Generalized Minimum Spanning Tree Problem Research Articles

The multicolored graph realization problem

We introduce the multicolored graph realization problem (MGR). The input to this problem is a colored graph (G,φ), i.e., a graph G together with a coloring φ on its vertices. We associate each colored graph (G,φ) with a cluster graph (Gφ) in which, after collapsing all vertices with the same color to a node, we remove multiple edges and self-loops. A set of vertices S is multicolored when S has exactly one vertex from each color class. The MGR problem is to decide whether there is a multicolored set S so that, after identifying each vertex in S with its color class, G[S] coincides with Gφ.The MGR problem is related to the well-known class of generalized network problems, most of which are NP-hard, like the generalized Minimum Spanning Tree problem. The MGR is a generalization of the multicolored clique problem, which is known to be W[1]-hard when parameterized by the number of colors. Thus, MGR remains W[1]-hard, when parameterized by the size of the cluster graph. These results imply that the MGR problem is W[1]-hard when parameterized by any graph parameter on Gφ, among which lies treewidth. Consequently, we look at the instances of the problem in which both the number of color classes and the treewidth of Gφ are unbounded. We consider three natural such graph classes: chordal graphs, convex bipartite graphs and 2-dimensional grid graphs. We show that MGR is NP-complete when Gφ is either chordal, biconvex bipartite, complete bipartite or a 2-dimensional grid. Our reductions show that the problem remains hard even when the maximum number of vertices in a color class is 3. In the case of the grid, the hardness holds even for graphs with bounded degree. We provide a complexity dichotomy with respect to cluster size.

A rapid learning automata-based approach for generalized minimum spanning tree problem

Generalized minimum spanning tree problem, which has several real-world applications like telecommunication network designing, is related to combinatorial optimization problems. This problem belongs to the NP-hard class and is a minimum tree on a clustered graph spanning one node from each cluster. Although exact and metaheuristic algorithms have been applied to solve the problems successfully, obtaining an optimal solution using these approaches and other optimization tools has been a challenge. In this paper, an attempt is made to achieve a sub-optimal solution using a network of learning automata (LA). This algorithm assigns an LA to every cluster so that the number of actions is the same as that of nodes in the corresponding cluster. At each iteration, LAs select one node from their clusters. Then, the weight of the constructed generalized spanning tree is considered as a criterion for rewarding or penalizing the selected actions. The experimental results on a set of 20 benchmarks of TSPLIB demonstrate that the proposed approach is significantly faster than the other mentioned algorithms. The results indicate that the new algorithm is competitive in terms of solution quality.

A two-level solution approach for solving the generalized minimum spanning tree problem

In this paper, we are addressing the generalized minimum spanning tree problem, denoted by GMSTP, which is a variant of the classical minimum spanning tree (MST) problem. The main characteristic of this problem is the fact that the vertices of the graph are partitioned into a given number of clusters and we are looking for a minimum-cost tree spanning a subset of vertices which includes exactly one vertex from each cluster. We describe a two-level solution approach for solving the GMSTP obtained by decomposing the problem into two logical and natural smaller subproblems: an upper-level (global) subproblem and a lower-level (local) subproblem and solving them separately. The goal of the first subproblem is to determine (global) trees spanning the clusters using a genetic algorithm with a diploid representation of the individuals, while the goal of the second subproblem is to determine the best tree (w.r.t. cost minimization), for the above mentioned global trees, spanning exactly one vertex from each cluster. The second subproblem is solved optimally using dynamic programming. Extensive computational results are reported and discussed for an often used set of benchmark instances. The obtained results show an improvement in the quality of the achieved solutions, and demonstrate the efficiency of our approach compared to the existing methods from the literature.

Automatically Produced Algorithms for the Generalized Minimum Spanning Tree Problem

The generalized minimum spanning tree problem consists of finding a minimum cost spanning tree in an undirected graph for which the vertices are divided into clusters. Such spanning tree includes only one vertex from each cluster. Despite the diverse practical applications for this problem, the NP-hardness continues to be a computational challenge. Good quality solutions for some instances of the problem have been found by combining specific heuristics or by including them within a metaheuristic. However studied combinations correspond to a subset of all possible combinations. In this study a technique based on a genotype-phenotype genetic algorithm to automatically construct new algorithms for the problem, which contain combinations of heuristics, is presented. The produced algorithms are competitive in terms of the quality of the solution obtained. This emerges from the comparison of the performance with problem-specific heuristics and with metaheuristic approaches.

A multi-operator genetic algorithm for the generalized minimum spanning tree problem

The generalized minimum spanning tree problem, with applications in the field of communication networks, is a computational challenge due essentially to its NP-hardness. The problem consists of finding a minimum cost spanning tree in an undirected graph whose vertices are grouped in clusters, such that the spanning tree contains only one vertex of each cluster. The algorithms that have provided the best results still do not optimally solve all instances in the literature. One of the most widely studied approaches to the problem is the use of genetic algorithms that, in all cases, use only single operators for crossover and mutation, disregarding the potential synergy of multi-operators. We present a multi-operator genetic algorithm of the genotype-phenotype class, in which the genotype is a chain of integers that represents a cluster's selected vertex. Therefore, the phenotype is a minimum cost spanning tree that is generated by means of Kruskal's algorithm and joins the vertices selected from each cluster. Two operators are used for crossover and five for mutation, three of which are local search operators. The performance of the resultant algorithm is evaluated using the most challenging instances in the literature, the results of which are compared with those of other mono-operator genetic algorithms and with the best existing results. With the 101 instances that are considered, an average error of 0.0142% is achieved, and in 83 instances, the best solution cost is obtained. Such performance is due both to the synergistic effect produced among the operators and the mutation operators working as local searches. Additionally, the results suggest that for many other combinatorial optimization problems, which have been addressed with a genetic algorithm, better results could possibly be obtained simply by using a greater number of variation operators.

Generalized minimum spanning tree games

The minimum-cost spanning tree game is a special class of cooperative games defined on a graph with a set of vertices and a set of edges, where each player owns a vertex. Solutions of the game represent ways to distribute the total cost of a minimum-cost spanning tree among all the players. When the graph is partitioned into clusters, the generalized minimum spanning tree problem is to determine a minimum-cost tree including exactly one vertex from each cluster. This paper introduces the generalized minimum spanning tree game and studies some properties of this game. The paper also describes a constraint generation algorithm to calculate a stable payoff distribution and presents computational results obtained using the proposed algorithm.

A GRASP-based approach to the generalized minimum spanning tree problem

Given a multipartite graph G the generalized minimum spanning tree problem is to find a tree of minimal cost that includes a vertex from each part. This paper proposes several versions of the GRASP metaheuristic for the problem. The GRASP approach is based on constructive heuristics as well as on additional improvement mechanisms such as path-relinking and iterated local search. Several computational experiments are performed over a set of existing instances. A cut generation algorithm is proposed that is able to find lower bounds, based on a formulation for Steiner’s problem in directed graphs. The computational results show that the best versions of the GRASP approach use improvement mechanisms. The solutions found are better than most of the known solutions in the literature and require significantly less computer time. Furthermore, a set of rules is defined for pre-processing the instances, based on the Bottleneck distance concept. Using those rules, it was possible to reduce the size of the instances to an average of 14% of the number of edges in relation to the original graphs.

Generalized network design polyhedra

AbstractIn recent years, there has been an increased literature on so‐called generalized network design problems (GNDPs), such as the generalized minimum spanning tree problem and the generalized traveling salesman problem. In a GNDP, the node set of a graph is partitioned into “clusters,” and the feasible solutions must contain one node from each cluster. Up to now, the polyhedra associated with different GNDPs have been studied independently. The purpose of this article is to show that it is possible, to a certain extent, to derive polyhedral results for all GNDPs simultaneously. Along the way, we point out some interesting connections to other polyhedra, such as the quadratic semiassignment polytope and the boolean quadric polytope. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011

Minimum Spanning Tree Problem with Label Selection

In this paper, we study the minimum spanning tree problem with label selection, that is, the problem of finding a minimum spanning tree of a vertex-labeled graph where the weight of each edge may vary depending on the selection of labels of vertices at both ends. The problem is especially important as the application to mathematical OCR. It is shown that the problem is NP-hard. However, for the application to mathematical OCR, it is sufficient to deal with only graphs with small tree-width. In this paper, a linear-time algorithm for series-parallel graphs is presented. Since the minimum spanning tree problem with label selection is closely related to the generalized minimum spanning tree problem, their relation is discussed.

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.

Combining variable neighborhood search with integer linear programming for the generalized minimum spanning tree problem

We consider the generalized version of the classical Minimum Spanning Tree problem where the nodes of a graph are partitioned into clusters and exactly one node from each cluster must be connected. We present a Variable Neighborhood Search (VNS) approach which uses three different neighborhood types. Two of them work in complementary ways in order to maximize search effectivity. Both are large in the sense that they contain exponentially many candidate solutions, but efficient polynomial-time algorithms are used to identify best neighbors. For the third neighborhood type we apply Mixed Integer Programming to optimize local parts within candidate solution trees. Tests on Euclidean and random instances with up to 1280 nodes indicate especially on instances with many nodes per cluster significant advantages over previously published metaheuristic approaches.

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.

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 \({\cal{NP}}\)-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 geometric generalized minimum spanning tree problem with grid clustering

This paper is concerned with a special case of the generalized minimum spanning tree problem. The problem is defined on an undirected graph, where the vertex set is partitioned into clusters, and non-negative costs are associated with the edges. The problem is to find a tree of minimum cost containing at least one vertex in each cluster. We consider a geometric case of the problem where the graph is complete, all vertices are situated in the plane, and Euclidean distance defines the edge cost. We prove that the problem is strongly NP-hard even in the case of a special structure of the clustering called grid clustering. We construct an exact exponential time dynamic programming algorithm and, based on this dynamic programming algorithm, we develop a polynomial time approximation scheme for the problem with grid clustering.

Upper and lower bounding strategies for the generalized minimum spanning tree problem

We address the generalized minimum spanning tree problem (GMST) which requires spanning at least one vertex out of every set of disjoint vertices in a graph. We show that the geometric version of this problem is NP -hard, and we propose two stochastic heuristics. The first one is a very fast randomized greedy search algorithm and the second one being a genetic algorithm. Also, we investigate some existing integer programming formulations and present an new one. A new Lagrangian based lower bound is proposed and implemented to assess the performance of the heuristics. Computational experiments performed on a large set of randomly generated instances with up to 1000 vertices and 10,000 edges provide evidence of the good performance of the proposed heuristics.

A new relaxation method for the generalized minimum spanning tree problem

We consider a generalization of the minimum spanning tree problem, called the generalized minimum spanning tree problem, denoted by GMST. It is known that the GMST problem is NP -hard. We present several mixed integer programming formulations of the problem. Based on a new formulation of the problem we give a new solution procedure that finds the optimal solution of the GMST problem for graphs with nodes up to 240. We discuss the advantages of our approach in comparison with earlier methods.

An approximation algorithm for the at least version of the generalized minimum spanning tree problem

We consider the at least version of the Generalized Minimum Spanning Tree Problem, denoted by L-GMSTP, which consists in finding a minimum cost tree spanning at least one node from each node set of a complete graph with the nodes partitioned into a given number of node sets called clusters. We assume that the cost function attached to edges satisfies the triangle inequality and the clusters have sizes bounded by \(\rho\). Under these assumptions we present a 2\(\rho\) approximation algorithm.The algorithm works by rounding an optimal fractional solution to a linear programming relaxation. Our technique is based on properties of optimal solutions to the linear programming formulation of the minimum spanning tree problem and the parsimonious property of Goemans and Bertsimas.

Heuristic Search for the Generalized Minimum Spanning Tree Problem

The generalized minimum spanning tree (GMST) problem occurs in telecommunications network planning, where a network of node clusters needs to be connected via a tree architecture using exactly one node per cluster. The problem is known to be NP-hard, and even finding a constant factor approximation algorithm is NP-hard. In this paper, we present two heuristic search approaches for the GMST problem: local search and a genetic algorithm. Our computational experiments show that these heuristics rapidly provide high-quality solutions for the GMST and outperform some previously suggested heuristics for the problem. In our computational tests on 211 test problems (including 169 problems from the TSPLIB set), our local-search heuristic found the optimal solution in 179 instances and our genetic-algorithm procedure found the optimal solution in 185 instances (out of the 211 instances, the optimal solution is known in 187 instances). Further, on each of the 19 unsolved instances from TSPLIB, both our local-search heuristic and genetic-algorithm procedure improved upon the best previously known solution.

Solving the generalized minimum spanning tree problem by a branch-and-bound algorithm

We present an exact algorithm for solving the generalized minimum spanning tree problem (GMST). Given an undirected connected graph and a partition of the graph vertices, this problem requires finding a least-cost subgraph spanning at least one vertex out of every subset. In this paper, the GMST is formulated as a minimum spanning tree problem with side constraints and solved exactly by a branch-and-bound algorithm. Lower bounds are derived by relaxing, in a Lagrangian fashion, complicating constraints to yield a modified minimum cost spanning tree problem. An efficient preprocessing algorithm is implemented to reduce the size of the problem. Computational tests on a large set of randomly generated instances with as many as 250 vertices, 1000 edges, and 25 subsets provide evidence that the proposed solution approach is very effective.

Approximation theory in combinatorial optimization. Application to the generalized minimum spanning tree problem

We present an overview of the approximation theory in
 combinatorial optimization. As an application we consider the 
 Generalized Minimum Spanning Tree (GMST) problem which is defined on an undirected complete graph with the nodes partitioned into
 clusters and non-negative costs are associated to the edges. This
 problem is NP-hard and it is known that a
 polynomial approximation algorithm cannot exist. We present an
 in-approximability result for the GMST problem and under special
 assumptions: cost function satisfying the triangle inequality and
 with cluster sizes bounded by \(\rho\), we give an approximation
 algorithm with ratio \(2 \rho\).