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.     

RAMP for the capacitated minimum spanning tree problem

This paper introduces dual and primal-dual RAMP algorithms for the solution of the capacitated minimum spanning tree problem (CMST). A surrogate constraint relaxation incorporating cutting planes is proposed to explore the dual solution space. In the dual RAMP approach, primal-feasible solutions are obtained by simple tabu searches that project dual solutions onto primal feasible space. A primal-dual approach is achieved by including a scatter search procedure that further exploits the adaptive memory framework. Computational results from applying the methods to a standard set of benchmark problems disclose that the dual RAMP algorithm finds high quality solutions very efficiently and that its primal-dual enhancement is still more effective.     

Modeling hop-constrained and diameter-constrained minimum spanning tree problems as Steiner tree problems over layered graphs

The hop-constrained minimum spanning tree problem (HMSTP) is an NP-hard problem arising in the design of centralized telecommunication networks with quality of service constraints. We show that the HMSTP is equivalent to a Steiner tree problem (STP) in an appropriate layered graph. We prove that the directed cut model for the STP defined in the layered graph, dominates the best previously known models for the HMSTP. We also show that the Steiner directed cuts in the extended layered graph space can be viewed as being a stronger version of some previously known HMSTP cuts in the original design space. Moreover, we show that these strengthened cuts can be combined and projected into new families of cuts, including facet defining ones, in the original design space. We also adapt the proposed approach to the diameter-constrained minimum spanning tree problem (DMSTP). Computational results with a branch-and-cut algorithm show that the proposed method is significantly better than previously known methods on both problems.     

Heuristics for the multi-level capacitated minimum spanning tree problem

The capacitated minimum spanning tree (CMST) problem is fundamental to the design of centralized communication networks. In this paper we consider the multi-level capacitated minimum spanning tree problem, a generalization of the well-known CMST problem. Based on work previously done in the field, three heuristics are presented, addressing unit and non-unit demand cases. The proposed heuristics have been also integrated into a mixed integer programming solver. Evaluation results are presented, for an extensive set of experiments, indicating the improvements that the heuristics bring to the particular problem.     

On the prize-collecting generalized minimum spanning tree problem

The prize-collecting generalized minimum spanning tree problem (PC-GMSTP), is a generalization of the generalized minimum spanning tree problem (GMSTP) and belongs to the hard core of -hard problems. We describe an exact exponential time algorithm for the problem, as well we present several mixed integer and integer programming formulations of the PC-GMSTP. Moreover, we establish relationships between the polytopes corresponding to their linear relaxations and present an efficient solution procedure that finds the optimal solution of the PC-GMSTP for graphs with up 240 nodes.     

The prize-collecting generalized minimum spanning tree problem

We introduce the prize-collecting generalized minimum spanning tree problem. In this problem a network of node clusters needs to be connected via a tree architecture using exactly one node per cluster. Nodes in each cluster compete by offering a payment for selection. This problem is NP-hard, and we describe several heuristic strategies, including local search and a genetic algorithm. Further, we present a simple and computationally efficient branch-and-cut algorithm. Our computational study indicates that our branch-and-cut algorithm finds optimal solutions for networks with up to 200 nodes within two hours of CPU time, while the heuristic search procedures rapidly find near-optimal solutions for all of the test instances.     

Multi-exchange neighborhood structures for the capacitated minimum spanning tree problem

The capacitated minimum spanning tree (CMST) problem is to find a minimum cost spanning tree with an additional cardinality constraint on the sizes of the subtrees incident to a given root node. The CMST problem is an NP-complete problem, and existing exact algorithms can solve only small size problems. Currently, the best available heuristic procedures for the CMST problem are tabu search algorithms due to Amberg et al. and Sharaiha et al. These algorithms use two-exchange neighborhood structures that are based on exchanging a single node or a set of nodes between two subtrees. In this paper, we generalize their neighborhood structures to allow exchanges of nodes among multiple subtrees simultaneously; we refer to such neighborhood structures as multi-exchange neighborhood structures. Our first multi-exchange neighborhood structure allows exchanges of single nodes among several subtrees. Our second multi-exchange neighborhood structure allows exchanges that involve multiple subtrees. The size of each of these neighborhood structures grows exponentially with the problem size without any substantial increase in the computational times needed to find improved neighbors. Our approach, which is based on the cyclic transfer neighborhood structure due to Thompson and Psaraftis and Thompson and Orlin transforms a profitable exchange into a negative cost subset-disjoint cycle in a graph, called an improvement graph, and identifies these cycles using variants of shortest path label-correcting algorithms. Our computational results with GRASP and tabu search algorithms based on these neighborhood structures reveal that (i) for the unit demand case our algorithms obtained the best available solutions for all benchmark instances and improved some; and (ii) for the heterogeneous demand case our algorithms improved the best available solutions for most of the benchmark instances with improvements by as much as 18%. The running times our multi-exchange neighborhood search algorithms are comparable to those taken by two-exchange neighborhood search algorithms. Received: September 1998 / Accepted: March 2001?Published online May 18, 2001     

A GRASP algorithm for the multi-criteria minimum spanning tree problem

This paper proposes a GRASP (Greedy Randomized Adaptive Search Procedure) algorithm for the multi-criteria minimum spanning tree problem, which is NP-hard. In this problem a vector of costs is defined for each edge of the graph and the problem is to find all Pareto optimal or efficient spanning trees (solutions). The algorithm is based on the optimization of different weighted utility functions. In each iteration, a weight vector is defined and a solution is built using a greedy randomized constructive procedure. The found solution is submitted to a local search trying to improve the value of the weighted utility function. We use a drop-and-add neighborhood where the spanning trees are represented by Prufer numbers. In order to find a variety of efficient solutions, we use different weight vectors, which are distributed uniformly on the Pareto frontier. The proposed algorithm is tested on problems with r=2 and 3 criteria. For non-complete graphs with n=10, 20 and 30 nodes, the performance of the algorithm is tested against a complete enumeration. For complete graphs with n=20, 30 and 50 nodes the performance of the algorithm is tested using two types of weighted utility functions. The algorithm is also compared with the multi-criteria version of the Kruskal’s algorithm, which generates supported efficient solutions. This work was funded by the Municipal Town Hall of Campos dos Goytacazes city. The used computer was acquired with resource of CNPq.     

A greedy heuristic for the capacitated minimum spanning tree problem

This paper develops a greedy heuristic for the capacitated minimum spanning tree problem (CMSTP), based on the two widely known methods of Prim and of Esau–Williams. The proposed algorithm intertwines two-stages: an enhanced combination of the Prim and Esau–Williams approaches via augmented and synthetic node selection criteria, and an increase of the feasible solution space by perturbing the input data using the law of cosines. Computational tests on benchmark problems show that the new heuristic provides extremely good performance results for the CMSTP, justifying its effectiveness and robustness. Furthermore, excluding the feasible space expansion, we show that we can still obtain good quality solutions in very short computational times.     

The minimum spanning tree problem with fuzzy costs

In this paper the minimum spanning tree problem in a given connected graph is considered. It is assumed that the edge costs are not precisely known and they are specified as fuzzy intervals. Possibility theory is applied to characterize the optimality of edges of the graph and to choose a spanning tree under fuzzy costs.     

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.     

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.     

A hybrid heuristic for the diameter constrained minimum spanning tree problem

We propose a hybrid GRASP and ILS based heuristic for the diameter constrained minimum spanning tree problem. The latter typically models network design applications where, under a given quality requirement, all vertices must be connected at minimum cost. An adaptation of the one time tree heuristic is used to build feasible diameter constrained spanning trees. Solutions thus obtained are then attempted to be improved through local search. Four different neighborhoods are investigated, in a scheme similar to VND. Upper bounds within 2% of optimality were obtained for problems in two test sets from the literature. Additionally, upper bounds stronger than those previously obtained in the literature are reported for OR-Library instances.     

Modeling and solving the bi-objective minimum diameter-cost spanning tree problem

The bi-objective minimum diameter-cost spanning tree problem (bi-MDCST) seeks spanning trees with minimum total cost and minimum diameter. The bi-objective version generalizes the well-known bounded diameter minimum spanning tree problem. The bi-MDCST is a NP-hard problem and models several practical applications in transportation and network design. We propose a bi-objective multiflow formulation for the problem and effective multi-objective metaheuristics: a multi-objective evolutionary algorithm and a fast nondominated sorting genetic algorithm. Some guidelines on how to optimize the problem whenever a priority order can be established between the two objectives are provided. In addition, we present bi-MDCST polynomial cases and theoretical bounds on the search space. Results are reported for four representative test sets.     

On the value of a random minimum spanning tree problem

Suppose we are given a complete graph on n vertices in which the lenghts of the edges are independent identically distributed non-negative random variables. Suppose that their common distribution function F is differentiable at zero and D = F′ (0) > 0 and each edge length has a finite mean and variance. Let L_n be the random variable whose value is the length of the minimum spanning tree in such a graph. Then we will prove the following: lim_{n → ∞}E(L_n) = ζ(3)/D where ζ(3) = Σ_{k = 1}^∞ 1/k³ = 1.202… and for any ε > 0 lim_{n → ∞} Pr(|L_n^?ζ(3)/D|) > ε) = 0.     

Genetic algorithm approach on multi-criteria minimum spanning tree problem

Minimum Spanning Tree (MST) problem is of high importance in network optimization. The multi-criteria MST (mc-MST) is a more realistic representation of the practical problem in the real world, but it is difficult for the traditional network optimization technique to deal with. In this paper, a genetic algorithm (GA) approach is developed to deal with this problem. Without neglecting its network topology, the proposed method adopts the Prüfer number as the tree encoding and applies the Multiple Criteria Decision Making (MCDM) technique and nondominated sorting technique to make the GA search give out all Pareto optimal solutions either focused on the region near the ideal point or distributed all along the Pareto frontier. Compared with the enumeration method of Pareto optimal solution, the numerical analysis shows the efficiency and effectiveness of the GA approach on the mc-MST problem.     

Restricted dynamic programming based neighborhoods for the hop-constrained minimum spanning tree problem

In this paper we develop, study and test new neighborhood structures for the Hop-constrained Minimum Spanning Tree Problem (HMSTP). These neighborhoods are defined by restricted versions of a new dynamic programming formulation for the problem and provide a systematic way of searching neighborhood structures based on node-level exchanges. We have also developed several local search methods that are based on the new neighborhoods. Computational experiments for a set of benchmark instances with up to 80 nodes show that the more elaborate methods produce in a quite fast way, heuristic solutions that are, for all cases, within 2% of the optimum.