Degree-constrained Minimum Spanning Tree Problem Research Articles

A hybrid genetic algorithm for the degree-constrained minimum spanning tree problem

Given an undirected, connected, edge-weighted graph G and a positive integer d, the degree-constrained minimum spanning tree (dc-MST) problem aims to find a minimum spanning tree T on G subject to the constraint that each vertex is either a leaf vertex or else has degree at most d in T, where d is a given positive integer. The dc-MST is $$\mathcal {NP}$$-hard problem for d$$\ge $$ 2 and finds several real-world applications. This paper proposes a hybrid approach ($$\mathcal {H}$$SSGA) combining a steady-state genetic algorithm and local search strategies for the this problem. An additional step (based on perturbation strategy at a regular interval of time) in the replacement strategy is applied in order to maintain diversity in the population throughout the search process. On a set of available 107 benchmark instances, computational results show the superiority of our proposed $$\mathcal {H}$$SSGA in comparison with the state-of-the-art metaheuristic techniques.

A New Crossover Algebra of GA for Solving the Degree Constrained Minimum Spanning Tree Problems

The degree constrained minimum spanning tree (DCMST) problem seeks a spanning tree of a connected graph with the smallest weight. It is of crucial importance to the design of communication networks. There is no polynomial algorithm for solving DCMST problem, heuristic algorithm is usually used to find an approximate solution. The heuristic algorithms, such as genetic algorithm (GA), face the problems on tree heritage, which means the offspring may not be a feasible solution. To address this problem, we propose a novel crossover algebra which considers both the edge learning and structure learning, and can guarantee the offspring be a feasible solution. Experiment results validate the effectiveness of the proposed algorithm.

ILP formulation of the degree-constrained minimum spanning hierarchy problem

Given a connected edge-weighted graph G and a positive integer B, the degree-constrained minimum spanning tree problem (DCMST) consists in finding a minimum cost spanning tree of G such that the degree of each vertex in the tree is less than or equal to B. This problem, which has been extensively studied over the last few decades, has several practical applications, mainly in networks. However, some applications do not especially impose a subgraph as a solution. For this purpose, a more flexible so-called hierarchy structure has been proposed. Hierarchy, which can be seen as a generalization of trees, is defined as a homomorphism of a tree in a graph. In this paper, we discuss the degree-constrained minimum spanning hierarchy (DCMSH) problem which is NP-hard. An integer linear program (ILP) formulation of this new problem is given. Properties of the solution are analysed, which allows us to add valid inequalities to the ILP. To evaluate the difference of cost between trees and hierarchies, the exact solution of DCMST and z problems are compared. It appears that, in sparse random graphs, the average percentage of improvement of the cost varies from 20 to 36% when the maximal authorized degree of vertices B is equal to 2, and from 11 to 31% when B is equal to 3. The improvement increases as the graph size increases.

Degree-constrained minimum spanning tree problem of uncertain random network

A degree-constrained minimum spanning tree (DCMST) problem involving any network aims to find the least weighted spanning tree of that network, subject to constraints on node degrees. In this paper, we first define a DCMST problem in an uncertain random network, where some weights are uncertain variables and others are random variables. We also introduce the concept of an ideal chance distribution for DCMST problem. In order to seek out the degree-constrained spanning tree (DCST) closest to the ideal chance distribution, an uncertain random programming model is formulated. An algorithm is presented to solve the DCMST problem. The effectiveness of our method and algorithm are exhibited by solving a numerical example.

Network design through 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 problem instances which we optimize using different evolutionary computation algorithms encoding individuals of the population using the proposed representation. To illustrate the applicability of our approach, we formulate the trans-European transport network as a DRCMST problem. In this network design, we simultaneously optimize nine transport corridors and show that it is straightforward using the proposed representation to add constraints depending on the specific characteristics of the network.

Degree-constrained minimum spanning tree problem with uncertain edge weights

Uncertainty theory has shown great advantages in solving many nondeterministic problems, one of which is the degree-constrained minimum spanning tree (DCMST) problem in uncertain networks. Based on different criteria for ranking uncertain variables, three types of DCMST models are proposed here: uncertain expected value DCMST model, uncertain α-DCMST model and uncertain most chance DCMST model. In this paper, we give their uncertainty distributions and fully characterize uncertain expected value DCMST and uncertain α-DCMST in uncertain networks. We also discover an equivalence relation between the uncertain α-DCMST of an uncertain network and the DCMST of the corresponding deterministic network. Finally, a related genetic algorithm is proposed here to solve the three models, and some numerical examples are provided to illustrate its effectiveness.

Branch-and-cut-and-price algorithms for the Degree Constrained Minimum Spanning Tree Problem

Assume that a connected undirected edge weighted graph G is given. The Degree Constrained Minimum Spanning Tree Problem (DCMSTP) asks for a minimum cost spanning tree of G where vertex degrees do not exceed given pre-defined upper bounds. In this paper, three exact solution algorithms are investigated for the problem. All of them are Branch-and-cut based and rely on the strongest formulation currently available for the problem. Additionally, to speed up the computation of dual bounds, they all use column generation, in one way or another. To test the algorithms, new hard to solve DCMSTP instances are proposed here. These instances, combined with additional ones taken from the literature, are then used in computational experiments. The experiments compare the new algorithms among themselves and also against the best algorithms currently available in the literature. As an outcome of them, one of the new algorithms stands out on top.

The salesman and the tree: the importance of search in CP

The traveling salesman problem (TSP) is a challenging optimization problem for CP and OR that has many industrial applications. Its generalization to the degree constrained minimum spanning tree problem (DCMSTP) is being intensively studied by the OR community. In particular, classical solution techniques for the TSP are being progressively generalized to the DCMSTP. Recent work on cost-based relaxations has improved CP models for the TSP. However, CP search strategies have not yet been widely investigated for these problems. The contributions of this paper are twofold. We first introduce a natural generalization of the weighted cycle constraint (WCC) to the DCMSTP. We then provide an extensive empirical evaluation of various search strategies. In particular, we show that significant improvement can be achieved via our graph interpretation of the state-of-the-art Last Conflict heuristic.

Disjunctive combinatorial branch in a subgradient tree algorithm for the DCMST problem with VNS-Lagrangian bounds

Abstract Consider an undirected graph G = ( V , E ) with a set of nodes V and a set of weighted edges E. The weight of an edge e ∈ E is denoted by c e ⩾ 0 . The degree constrained minimum spanning tree (DCMST) problem consists in finding a minimum spanning tree of G subject to maximum degree constraints d v ∈ N on the number of edges connected to each node v ∈ V . We propose a Variable Neighborhood Search (VNS)-Lagrangian heuristic that outperforms the best known heuristics in the literature for this problem. We use an exact subgradient tree (SGT) algorithm in order to prove solution optimality. The SGT algorithm uses a combinatorial relaxation to evaluate lower bounds on each relaxed solution. Thus, we propose a new branching scheme that separates an integer relaxed solution from the domain of spanning trees while generating new disjoint SGT-node partitions. We prove optimality for many benchmark instances and improve lower and upper bounds for the instances whose optimal solutions remain unknown.

Degree constrained minimum spanning tree problem: a learning automata approach

Degree-constrained minimum spanning tree problem is an NP-hard bicriteria combinatorial optimization problem seeking for the minimum weight spanning tree subject to an additional degree constraint on graph vertices. Due to the NP-hardness of the problem, heuristics are more promising approaches to find a near optimal solution in a reasonable time. This paper proposes a decentralized learning automata-based heuristic called LACT for approximating the DCMST problem. LACT is an iterative algorithm, and at each iteration a degree-constrained spanning tree is randomly constructed. Each vertex selects one of its incident edges and rewards it if its weight is not greater than the minimum weight seen so far and penalizes it otherwise. Therefore, the vertices learn how to locally connect them to the degree-constrained spanning tree through the minimum weight edge subject to the degree constraint. Based on the martingale theorem, the convergence of the proposed algorithm to the optimal solution is proved. Several simulation experiments are performed to examine the performance of the proposed algorithm on well-known Euclidean and non-Euclidean hard-to-solve problem instances. The obtained results are compared with those of best-known algorithms in terms of the solution quality and running time. From the results, it is observed that the proposed algorithm significantly outperforms the existing method.

Efficient Forest Data Structure for Evolutionary Algorithms Applied to Network Design

The design of a network is a solution to several engineering and science problems. Several network design problems are known to be NP-hard, and population-based metaheuristics like evolutionary algorithms (EAs) have been largely investigated for such problems. Such optimization methods simultaneously generate a large number of potential solutions to investigate the search space in breadth and, consequently, to avoid local optima. Obtaining a potential solution usually involves the construction and maintenance of several spanning trees, or more generally, spanning forests. To efficiently explore the search space, special data structures have been developed to provide operations that manipulate a set of spanning trees (population). For a tree with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> nodes, the most efficient data structures available in the literature require time <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ) to generate a new spanning tree that modifies an existing one and to store the new solution. We propose a new data structure, called node-depth-degree representation (NDDR), and we demonstrate that using this encoding, generating a new spanning forest requires average time <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (√ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ). Experiments with an EA based on NDDR applied to large-scale instances of the degree-constrained minimum spanning tree problem have shown that the implementation adds small constants and lower order terms to the theoretical bound.

An Improved Ant-Based Algorithm for the Degree-Constrained Minimum Spanning Tree Problem

The degree-constrained minimum spanning tree (DCMST) problem is the problem of finding the minimum cost spanning tree in an edge weighted complete graph such that each vertex in the spanning tree has degree ≤ d for some d ≥ 2. The DCMST problem is known to be NP-hard. This paper presents an ant-based algorithm to find low cost degree-constrained spanning trees (DCST). The algorithm employs a set of ants which traverse the graph and identify a set of candidate edges, from which a DCST is constructed. Local optimization algorithms are then used to further improve the DCST. Extensive experiments using 612 problem instances show many improvements over existing algorithms.

DEGREE-CONSTRAINED MINIMUM SPANNING TREE PROBLEM IN STOCHASTIC GRAPH

A degree-constrained minimum spanning tree (DCMST) problem is an NP-hard combinatorial optimization problem in graph theory seeking the minimum cost spanning tree with the additional constraint on the vertex degree. Several different approaches have been proposed in the literature to solve this problem using a deterministic graph. However, to the best of the author's knowlege, no work has been performed on solving the problem using stochastic edge-weighted graphs. In this article, a learning automata–based algorithm is proposed to find a near optimal solution of the DCMST problem using a stochastic graph, where the cost associated with the graph edge is a random variable with a priori unknown probability distribution. The convergence of the proposed algorithm to the optimal solution is theoretically proved based on the Martingale theorem. To show the performance of the proposed algorithm, several simulation experiments are conducted on stochastic Euclidean graph instances. Numerical results are compared with those of the standard sampling method (SSM). The numerical results confirm the superiority of the proposed sampling technique over the SSM both in terms of the sampling rate and solution optimality.

Optimization Algorithm for Solving Degree-Constrained Minimum Spanning Tree Problem

为求解大规模结点度约束最小生成树问题,提出一种带有嫁接和剪接算子操作的优化算法.通过借鉴花草果树种植技术,建立一种以基本遗传算子为基础、带有加速和调节算子作为激励的进化计算体系;嫁接以一种贪婪的思想加速搜索,按收益最大化原则进行剪接.对可能陷入局部极值引起冲突的现象及冲突检测的方法进行分析,并提出了冲突的若干解决方法.针对DCMST问题求解中的复杂性,提出了几种有效的嫁接和剪接的策略,并对算法的收敛性和计算复杂度进行了分析.通过该算法对结点数为50~500之间的Euclidean问题和按均匀随机方式产生的non-Euclidean度约束最小生成树问题进行求解.与现有文献的实验结果对比表明,该方法在求解最好解的精度和收敛速度上均有一定的优势.;To solve the degree-constrained spanning minimum tree (DCMST) problems with a large scale of nodes, an optimization algorithm based on grafting and pruning operator is proposed. Learning from the flower planting techniques, this paper establishes, an evolutionary computation framework containing accelerating and adjusting operators based on conventional genetic operators. The grafting and pruning are performed by a greedy strategy and gain maximization respectively. The collision caused by possible local minima is analyzed and detected, and several methods dealing with the collision are discussed. To tackle the complexity of DCMST problems, some strategies of grafting and pruning are proposed. The convergence of the proposed algorithm and the computation complexity are analyzed. For DCMST problems of Euclidean and uniform random non-Euclidean instances from 50 to 500 nodes, the experiments show that the quality and convergence rate of the proposed method are the best compared with the known results.

Spanning Trees with Node Degree Dependent Costs and Knapsack Reformulations

Abstract The Degree constrained Minimum Spanning Tree Problem (DMSTP) consists in finding a minimal cost spanning tree such that every node has a degree no greater than a fixed value. We consider a generalization of the DMSTP with a more general objective function that includes modular costs associated to the degree of each node. We show how the problem can be viewed as the intersection of a spanning tree problem and a knapsack problem. We present several linear programming models, based on the previous decompositions, together with some valid inequalities and compare their respective linear programming relaxations using random instances.

Lower and upper bounds for the degree‐constrained minimum spanning tree problem

AbstractIn this paper, we propose a Lagrangian Non‐Delayed Relax‐and‐Cut algorithm for the Degree‐Constrained Minimum Spanning Tree Problem (DCMSTP). Since degree‐constrained trees are the common vertices of the Spanning Tree and the b‐Matching polytopes, strengthened DCMSTP lower bounds are generated through the use of Blossom Inequalities (BIs). BIs are facet defining for the b‐Matching Polytope and, to the best of our knowledge, have never been used before for DCMSTP. Lagrangian information is also used here, in a Kruskal style algorithm (followed by local search) to obtain valid DCMSTP upper bounds. Whenever optimality could not be proven by Relax‐and‐Cut alone, Lagrangian lower bounds are carried over to a cutting plane algorithm, without the need for solving any separation problem. This is accomplished after identifying, at Relax‐and‐Cut, a small set of attractive Subtour Elimination Constraints and BIs. Computational results, on Euclidean and on random cost DCMSTP instances, indicate that our Relax‐and‐Cut lower bounds either dominate or else are competitive with linear programming and Lagrangian relaxation lower bounds found in the literature. Furthermore, our upper bounds appear competitive with the best in the literature. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 55–66 2007

FUZZY RANDOM DEGREE-CONSTRAINED MINIMUM SPANNING TREE PROBLEM

The degree-constrained minimum spanning tree problem (dc-MST) is to find a minimum spanning tree of the given graph, subject to constraints on node degrees. This paper investigates the dc-MST problem with fuzzy random weights. Three concepts are presented: expected fuzzy random dc-MST, (α,β)-dc-MST and the most chance dc-MST according to different optimization requirement. Correspondingly, by using the concepts as decision criteria, three fuzzy random programming models for dc-MST are given. Finally, a hybrid intelligent algorithm is designed to solve these models, and some numerical examples are provided to illustrate its effectiveness.

Biased mutation operators for subgraph-selection problems

Many graph problems seek subgraphs of minimum weight that satisfy a set of constraints. Examples include the minimum spanning tree problem (MSTP), the degree-constrained minimum spanning tree problem (d-MSTP), and the traveling salesman problem (TSP). Low-weight edges predominate in optimum solutions to such problems, and the performance of evolutionary algorithms (EAs) is often improved by biasing variation operators to favor these edges. We investigate the impact of biased edge-exchange mutation. In a large-scale empirical investigation on Euclidean and uniform random instances, we describe the distributions of edges in optimum solutions of the MSTP, the d-MSTP, and the TSP in terms of the edges' weight-based ranks. We approximate these distributions by exponential functions and derive approximately optimal probabilities for selecting edges to be incorporated into candidate solutions during mutation. A theoretical analysis of the expected running time of a (1+1)-EA on nondegenerate instances of the MSTP shows that when using the derived probabilities for edge selection in mutation, the (1+1)-EA is asymptotically as fast as a classical implementation of Kruskal's minimum spanning tree algorithm. In experiments on the MSTP, d-MSTP, and the TSP, we compare the new edge-selection strategy to four alternative methods. The results of a (1+1)-EA on instances of the MSTP support the theory and indicate that the new strategy is superior to the other methods in practice. On instances of the d-MSTP, a more sophisticated EA with a larger population and unbiased recombination performs better with the new biased mutation than with alternate mutations. On the TSP, the advantages of weight-biased mutation are generally smaller, because the insertion of a specific new edge into a tour requires the insertion of a second dependent edge as well. Although we considered Euclidean and uniform random instances only, we conjecture that the same biasing toward low-weight edges also works well on other instance classes structured in different ways.

Algorithms for the degree-constrained minimum spanning tree problem

Algorithms for the degree-constrained minimum spanning tree problem

Optimization in the migration problem of mobile agents in distributed information retrieval systems

In this paper, we employ genetic algorithms to solve the migration problem (MP). We propose a new encoding scheme to represent trees, which is composed of two parts: the pre-ordered traversal sequence of tree vertices and the children number sequence of corresponding tree vertices. The proposed encoding scheme has the advantages of simplicity for encoding and decoding, ease for GA operations, and better equilibrium between exploration and exploitation. It is also adaptive in that, with few restrictions on the length of code, it can be freely lengthened or shortened according to the characteristics of the problem space. Furthermore, the encoding scheme is highly applicable to the degree-constrained minimum spanning tree problem because it also contains the degree information of each node. The simulation results demonstrate the higher performance of our algorithm, with fast convergence to the optima or sub-optima on various problem sizes. Comparing with the binary string encoding of vertices, when the problem size is large, our algorithm runs remarkably faster with comparable search capability.