A filter-and-fan algorithm for the capacitated minimum spanning tree problem (original) (raw)
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A greedy heuristic for the capacitated minimum spanning tree problem
Journal of the Operational Research Society
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.
A hybrid VNS algorithm for solving the multi-level capacitated minimum spanning tree problem
Electronic Notes in Discrete Mathematics
This work addresses the multi-level capacitated minimum spanning tree (MLCMST) problem. It consists of finding a minimal cost spanning tree such that the flow to be transferred from a central node (root) to the other nodes is bounded by the edge capacities. In this paper, a hybrid algorithm, combining the Variable Neighborhood Search (VNS) metaheuristic and one mathematical programming formulation of the literature, is used for solving it. The formulation is used to give an initial solution to VNS. Five neighborhoods are used for exploring the solution space. Results show that the VNS is able to improve the initial solutions and to obtain small gap solutions for all instance sets.
Journal of Heuristics, 2008
We propose a GRASP using an hybrid heuristic-subproblem optimization approach for the Multi-Level Capacitated Minimum Spanning Tree (MLCMST) problem. The motivation behind such approach is that to evaluate moves rearranging the configuration of a subset of nodes may require to solve a smaller-sized MLCMST instance. We thus use heuristic rules to define, in both the construction and the local search phases, subproblems which are in turn solved exactly by employing an integer programming model. We report numerical results obtained on benchmark instances from the literature, showing the approach to be competitive in terms of solution quality. The proposed GRASP have in fact improved the best known upper bounds for almost all of the considered instances.
Branch-and-cut and hybrid local search for the multi-level capacitated minimum spanning tree problem
Networks, 2012
We propose algorithms to compute tight lower bounds and high quality upper bounds for the Multi-Level Capacitated Minimum Spanning Tree problem. We first develop a branch-and-cut algorithm, introducing some new features: (i) the exact separation of cuts corresponding to some master equality polyhedra found in the formulation; (ii) the separation of Fenchel cuts, solving LPs considering all the possible solutions restricted to small portions of the graph. We then use that branch-and-cut within a GRASP that performs moves by solving to optimality subproblems corresponding to partial solutions. The computational experiments were conducted on 450 benchmark instances from the literature. Numerical results show improved best known upper bounds for almost all instances that could not be solved to optimality.
VNS and second order heuristics for the min-degree constrained minimum spanning tree problem
Computers & Operations Research, 2009
Given an undirected graph with weights associated with its edges, the min-degree constrained minimum spanning tree (md-MST) problem consists in finding a minimum spanning tree of the given graph, imposing minimum degree constraints in all nodes except the leaves. This problem was recently proposed in Almeida et al. [Min-degree constrained minimum spanning tree problem: Complexity, proprieties and formulations. Operations Research Center, University of Lisbon, Working-paper no. 6; 2006], where its theoretical complexity was characterized and showed to be NP-hard. The present paper discusses variable neighborhood search (VNS) metaheuristics addressing the md-MST. A so-called enhanced version of a second order (ESO) repetitive technique is also considered, in order to guide the search in both shaking and improvement phases of the VNS method. A Kruskal based greedy heuristic adapted to the md-MST is also presented, being used within the ESO framework. VNS randomized methodologies are also discussed. These are the first heuristics to the md-MST ever proposed in the literature. Computational experiments are conducted on instances adapted from benchmark ones used in the context of the well-known degree constraint minimum spanning tree problem. These experiments have shown that randomized VNS methods enclosing an ESO algorithm can produce very interesting results. In particular, that a simpler VNS randomized methodology might be taken into account when very high dimensional instances are under consideration.
Operations Research Letters, 2003
The capacitated minimum spanning tree (CMST) problem is to ÿnd a minimum cost spanning tree in a network where nodes have speciÿed demands, with an additional capacity constraints on the subtrees incident to a given source node s. The capacitated minimum spanning tree problem arises as an important subproblem in many telecommunication network design problems. In a recent paper, Ahuja et al. (Math. Program. 91 proposed two very large-scale neighborhood search algorithms for the capacitated minimum spanning tree problem. Their ÿrst node-based neighborhood structure is obtained by performing multi-exchanges involving several trees where each tree contributes a single node. Their second tree-based neighborhood structure is obtained by performing multi-exchanges where each tree contributes a subtree. The computational investigations found that node-based multi-exchange neighborhood gives the best performance for the homogenous demand case (when all nodes have the same demand), and the tree-based multi-exchange neighborhood gives the best performance for the heterogeneous demand case (when nodes may have di erent demands). In this paper, we propose a composite neighborhood structure that subsumes both the node-based and tree-based neighborhoods, and outperforms both the previous neighborhood search algorithms for solving the capacitated minimum spanning tree problem on standard benchmark instances. We also develop improved dynamic programming based exact algorithms for searching the composite neighborhood.
We propose efficient algorithms to compute tight lower bounds and high quality upper bounds for the Multi-Level Capacitated Minimum Spanning Tree problem. We first develop a branch-andcut algorithm for the problem. This algorithm is able to solve instances of medium size and to provide tight lower bounds for larger ones. We then use the branch-and-cut within GRASP to evaluate subproblems during the search. The computational experiments conducted have improved best known lower and upper bounds on benchmark instances.
A branch and bound algorithm for the capacitated minimum spanning tree problem
Networks, 1993
Given an undirected graph G = (N, E ) with a cost associated with each edge C: E R , and a demand associated with each node A: N + R,. A special node is designated as the center. The capacitated minimum spanning tree (CMST) problem is to find a minimum spanning tree for graph G such that the sum of demands on each branch stem from the center does not exceed a given capacity. The CMST problem has many applications in network design, centralized telecommunications, and vehicle routing. In this paper, we present a new formulation and a full optimization algorithm by branch and bound. The lower bounds are generated by Lagrangean relaxation with tightening constraints. Computational results based upon the methodology presented are shown. 0 7993 by John Wiley & Sons, Inc.
Multi-exchange neighborhood structures for the capacitated minimum spanning tree problem
Mathematical Programming, 2001
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.
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.