A two-level solution approach for solving the generalized minimum spanning tree problem (original) (raw)
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HEURISTIC ALGORITHMS FOR THE GENERALIZED MINIMUM SPANNING TREE PROBLEM
We consider a generalization of the classical minimum spanning tree problem called the generalized minimum spanning tree problem and denoted by GMST problem. The aim of this paper is to present an exact exponential time algorithm for the GMST problem as well three heuristic algorithms, two of them based on Prim's and Kruskal's algorithms for the minimum spanning tree problem and one based on the local global approach. These three algorithms are implemented and computational results are reported for many instances of the problem.
Raf. J. of Comp. & Math ’ s. , 2011
The paper tackled a survey of two optimization methods to study spanning tree problem by modifying the spanning tree problem to generate all of possible solutions in undirected tree graph with simulated annealing algorithm and ant colony optimization algorithm. These algorithms are two of the optimization methods to find optimal solution from many of solutions in search space. A program is written in MATLAB 6.5 language to simulate these two algorithms with spanning tree problem. The experimental results in this paper show the effectiveness and easy implementation of each algorithm to find optimal solution, and to perform significantly better than the manual method.
Maǧallaẗ al-rāfidayn li-ʿulūm al-ḥāsibāt wa-al-riyāḍiyyāẗ, 2011
The paper tackled a survey of two optimization methods to study spanning tree problem by modifying the spanning tree problem to generate all of possible solutions in undirected tree graph with simulated annealing algorithm and ant colony optimization algorithm. These algorithms are two of the optimization methods to find optimal solution from many of solutions in search space. A program is written in MATLAB 6.5 language to simulate these two algorithms with spanning tree problem. The experimental results in this paper show the effectiveness and easy implementation of each algorithm to find optimal solution, and to perform significantly better than the manual method.
2009
In this paper, a new hybrid genetic algorithmknown as HGA -is proposed for solving the Bounded Diameter Minimum Spanning Tree (BDMST) problem. We experiment with HGA on two sets of benchmark problem instances, both Euclidean and Non-Euclidean. On the Euclidean problem instances, HGA is shown to outperform the previous best two Genetic Algorithms (GAs) reported in the BDMST literature, while on the Non-Euclidean problem instance, HGA performs comparably with these two GAs.
A new hybrid Genetic Algorithm for solving the Bounded Diameter Minimum Spanning Tree problem
2008
In this paper, a new hybrid genetic algorithmknown as HGA -is proposed for solving the Bounded Diameter Minimum Spanning Tree (BDMST) problem. We experiment with HGA on two sets of benchmark problem instances, both Euclidean and Non-Euclidean. On the Euclidean problem instances, HGA is shown to outperform the previous best two Genetic Algorithms (GAs) reported in the BDMST literature, while on the Non-Euclidean problem instance, HGA performs comparably with these two GAs.
Proceedings of the fourteenth international conference on Genetic and evolutionary computation conference - GECCO '12, 2012
We consider the recently proposed concept of enhancing an evolutionary algorithm (EA) with a complete solution archive. It stores evaluated solutions during the optimization in order to detect duplicates and to efficiently transform them into yet unconsidered solutions. For this approach we introduce the so-called bounding extension in order to identify and prune branches in the trie-based archive which only contain inferior solutions. This extension enables the EA to concentrate the search on promising areas of the solution space. Similarly to the classical branch-and-bound technique, bounds are obtained via primal and dual heuristics. As an application we consider the generalized minimum spanning tree problem where we are given a graph with nodes partitioned into clusters and exactly one node from each cluster must be connected in the cheapest way. As the EA uses operators based on two dual representations, we exploit two corresponding tries that complement each other. Test results on TSPlib instances document the strength of this concept and that it can compete with the leading metaheuristics for this problem in the literature.
The generalized minimum spanning tree problem: Polyhedral analysis and branch-and-cut algorithm
Networks, 2004
This article presents a branch-and-cut algorithm for the Generalized Minimum Spanning Tree Problem (GMSTP). Given an undirected graph whose vertex set is partitioned into clusters, the GMSTP consists of determining a minimum cost tree including exactly one vertex per cluster. Applications of the GMSTP are encountered in telecommunications. An integer linear programming formulation is presented and new classes of valid inequalities are developed, several of which are proved to be facet defining. A branch-and-cut algorithm and a tabu search heuristic are developed. Extensive computational experiments show that instances involving up to 160 or 200 vertices can be solved to optimality, depending on whether edge costs are Euclidean or random.
A Lagrangean Heuristic For The Degree Constrained Minimal Spanning Tree Problem
Review of Business Information Systems (RBIS), 2010
In this paper we present a new heuristic procedure to solve the degree constrained minimal spanning tree problem. This procedure uses Lagrangian relaxation of the integer programming formulation of the problem to get a lower bound for the optimal objective function value. A subgradient optimization method is used to find multipliers that give good lower bounds. A branch exchange procedure is used after each iteration of the subgradient optimization to generate a feasible solution from an infeasible Lagrangean solution. Computational results are given for problems with up to 300 nodes. The heuristic procedure presented here gives optimal solutions in most instances. For problem sets that were not solved optimally, the gap between the lower bound and the feasible solution was less than 10-2 percent.
Improved heuristics for the bounded-diameter minimum spanning tree problem
Soft Computing, 2007
Given an undirected, connected, weighted graph and a positive integer k, the bounded-diameter minimum spanning tree (BDMST) problem seeks a spanning tree of the graph with smallest weight, among all spanning trees of the graph, which contain no path with more than k edges. In general, this problem is NP-Hard for 4 ≤ k < n − 1, where n is the number of vertices in the graph. This work is an improvement over two existing greedy heuristics, called randomized greedy heuristic (RGH) and centre-based tree construction heuristic (CBTC), and a permutation-coded evolutionary algorithm for the BDMST problem. We have proposed two improvements in RGH/CBTC. The first improvement iteratively tries to modify the bounded-diameter spanning tree obtained by RGH/CBTC so as to reduce its cost, whereas the second improves the speed. We have modified the crossover and mutation operators and the decoder used in permutation-coded evolutionary algorithm so as to improve its performance. Computational results show the effectiveness of our approaches. Our approaches obtained better quality solutions in a much shorter time on all test problem instances considered.