight function. As a special case, it contains the hierarchical network design problem. With the notion of "bottleneck length/distance", a min-max measure, we analyze the interaction between differently weighted edges in a solution. Combining the results with known methods for the Steiner problem in graphs and the hierarchical network design problem, two heuristics for the MWS are developed, one based on weight modifications and the other on exchanging edges. Both are of time complexityO(kv ²), withv the number of nodes andk the number of special nodes in the graph. The first is also suited for thedirected MWS; the second is expected to perform better on the undirected version. Before actually solving the Steiner problem in graphs and the hierarchical network design problem, preprocessing techniques exploiting tests to reduce the problem graphs have proven to be valuable. We adapt three prominent tests for use in the MWS.     

Local search for the minimum label spanning tree problem with bounded color classes

In the Minimum Label Spanning Tree problem, the input consists of an edge-colored undirected graph, and the goal is to find a spanning tree with the minimum number of different colors. We investigate the special case where every color appears at most r times in the input graph. This special case is polynomially solvable for r=2, and NP- and APX-complete for any fixed r?3.We analyze local search algorithms that are allowed to switch up to k of the colors used in a feasible solution. We show that for k=2 any local optimum yields an (r+1)/2-approximation of the global optimum, and that this bound is tight. For every k?3, there exist instances for which some local optima are a factor of r/2 away from the global optimum.     

A constraint-based parallel local search for the edge-disjoint rooted distance-constrained minimum spanning tree problem

Many network design problems arising in areas as diverse as VLSI circuit design, QoS routing, traffic engineering, and computational sustainability require clients to be connected to a facility under path-length constraints and budget limits. These problems can be seen as instances of the rooted distance-constrained minimum spanning-tree problem (RDCMST), which is NP-hard. An inherent feature of these networks is that they are vulnerable to a failure. Therefore, it is often important to ensure that all clients are connected to two or more facilities via edge-disjoint paths. We call this problem the edge-disjoint RDCMST (ERDCMST). Previous work on the RDCMST has focused on dedicated algorithms and therefore it is difficult to use these algorithms to tackle the ERDCMST. We present a constraint-based parallel local search algorithm for solving the ERDCMST. Traditional ways of extending a sequential algorithm to run in parallel perform either portfolio-based search in parallel or parallel neighbourhood search. Instead, we exploit the semantics of the constraints of the problem to perform multiple moves in parallel by ensuring that they are mutually independent. The ideas presented in this paper are general and can be adapted to other problems as well. The effectiveness of our approach is demonstrated by experimenting with a set of problem instances taken from real-world passive optical network deployments in Ireland, Italy, and the UK. Our results show that performing moves in parallel can significantly reduce the elapsed time and improve the quality of the solutions of our local search approach.     

Decomposition methods for the two-stage stochastic Steiner tree problem

A new algorithmic approach for solving the stochastic Steiner tree problem based on three procedures for computing lower bounds (dual ascent, Lagrangian relaxation, Benders decomposition) is introduced. Our method is derived from a new integer linear programming formulation, which is shown to be strongest among all known formulations. The resulting method, which relies on an interplay of the dual information retrieved from the respective dual procedures, computes upper and lower bounds and combines them with several rules for fixing variables in order to decrease the size of problem instances. The effectiveness of our method is compared in an extensive computational study with the state-of-the-art exact approach, which employs a Benders decomposition based on two-stage branch-and-cut, and a genetic algorithm introduced during the DIMACS implementation challenge on Steiner trees. Our results indicate that the presented method significantly outperforms existing ones, both on benchmark instances from literature, as well as on large-scale telecommunication networks.     

Russian doll search for the Steiner triple covering problem

Russian doll search is applied to finding maximum independent sets in hypergraphs, focusing on a particular subproblem of the hitting set problem, the Steiner triple covering problem. An instance denoted A ₁₃₅ is solved considerably faster with Russian doll search than with integer linear programming and a state-of-the-art optimization tool (using otherwise a similar established approach to split the problem into subproblems). In addition, the improvement in speed makes it possible to carry out a search proving that all optimal solutions for A ₁₃₅ are isomorphic.     

Variable neighbourhood search: methods and applications

Variable neighbourhood search (VNS) is a metaheuristic, or a framework for building heuristics, based upon systematic changes of neighbourhoods both in descent phase, to find a local minimum, and in perturbation phase to emerge from the corresponding valley. It was first proposed in 1997 and has since then rapidly developed both in its methods and its applications. In the present paper, these two aspects are thoroughly reviewed and an extensive bibliography is provided. Moreover, one section is devoted to newcomers. It consists of steps for developing a heuristic for any particular problem. Those steps are common to the implementation of other metaheuristics.      

Combining variable neighborhood search with integer linear programming for the generalized minimum spanning tree problem

We consider the generalized version of the classical Minimum Spanning Tree problem where the nodes of a graph are partitioned into clusters and exactly one node from each cluster must be connected. We present a Variable Neighborhood Search (VNS) approach which uses three different neighborhood types. Two of them work in complementary ways in order to maximize search effectivity. Both are large in the sense that they contain exponentially many candidate solutions, but efficient polynomial-time algorithms are used to identify best neighbors. For the third neighborhood type we apply Mixed Integer Programming to optimize local parts within candidate solution trees. Tests on Euclidean and random instances with up to 1280 nodes indicate especially on instances with many nodes per cluster significant advantages over previously published metaheuristic approaches. This work is supported by the RTN ADONET under grant 504438.     

A branch-and-cut algorithm for the Steiner tree problem with delays

In this paper, we investigate the Steiner tree problem with delays, which is a generalized version of the Steiner tree problem applied to multicast routing. For this challenging combinatorial optimization problem, we present an enhanced directed cut-based MIP formulation and an exact solution method based on a branch-and-cut approach. Our computational study reveals that the proposed approach can optimally solve hard dense instances.