A constrained minimum spanning tree problem

In the classical general framework of the minimum spanning tree problem for a weighted graph we consider the case in which a predetermined vertex has a certain fixed degree. In other words, given a weighted graph G, one of its vertices v0 and a positive integer k, we consider the problem of finding the minimum spanning tree of G in which the vertex v0 has degree k, that is the number of edges coming out of v0. We recall that among the various methods for the solution of the unconstrained problem an efficient way to find the minimum spanning tree is based on the simple procedure of choosing one after the other an edge of minimum weight that has not be chosen yet and does not create cycles if added to the previously chosen edges. This technique is known as the \u201cgreedy algorithm\u201d. There are problems for which the greedy algorithm works and problems for which it does not. We prove that for the solution of the one degree constrained minimum spanning tree problem the classical greedy algorithm finds a right solution

The Modified CW1 Algorithm for the Degree Restricted Minimum Spanning Tree Problem

Given edge weighted graph G (all weights are non-negative), The Degree Constrained Minimum Spanning Tree Problem is concerned with finding the minimum weight spanning tree T satisfying specified degree restrictions on the vertices. This problem arises naturally in communication networks where the degree of a vertex represents the number of line interfaces available at a terminal (center). The applications of the Degree Constrained Minimum Spanning Tree problems that may arise in real-life include: the design of telecommunication, transportation, and energy networks. It is also used as a subproblem in the design of networks for computer communication, transportation, sewage and plumbing. Since, apart from some trivial cases, the problem is computationally difficult (NP-complete), a number of heuristics have been proposed. In this paper we will discuss the modification of CW1 Algorithm that already proposed by Wamiliana and Caccetta (2003). The results on540 random table problems will be discussed

The Modified CW1 Algorithm For The Degree Restricted Minimum Spanning Tree Problem

Given edge weighted graph G (all weights are non-negative), The Degree Constrained Minimum Spanning Tree Problem is concerned with finding the minimum weight spanning tree T satisfying specified degree restrictions on the vertices. This problem arises naturally in communication networks where the degree of a vertex represents the number of line interfaces available at a terminal (center). The applications of the Degree Constrained Minimum Spanning Tree problems that may arise in real-life include: the design of telecommunication, transportation, and energy networks. It is also used as a subproblem in the design of networks for computer communication, transportation, sewage and plumbing. Since, apart from some trivial cases, the problem is computationally difficult (NP-complete), a number of heuristics have been proposed. In this paper we will discuss the modification of CW1 Algorithm that already proposed by Wamiliana and Caccetta (2003). The results on540 random table problems will be discussed

Determining hop-constrained spanning trees with repetitive heuristics, Journal of Telecommunications and Information Technology, 2007, nr 4

The hop-constrained minimum spanning tree problem is the problem of determining a rooted spanning tree of minimum cost in which each path from the root node to any other node contains at most H hops or edges. This problem relates to the design of centralized tree networks with quality of service requirements (in telecommunications) and has a close relation with other tree problems. In this paper we investigate the adaptation of some well-known “repetitive” heuristics used for the capacitated minimum spanning tree problem to the hop-constrained minimum spanning tree problem and investigate some simple look ahead mechanisms for enhancing the quality of a savings heuristic. Computational results for a set of benchmark tests with up to 80 nodes are presented

A polyhedral study of the diameter constrained minimum spanning tree problem

This paper provides a first polyhedral study of the diameter constrained minimum spanning tree problem (DMSTP). We introduce a new set of inequalities, the circular-jump inequalities which strengthen the well-known jump inequalities. These inequalities are further generalized in two ways: either by increasing the number of partitions defining a jump, or by combining jumps with cutsets. Most of the proposed new inequalities are shown to define facets of the DMSTP polytope under very mild conditions. Currently best known lower bounds for the DMSTP are obtained from an extended formulation on a layered graph using the concept of central nodes/edges. A subset of the new families of inequalities is shown to be not implied by this layered graph formulation

Network Flow Models for Designing Diameter-Constrained Minimum Spanning and Steiner Trees

The Diameter-Constrained Minimum Spanning Tree Problem seeks a least cost spanning tree subject to a (diameter) bound imposed on the number of edges in the tree between any node pair. A traditional multicommodity flow model with a commodity for every pair of nodes was unable to solve a 20-node and 100-edge problem after one week of computation. We formulate the problem as a directed tree from a selected central node or a selected central edge. Our model simultaneously finds a central node or a central edge and uses it as the source for the commodities in a directed multicommodity flow model with hop constraints. The new model has been able to solve the 20-node, 100-edge instance to optimality after less than four seconds. We also present model enhancements when the diameter bound is odd (these situations are more difficult). We show that the linear programming relaxation of the best formulations discussed in this paper always give an optimal integer solution for two special, polynomially-solvable cases of the problem. We also examine the Diameter Constrained Minimum Steiner Tree problem. We present computational experience in solving problem instances with up to 100 nodes and 1000 edges. The largest model contains more than 250,000 integer variables and more than 125,000 constraints

Analisis dan Penerapan Kombinasi Algoritma Lagrangian dan Particle Swarm Optimization untuk Pemecahan Masalah Degree-Constrained Minimum Spanning Tree

ABSTRAKSI: Permasalahan pencarian pohon merentang dengan cost minimum pada sebuah graf sering muncul terutama dengan adanya beberapa batasan seperti batasan derajat pada simpul. Permasalahan perancangan jaringan seperti pada jaringan telekomunikasi misalnya, terdapat batasan derajat pada nodes. Terdapat beberapa pendekatan yang dilakukan untuk menyelesaikan permasalahan tersebut, salah satunya pada penelitian ini digunakan pendekatan metode PSO yang digabungkan dengan fungsi lagrange.Kata Kunci : particle swarm optimization, lagrange, pohon merentang minimum, batasan derajatABSTRACT: For some problems a minimum spanning tree has constraints. The constraints are based on the problem. In the network design for telecommunication for example, there is degree-constrained for the nodes. We call the problem degree-constrained minimum spanning tree (DCMST) problem. There are some heuristics to solve the DCMST problem. In this paper the author uses a combinatorial lagrangian function and particle swarm optimization (COLAPSO).Keyword: particle swarm optimization, lagrangian, minimum spanning tree, degree-constraine

Evolutionary computation of forests with Degree- and Role-Constrained Minimum Spanning Trees

Finding the degree-constrained minimum spanning tree (DCMST) of a graph is a widely studied NP-hard problem. One of its most important applications is network design. Here we deal with a new variant of the DCMST problem, which consists of finding not only the degree- but also the role-constrained minimum spanning tree (DRCMST), i.e., we add constraints to restrict the role of the nodes in the tree to root, intermediate or leaf node. Furthermore, we do not limit the number of root nodes to one, thereby, generally, building a forest of DRCMSTs. The modeling of network design problems can benefit from the possibility of generating more than one tree and determining the role of the nodes in the network. We propose a novel permutation-based representation to encode these forests. In this new representation, one permutation simultaneously encodes all the trees to be built. We simulate a wide variety of DRCMST problems which we optimize using eight different evolutionary computation algorithms encoding individuals of the population using the proposed representation. The algorithms we use are: estimation of distribution algorithm, generational genetic algorithm, steady-state genetic algorithm, covariance matrix adaptation evolution strategy, differential evolution, elitist evolution strategy, non-elitist evolution strategy and particle swarm optimization. The best results are for the estimation of distribution algorithms and both types of genetic algorithms, although the genetic algorithms are significantly faster. -------------------------------------------------------------------------------------------------- Trabajo publicado en: Antón Sánchez, Laura; Bielza Lozoya, Maria Concepcion y Larrañaga Múgica, Pedro (2017). Network Design through Forests with Degree- and Role-constrained Minimum Spanning Trees. "Journal of Heuristics ", v. 23 (n. 1); pp. 31-51. ------------------------------------------

A dandelion-encoded evolutionary algorithm for the delay-constrained capacitated minimum spanning tree problem

This paper proposes an evolutionary algorithm with Dandelion-encoding to tackle the Delay-Constrained Capacitated Minimum Spanning Tree (DC-CMST) problem. This problem has been recently proposed, and consists of finding several broadcast trees from a source node, jointly considering traffic and delay constraints in trees. A version of the problem in which the source node is also included in the optimization process is considered as well in the paper. The Dandelion code used in the proposed evolutionary algorithm has been recently proposed as an effective way of encoding trees in evolutionary algorithms. Good properties of locality has been reported on this encoding, which makes it very effective to solve problems in which the solutions can be expressed in form of trees. In the paper we describe the main characteristics of the algorithm, the implementation of the Dandelion-encoding to tackled the DC-CMST problem and a modification needed to include the source node in the optimization. In the experimental section of this article we compare the results obtained by our evolutionary with that of a recently proposed heuristic for the DC-CMST. the Least Cost (LC) algorithm. We show that our Dandelion-encoded evolutionary algorithm is able to obtain better results that the LC in all the instances tackled. (C) 2008 Elsevier B.V. All rights reserved