Capacitated Minimum Spanning Tree Problem Research Articles

Design of capacitated minimum spanning tree with uncertain cost and demand parameters

The classical Capacitated Minimum Spanning Tree Problem (CMSTP) deals with finding a minimum-cost spanning tree so that the total demand of the vertices in each subtree does not exceed the capacity limitation. In most of the CMSTP models, the edge costs and the demands of the vertices in the network are assumed to be known with certainty. This paper considers the CMSTP model, where the edge costs and/or the demands are only approximately known. A fast approximate reasoning algorithm, which is based on the Esau–Williams savings heuristic and fuzzy logic rules, is proposed. The computational results of the study based on the proposed approach are also reported.

Robust branch-cut-and-price for the Capacitated Minimum Spanning Tree problem over a large extended formulation

This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arborescence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms: powerful new cuts expressed over a very large set of variables are added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very significant improvements over previous algorithms. Several open instances could be solved to optimality.

Enhanced second order algorithm applied to the capacitated minimum spanning tree problem

Given a centralized undirected graph with costs associated with its edges, the capacitated minimum spanning tree problem is to find a minimum cost spanning tree of the given graph, subject to a capacity constraint in all subtrees incident in the central node. As the problem is NP-hard, we propose an enhanced version of the well-known second order algorithm, described in [Karnaugh M. A new class of algorithms for multipoint network optimization. IEEE Transactions on Communications 1976;COM-24:500–5.]. The original version of this algorithm is based on a look-ahead strategy, used for a tentative inclusion of a constraint to the problem, performed in each iteration. In the new enhanced version, we propose the inclusion of look-behind steps, which can be seen as the reverse of the look-ahead procedure. Therefore and using some memory features, the method can continue even when facing the traditional stopping criterion of the original algorithm. Computational experiments showing the effectiveness of the new method on benchmark instances are reported.

Approximation algorithms for the capacitated minimum spanning tree problem and its variants in network design

Given an undirected graph G = ( V,E ) with nonnegative costs on its edges, a root node r V , a set of demands D V with demand v D wishing to route w(v) units of flow (weight) to r , and a positive number k , the Capacitated Minimum Steiner Tree (CMStT) problem asks for a minimum Steiner tree, rooted at r , spanning the vertices in D * { r }, in which the sum of the vertex weights in every subtree connected to r is at most k . When D = V , this problem is known as the Capacitated Minimum Spanning Tree (CMST) problem. Both CMsT and CMST problems are NP-hard. In this article, we present approximation algorithms for these problems and several of their variants in network design. Our main results are the following: ---We present a (³ Á ST + 2)-approximation algorithm for the CMStT problem, where ³ is the inverse Steiner ratio , and Á ST is the best achievable approximation ratio for the Steiner tree problem. Our ratio improves the current best ratio of 2Á ST + 2 for this problem. ---In particular, we obtain (³ + 2)-approximation ratio for the CMST problem, which is an improvement over the current best ratio of 4 for this problem. For points in Euclidean and rectilinear planes, our result translates into ratios of 3.1548 and 3.5, respectively. ---For instances in the plane, under the L p norm, with the vertices in D having uniform weights, we present a nontrivial (7/5Á ST + 3/2)-approximation algorithm for the CMStT problem. This translates into a ratio of 2.9 for the CMST problem with uniform vertex weights in the L p metric plane. Our ratio of 2.9 solves the long-standing open problem of obtaining any ratio better than 3 for this case. ---For the CMST problem, we show how to obtain a 2-approximation for graphs in metric spaces with unit vertex weights and k = 3,4. ---For the budgeted CMST problem, in which the weights of the subtrees connected to r could be up to ± k instead of k (± e 1), we obtain a ratio of ³ + 2/±.

Network flow models for the local access network expansion problem

In the local access network expansion problem we need to expand a given network (with tree topology) due to traffic demand increase. We can expand the network either by increasing the capacity of its edges and/or by installing equipments that concentrate the traffic of several demand points. This problem has received some attention in the literature. Here, we view the problem as an extension of the well-known capacitated minimum spanning tree problem. We adapt two types of flow-based formulations, aggregated and disaggregated formulations and present some valid inequalities to improve the linear programming bounds of the previous formulations. Computational results from a set of tests with 100 and 200 nodes that compare the lower bounds given by the different models, new and old, are reported. Our results demonstrate that the flow-based models are worth trying for these problems.

Least cost heuristic for the delay-constrained capacitated minimum spanning tree problem

This study deals with the Delay-Constrained Capacitated Minimum Spanning Tree (DC-CMST) problem arising in topology discovery of local network. While the traditional Capacitated Minimum Spanning Tree (CMST) problem deals with only traffic capacity constraint of a port of a source node, and Delay-Constrained Minimum Spanning Tree (DCMST) considers only the maximum end-to-end delay constraint, the DC-CMST problem deals with mean network delay and traffic capacity constraints simultaneously. The DC-CMST problem consists of finding a set of minimum cost spanning trees to link end-nodes to a source node which satisfy the traffic requirements at end-nodes and the required mean delay of a network. We formulate the DC-CMST problem and present the Least-Cost DC-CMST Heuristic, which consists of node exchange algorithm, node shift algorithm, and mean delay link capacity algorithm to solve the DC-CMST problem. Results from performance analysis show that the proposed heuristic can produce, in a very short computation time, better results than the previous CMST algorithms modified to solve the DC-CMST problem. The proposed Least-Cost DC-CMST Heuristic can be applied to the topological design of large local networks and to the construction of least cost broadcast and multicast trees for efficient routing algorithms.

Savings based ant colony optimization for the capacitated minimum spanning tree problem

The problem of connecting a set of client nodes with known demands to a root node through a minimum cost tree network, subject to capacity constraints on all links is known as the capacitated minimum spanning tree (CMST) problem. As the problem is NP-hard, we propose a hybrid ant colony optimization (ACO) algorithm to tackle it heuristically. The algorithm exploits two important problem characteristics: (i) the CMST problem is closely related to the capacitated vehicle routing problem (CVRP), and (ii) given a clustering of client nodes that satisfies capacity constraints, the solution is to find a MST for each cluster, which can be done exactly in polynomial time. Our ACO exploits these two characteristics of the CMST by a solution construction originally developed for the CVRP. Given the CVRP solution, we then apply an implementation of Prim's algorithm to each cluster to obtain a feasible CMST solution. Results from a comprehensive computational study indicate the efficiency and effectiveness of the proposed approach.

The capacitated minimum spanning tree problem: On improved multistar constraints

In this paper, we propose a new set of valid inequalities for the Capacitated Minimum Spanning Tree (CMST) problem with general node demands. These inequalities are stronger versions of the well-known multistar constraints (see Hall [Informs J. Comput. 8 (1996) 219]). We also propose and test one lagrangean relaxation scheme that dualizes the new constraints. Our computational results involving instances with up to 80 nodes plus the root node, show that the new inequalities are worth using for random cost and sparse instances. These instances appear to be more difficult to solve than Euclidean ones. We also propose a new arc elimination test for the CMST problem. Besides helping to reduce the size of the instances being solved, the new test improves the performance of the lagrangean relaxation proposed in this paper for the random cost instances.

A Multi-Exchange Neighborhood for Minimum Makespan Parallel Machine Scheduling Problems

We propose new local search algorithms for minimum makespan parallel machine scheduling problems, which perform multiple exchanges of jobs among machines. Inspired by the work of Thompson and Orlin (1989) on cyclic transfer neighborhood structures, we model multiple exchanges of jobs as special disjoint cycles and paths in a suitably defined improvement graph, by extending definitions and properties introduced in the context of vehicle routing problems (Thompson and Psaraftis, 1993) and of the capacitated minimum spanning tree problem (Ahuja et al., 2001). Several algorithms for searching the neighborhood are suggested.

The capacitated minimum spanning tree problem: revisiting hop-indexed formulations

The capacitated minimum spanning tree problem is to find a minimum spanning tree with an additional cardinality constraint on the number of nodes in any subtree off a given root node. In this paper we propose two improvements on a previous cutting-plane method proposed by Gouveia and Martins (Networks 35(1) (2000) 1) namely, a new set of inequalities that can be seen as hop-indexed generalization of the well known generalized subtour elimination (GSE) constraints and an improved separation heuristic for the original set of GSE constraints. Computational results show that the inclusion of the new separation routine and the inclusion of the new inequalities in Gouveia and Martins’ iterative method (see (Networks 35(1) (2000) 1)) produce improvements on previously reported lower bounds. Furthermore, with the improved method, several of previous unsolved instances have been solved to optimality.

The capacitated minimum spanning tree problem: revisiting hop-indexed formulations

The capacitated minimum spanning tree problem is to find a minimum spanning tree with an additional cardinality constraint on the number of nodes in any subtree off a given root node. In this paper we propose two improvements on a previous cutting-plane method proposed by Gouveia and Martins (Networks 35(1) (2000) 1) namely, a new set of inequalities that can be seen as hop-indexed generalization of the well known generalized subtour elimination (GSE) constraints and an improved separation heuristic for the original set of GSE constraints. Computational results show that the inclusion of the new separation routine and the inclusion of the new inequalities in Gouveia and Martins’ iterative method (see (Networks 35(1) (2000) 1)) produce improvements on previously reported lower bounds. Furthermore, with the improved method, several of previous unsolved instances have been solved to optimality.

A composite very large-scale neighborhood structure for the capacitated minimum spanning tree problem

The capacitated minimum spanning tree (CMST) problem is to find a minimum cost spanning tree in a network where nodes have specified 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 (2001) 71) proposed two very large-scale neighborhood search algorithms for the capacitated minimum spanning tree problem. Their first 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 different 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.

Multistars and directed flow formulations

AbstractThe Capacitated Minimum Spanning Tree Problem seeks a least‐cost spanning tree subject to a bound imposed on the number of nodes in each subtree pending from a given root node. Araque et al. (Technical Report SOR‐90‐12, Princeton University, 1990) introduced several classes of facet‐defining inequalities for the undirected version of the problem, most of which have straightforward analogs to the directed version and are also facet‐defining in that case (see Zhang, Master's thesis, 1993). The multistar constraints are one such class. Gouveia [Telecommun Syst 1 (1993), 51–56] showed that a directed flow formulation gives a polynomial representation of the class of directed multistar constraints. This equivalence shows how to obtain a polynomial‐time separation algorithm for this class of inequalities. In this paper, we show that the previous equivalence result implies that we can also separate in polynomial time the exponential‐sized class of undirected multistar constraints. We also show that “using a directed model” plays a key role in obtaining a polynomial‐time separation algorithm for this class of inequalities, that is, using a directed flow model seems to be crucial for obtaining a polynomial‐time separation algorithm for the class of undirected multistar constraints. © 2002 Wiley Periodicals, Inc.

A simple enhancement of the Esau–Williams heuristic for the capacitated minimum spanning tree problem

The Esau–Williams algorithm is one of the best known heuristics for the capacitated minimum spanning tree problem. This paper describes a simple enhancement of this heuristic. On benchmark test problems, the modified method does not result in much larger computation times and almost always produces lower solution values than does the Esau–Williams algorithm. The improvements are often significant.

A simple enhancement of the Esau–Williams heuristic for the capacitated minimum spanning tree problem

A simple enhancement of the Esau–Williams heuristic for the capacitated minimum spanning tree problem

Snow Emergency Vehicle Routing with Route Continuity Constraints

New results are presented from continuing research dealing with development of a decision support system to assist the Maryland State Highway Administration Office of Maintenance staff in designing snow emergency routes for Calvert County, Maryland. By taking into account some of the more realistic constraints, an attempt is made to solve two problems. One involves minimizing the total number of trucks, and the second one involves minimizing the total deadhead distance given the number of trucks. The two problems do not result in identical solutions in general. The first problem is formulated as the well-known capacitated minimum spanning tree (CMST) problem. The second problem is also formulated as a CMST problem but with an approximate function for total deadhead distance. Some application results are also reported, which indicate that using such a system can achieve improvements in service and savings in operational costs.

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.

Very large‐scale neighborhood search

AbstractNeighborhood search algorithms are often the most effective approaches available for solving partitioning problems, a difficult class of combinatorial optimization problems arising in many application domains including vehicle routing, telecommunications network design, parallel machine scheduling, location theory, and clustering. A critical issue in the design of a neighborhood search algorithm is the choice of the neighborhood structure, that is, the manner in which the neighborhood is defined. Currently, the two‐exchange neighborhood is the most widely used neighborhood for solving partitioning problems. The paper describes the cyclic exchange neighborhood, which is a generalization of the two‐exchange neighborhood in which a neighbor is obtained by performing a cyclic exchange. The cyclic exchange neighborhood has substantially more neighbors compared to the two‐exchange neighborhood. This paper outlines a network optimization based methodology to search the neighborhood efficiently and presents a proof of concept by applying it to the capacitated minimum spanning tree problem, an important problem in telecommunications network design.

Very large-scale neighborhood search

Neighborhood search algorithms are often the most effective approaches available for solving partitioning problems, a difficult class of combinatorial optimization problems arising in many application domains including vehicle routing, telecommunications network design, parallel machine scheduling, location theory, and clustering. A critical issue in the design of a neighborhood search algorithm is the choice of the neighborhood structure, that is, the manner in which the neighborhood is defined. Currently, the two-exchange neighborhood is the most widely used neighborhood for solving partitioning problems. The paper describes the cyclic exchange neighborhood, which is a generalization of the two-exchange neighborhood in which a neighbor is obtained by performing a cyclic exchange. The cyclic exchange neighborhood has substantially more neighbors compared to the two-exchange neighborhood. This paper outlines a network optimization based methodology to search the neighborhood efficiently and presents a proof of concept by applying it to the capacitated minimum spanning tree problem, an important problem in telecommunications network design.

Heuristic procedure neural networks for the CMST problem

Combinatorial optimization problems are by nature very difficult to solve, and the Capacitated Minimum Spanning Tree problem is one such problem. Much work has been done in the management sciences to develop heuristic solution procedures that suboptimally solve large instances of the Capacitated Minimum Spanning Tree problem in a reasonable amount of time. The Capacitated Minimum Spanning Tree problem is used in this paper to develop and demonstrate a hybrid neural network methodology that incorporates heuristic methods into the neural network topological design. The heuristic procedure is embedded into the neural network topological design, and an iterative improvement process is performed using the neural network. The semi-relaxed energy function of the problem is used to develop a neural network weight adjustment procedure that modifies the problem costs. In three-quarters (75%) of our experiments, the hybrid neural networks produced better results than any of the traditional procedures tested. Scope and purpose For solving combinatorial optimization problems, neural networks have traditionally been outperformed by traditional heuristic techniques developed specifically for the problem in question. This research is a step toward integrating the problem specific knowledge embedded in a traditional heuristic with the adaptive capabilities of neural networks. This is accomplished by creating a neural network topological design that embeds the steps of the traditional heuristic. The neural network learning then improves upon the performance of the embedded heuristic by modifying the neural weights attached to the embedded heuristic.