Comparison of Algorithms for the Degree Constrained Minimum Spanning Tree

The Degree Constrained Minimum Spanning Tree (DCMST) on a graph is the problem of generating a minimum spanning tree with constraints on the number of arcs that can be incident to vertices of the graph. In this paper we develop three heuristics for the DCMST, including simulated annealing, a genetic algorithm and a method based on problem space search. We propose alternative tree representations to facilitate the neighbourhood searches for the genetic algorithm. The tree representation that we use for the genetic algorithm can be generalised to other tree optimisation problems as well. We compare the computational performance of all of these approaches against the performance of an exact solution approach in the literature. In addition, we also develop a new exact solution approach based on the combinatorial structure of the problem. We test all of these approaches using standard problems taken from the literature and some new test problems that we generate.     

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.     

A multi-population hybrid biased random key genetic algorithm for hop-constrained trees in nonlinear cost flow networks

Genetic algorithms and other evolutionary algorithms have been successfully applied to solve constrained minimum spanning tree problems in a variety of communication network design problems. In this paper, we enlarge the application of these types of algorithms by presenting a multi-population hybrid genetic algorithm to another communication design problem. This new problem is modeled through a hop-constrained minimum spanning tree also exhibiting the characteristic of flows. All nodes, except for the root node, have a nonnegative flow requirement. In addition to the fixed charge costs, nonlinear flow dependent costs are also considered. This problem is an extension of the well know NP-hard hop-constrained Minimum Spanning Tree problem and we have termed it hop-constrained minimum cost flow spanning tree problem. The efficiency and effectiveness of the proposed method can be seen from the computational results reported.     

On the complexity of some arborescences finding problems on a multihop radio network

An arborescence of a multihop radio network is a directed spanning tree (with rootx) such that the edges are directed away from the root. Based upon an arborescence,x canbroadcast a message to other nodes according to the directed edges of the spanning tree. The minimum transmission power arborescence problem is to find an arborescence such that the message can be broadcasted to other nodes by using a minimal amount of transmission power. The minimum delay arborescence problem is to find an arborescence such that a message can be broadcasted to other nodes by using a minimal number of broadcast transmission. In this paper we show that both these problems areNP-complete. The reductions are from the maximum leaf spanning tree problem.Areverse arborescence is similar to an arborescence except that the edges are directed toward the root. Based upon a reverse arborescence, the root node cancollect information from other nodes. In this paper we also show that the reverse minimum transmission power arborescence problem can be solved with the same computational complexity as that of finding a minimum cost spanning tree, and the reverse minimum delay arborescence problem can be solved with the same computational complexity as that of finding a spanning tree.     

On the inverse problem of minimum spanning tree with partition constraints

In this paper we first discuss the properties of minimum spanning tree and minimum spanning tree with partition constraints. We then concentrate on the inverse problem of minimum spanning tree with partition constraints in which we need to adjust the weights of the edges in a network as less as possible so that a given spanning tree becomes the minimum one among all spanning trees that satisfy the partition restriction. Based on the calculation of maximum cost flow in networks, we propose a strongly polynomial algorithm for solving the problem.The author gratefully acknowledges the partial support of Croucher Foundation.     

A genetic algorithm approach on tree-like telecommunication network design problem

In the context of telecommunication networks, the network terminals involve certain constraints that are either related with the performance of the corresponding network or with the availability of some classes of devices. In this paper, we discuss a tree-like telecommunication network design problem with the constraint limiting the number of terminals. First, this problem is formulated as a leaf-constrained minimum spanning tree (lc-MST). Then we develop a tree-based genetic representation to encode the candidate solutions of the lc-MST problem. Compared with the existing heuristic algorithm, the numerical results show the high effectiveness of the proposed GA approach on this problem.     

An addendum to the hierarchical network design problem

The hierarchical network design problem is the problem to find a spanning tree of minimum total weight, when the edges of the path between two given nodes are weighted by an other cost function than the tree edges not on this path. We point out that a dynamic programming oriented heuristic can already be found in literature. Further we report on possible extensions and improvements.     

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.     

Modeling and solving the bi-objective minimum diameter-cost spanning tree problem

The bi-objective minimum diameter-cost spanning tree problem (bi-MDCST) seeks spanning trees with minimum total cost and minimum diameter. The bi-objective version generalizes the well-known bounded diameter minimum spanning tree problem. The bi-MDCST is a NP-hard problem and models several practical applications in transportation and network design. We propose a bi-objective multiflow formulation for the problem and effective multi-objective metaheuristics: a multi-objective evolutionary algorithm and a fast nondominated sorting genetic algorithm. Some guidelines on how to optimize the problem whenever a priority order can be established between the two objectives are provided. In addition, we present bi-MDCST polynomial cases and theoretical bounds on the search space. Results are reported for four representative test sets.     

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.     

Paths, trees and matchings under disjunctive constraints

We study the minimum spanning tree problem, the maximum matching problem and the shortest path problem subject to binary disjunctive constraints: A negative disjunctive constraint states that a certain pair of edges cannot be contained simultaneously in a feasible solution. It is convenient to represent these negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the edges of the underlying graph, and whose edges encode the constraints.We prove that the minimum spanning tree problem is strongly NP-hard, even if every connected component of the conflict graph is a path of length two. On the positive side, this problem is polynomially solvable if every connected component is a single edge (that is, a path of length one). The maximum matching problem is NP-hard for conflict graphs where every connected component is a single edge.Furthermore we will also investigate these graph problems under positive disjunctive constraints: In this setting for certain pairs of edges, a feasible solution must contain at least one edge from every pair. We establish a number of complexity results for these variants including APX-hardness for the shortest path problem.     

A method for maximizing the reliability coefficient of a communications network

The most common idea of network reliability in the literature is a numerical parameter calledoverall network reliability, which is the probability that the network will be in a successful state in which all nodes can mutually communicate. Most papers concentrate on the problem of calculating the overall network reliability which is known to be an NP hard problem. In the present paper, the question asked is how to find a method for determining a reliable subnetwork of a given network. Givenn terminals and one central computer, the problem is to construct a network that links each terminal to the central computer, subject to the following conditions: (1) each link must be economically feasible; (2) the minimum number of links should be used; and (3) the reliability coefficient should be maximized. We argue that the network satisfying condition (2) is a spanning arborescence of the network defined by condition (1). We define the idea of thereliability coefficient of a spanning arborescence of a network, which is the probability that a node at average distance from the root of the arborescence can communicate with the root. We show how this coefficient can be calculated exactly when there are no degree constraints on nodes of the spanning arborescence, or approximately when such degree constraints are present. Computational experience for networks consisting of up to 900 terminals is given.This report was prepared as part of the activities of the Management Science Research Group, Carnegie-Mellon University, under Contract No. N00014-82-K-0329 NR 047–048 with the U.S. Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.     

Optimal Direct and Indirect Covering Trees

The maximal covering subtree problem has applications in transportation network design and extensive facility location. A subtree of a network is a tree that is not a full spanning tree. Finding an optimal subtree may involve the two objectives, minimizing the total arc cost or distance of the subtree, and maximizing the subtree's coverage of population or demand at nodes. Coverage may be defined as direct or indirect: indirect coverage requires that a node be within a distance threshold s>0 of the subtree, and direct coverage requires that a node be connected to the subtree (s=0). Previous approaches to this problem have sought to identify optimal subtrees of a parent network that is itself a tree (e.g., a minimum spanning tree). In this paper four new bi-objective zero–one programming models are presented. The first two are models for the problem of finding optimal subtrees on a single spanning tree parent under conditions of (1) direct and (2) indirect coverage. These problems have been addressed previously in the literature. In the third and fourth models, the subtree can be selected from among multiple candidate parent spanning trees simultaneously. The latter models address a new generalization of the first problem and offer both increased flexibility and the potential for a more diverse array of solutions. The models have integer-friendly solution properties and are relatively small in terms of the number of decision variables and constraints. Computational experience is reported for a demonstration problem and results are compared to the results of previous models.     

Opportune moment strategies for a cost spanning tree game

Cost spanning tree problems concern the construction of a tree which provides a connection with the source for every node of the network. In this paper, we address cost sharing problems associated to these situations when the agents located at the nodes act in a non-cooperative way. A class of strategies is proposed which produce minimum cost spanning trees and, at the same time, are strong Nash equilibria for a non-cooperative game associated to the problem. They are also subgame perfect Nash equilibria.     

New formulations of the Hop-Constrained Minimum Spanning Tree problem via Miller-Tucker-Zemlin constraints

Given an undirected network with positive edge costs and a natural number p, the Hop-Constrained Minimum Spanning Tree problem (HMST) is the problem of finding a spanning tree with minimum total cost such that each path starting from a specified root node has no more than p hops (edges). In this paper, we develop new formulations for HMST. The formulations are based on Miller-Tucker-Zemlin (MTZ) subtour elimination constraints, MTZ-based liftings in the literature offered for HMST, and a new set of topology-enforcing constraints. We also compare the proposed models with the MTZ-based models in the literature with respect to linear programming relaxation bounds and solution times. The results indicate that the new models give considerably better bounds and solution times than their counterparts in the literature and that the new set of constraints is competitive with liftings to MTZ constraints, some of which are based on well-known, strong liftings of Desrochers and Laporte (1991).     

Modeling hop-constrained and diameter-constrained minimum spanning tree problems as Steiner tree problems over layered graphs

The hop-constrained minimum spanning tree problem (HMSTP) is an NP-hard problem arising in the design of centralized telecommunication networks with quality of service constraints. We show that the HMSTP is equivalent to a Steiner tree problem (STP) in an appropriate layered graph. We prove that the directed cut model for the STP defined in the layered graph, dominates the best previously known models for the HMSTP. We also show that the Steiner directed cuts in the extended layered graph space can be viewed as being a stronger version of some previously known HMSTP cuts in the original design space. Moreover, we show that these strengthened cuts can be combined and projected into new families of cuts, including facet defining ones, in the original design space. We also adapt the proposed approach to the diameter-constrained minimum spanning tree problem (DMSTP). Computational results with a branch-and-cut algorithm show that the proposed method is significantly better than previously known methods on both problems.     

Heuristics for the multi-level capacitated minimum spanning tree problem

The capacitated minimum spanning tree (CMST) problem is fundamental to the design of centralized communication networks. In this paper we consider the multi-level capacitated minimum spanning tree problem, a generalization of the well-known CMST problem. Based on work previously done in the field, three heuristics are presented, addressing unit and non-unit demand cases. The proposed heuristics have been also integrated into a mixed integer programming solver. Evaluation results are presented, for an extensive set of experiments, indicating the improvements that the heuristics bring to the particular problem.     

Greedy heuristics for the diameter-constrained minimum spanning tree problem

The diameter-constrained minimum spanning tree problem is an NP-hard combinatorial optimization problem that seeks a minimum cost spanning tree with a limit D imposed upon the length of any path in the tree. We begin by presenting four constructive greedy heuristics and, secondly, we present some second-order heuristics, performing some improvements on feasible solutions, hopefully leading to better objective function values. We present a heuristic with an edge exchange mechanism, another that transforms a feasible spanning tree solution into a feasible diameter-constrained spanning tree solution, and finally another with a repetitive mechanism. Computational results show that repetitive heuristics can improve considerably over the results of the greedy constructive heuristics, but using a huge amount of computation time. To obtain computational results, we use instances of the problem corresponding to complete graphs with a number of nodes between 20 and 60 and with the value of D varying between 4 and 9.     

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. Received: September 1998 / Accepted: March 2001?Published online May 18, 2001     

Skewed VNS enclosing second order algorithm for the degree constrained minimum spanning tree problem

We develop ideas to enhance the performance of the variable neighborhood search (VNS) by guiding the search in the shaking phase, and by employing the Skewed strategy. For this purpose, Second Order algorithms and Skewed functions expressing structural differences are embedded in an efficient VNS proposed in the literature for the degree constrained minimum spanning tree problem. Given an undirected graph with weights associated with its edges, the degree constrained minimum spanning tree problem consists in finding a minimum spanning tree of the given graph, subject to constraints on node degrees. Computational experiments are conducted on the largest and hardest benchmark instances found in the literature to date. We report results showing that the VNS with the proposed strategies improved the best known solutions for all the hardest benchmark instances. Moreover, these new best solutions significantly reduced the gaps with respect to tight lower bounds reported in the literature.