Constrained Minimum Spanning Tree Problem Research Articles

The Use of Probability and Edge Analysis to Solve the Multi-Period Degree Constrained Minimum Spanning Tree Problem

The goal of the Multiperiod Degree Constrained Minimum Spanning Tree (MPDCMST) problem is to determine the smallest weight-spanning tree that satisfies the vertex installation criterion for each period and maintains the degree requirement in each vertex. This issue emerges as a network connection problem. The degree requirement indicates the reliability of each vertex, and the vertex connection/installation requirement denotes the priority vertices that must be inserted in the network within a specific time frame. The installation is split up into multiple phases/stages. This is because of various considerations such as severe weather, budgetary limitations, etc. In this research, two algorithms for solving the MPDCMST using probability hybridized with Prim’s modification, and edge analysis are proposed. The algorithms are implemented on the undirected complete graph of orders 10 to 100. The solutions are compared with some heuristics which are already in the literature. The results show that the proposed algorithms perform better.

On Performance of a Simple Multi-objective Evolutionary Algorithm on the Constrained Minimum Spanning Tree Problem

The constrained optimization problems can be transformed into multi-objective optimization problems, and thus can be optimized by multi-objective evolutionary algorithms. This method has been successfully used to solve the constrained optimization problems. However, little theoretical work has been done on the performance of multi-objective evolutionary algorithms for the constrained optimization problems. In this paper, we theoretically analyze the performance of a multi-objective evolutionary algorithms on the constrained minimum spanning tree problem. First, we theoretically prove that the multi-objective evolutionary algorithm is capable of finding a (2,1)-approximation solution for the constrained minimum spanning tree problem in a pseudopolynomial runtime. Then, this simple multi-objective evolutionary algorithm is shown to be efficient on a constructed instance of the problem.

Improved formulations and branch-and-cut algorithms for the angular constrained minimum spanning tree problem

The Angular Constrained Minimum Spanning Tree Problem (\(\alpha \)-MSTP) is defined in terms of a complete undirected graph \(G=(V,E)\) and an angle \(\alpha \in (0,2\pi ]\). Vertices of G define points in the Euclidean plane while edges, the line segments connecting them, are weighted by the Euclidean distance between their endpoints. A spanning tree is an \(\alpha \)-spanning tree (\(\alpha \)-ST) of G if, for any \(i \in V\), the smallest angle that encloses all line segments corresponding to its i-incident edges does not exceed \(\alpha \). \(\alpha \)-MSTP consists in finding an \(\alpha \)-ST with the least weight. In this work, we discuss families of \(\alpha \)-MSTP valid inequalities. One of them is a lifting of existing angular constraints found in the literature and the others come from the Stable Set polytope, a structure behind \(\alpha \)-STs disclosed here. We show that despite being already satisfied by the previously strongest known formulation, \({\mathcal {F}}_{xy}\), these lifted angular constraints are capable of strengthening another existing \(\alpha \)-MSTP model so that both become equally strong, at least for the instances tested here. Inequalities from the Stable Set polytope improve the best known Linear Programming Relaxation (LPRs) bounds by about 1.6%, on average, for the hardest instances of the problem. Additionally, we indicate how formulation \({\mathcal {F}}_{xy}\) can be more effectively used in Branch-and-cut (BC) algorithms, by reducing the number of variables explicitly enforced to be integer constrained and by eliminating constraints that do not change the quality of its LPR bounds. Extensive computational experiments conducted here suggest that the combination of the ideas above allows us to redefine the best performing \(\alpha \)-MSTP algorithms, for almost the entire spectrum of \(\alpha \) values, the exception being the easy instances, those with \(\alpha \ge \frac{2\pi }{3}\). In particular, for the hardest ones (corresponding to \(\alpha \in \{\frac{\pi }{2}, \frac{\pi }{3},\frac{2\pi }{5}\}\)) that could be solved to proven optimality, the best BC algorithm suggested here improves on average CPU times by factors of up to 5, on average.

A structured solution framework for fuzzy minimum spanning tree problem and its variants under different criteria

This paper develops a unified and structured solution framework for the minimum spanning tree (MST) problem and its variants (e.g., constrained MST problem and inverse MST problem) on networks with fuzzy link weights. It is applicable to any additive decision criterion under fuzziness (e.g., expected value, value at risk, and conditional value at risk), for generalized cases that the link weights may be represented by arbitrary types of fuzzy variables. It also applies to the entropy criterion while the link weights are continuous fuzzy variables. Following the optimality conditions of the fuzzy MST under different decision criteria proved first in this paper, it is shown that the MST problem and its variants on a fuzzy network can be converted into equivalent deterministic counterparts on their corresponding crisp networks. Consequently, these problems can be effectively solved via their deterministic counterparts without fuzzy simulation, and meanwhile, the performance of the trees under a specified criterion is precisely measured. The accuracy and efficiency are both significantly improved compared with other fuzzy simulation-based approaches. Numerical examples illustrate the superiority of the proposed solution framework. Furthermore, some new theoretical conclusions on the MST problem under fuzziness are also presented.

Computational Aspects of Some Algorithms for The Multiperiod Degree Constrained Minimum Spanning Tree Problem

Given a graph G(V,E), where V is the set of vertices and E is the set of edges connecting vertices in V, and for every edge eij there is an associated weight cij ≥0, The Multi Period Degree Constrained Minimum Spanning Tree (MPDCMST) is a problem of finding an MST while also considering the degree constrained on every vertex, and satisfying vertices installation on every period. The restriction on the vertex installation is needed due to some conditions such as fund limitation, harsh weather, and so on. In this research some algorithms developed to solve the MPDCMST Problem will be discussed and compared.

Modeling and solving the angular constrained minimum spanning tree problem

Assume one is given an angle α ∈ (0, 2π] and a complete undirected graph G=(V,E). The vertices in V represent points in the Euclidean plane. The edges in E represent the line segments between these points, with edge weights equal to segment lengths. Spanning trees of G are called α-spanning trees (α-STs) if, for any i ∈ V, the smallest angle that encloses all line segments corresponding to its i-incident edges does not exceed α. The Angular Constrained Minimum Spanning Tree Problem (α-MSTP) seeks an α-ST with the least weight. The problem arises in the design of wireless communication networks operating under directional antennas. We propose two α-MSTP formulations. One, Fx requiring, in principle, O(2|V|) inequalities to model the angular constraints (α-AC). For α ∈ (0, π), however, we show that just O(|V|3) of them suffice to attain not only a formulation but also the same Linear Programming relaxation (LPR) bound as the full blown model. The other formulation, Fxy, enlarges the set of Fx variables but manages to model α-AC, compactly. Furthermore, Fxy LPR bounds are proven to dominate their Fx counterparts. That dominance, however, is empirically shown to reduce as α increases. Finally, exact Branch-and-Cut algorithms implemented for the two formulations are shown, empirically, to exchange roles as top performer, throughout the [0, 2π) range of α values.

Approximation algorithm for receiver interference problem in dual power Wireless Sensor Networks

The problem of assigning power levels to the nodes of a wireless sensor network from a given a set of two power levels is called Dual power management problem and the underlying network is called Dual power network. We consider the problem of minimizing the maximum receiver interference of such a network. The interference disrupts the communication and forces the data packets to be retransmitted. The motivation is to conserve the energy by minimizing the interference and maintaining the connectivity of the dual power network. Receiver interference problem is proved to be NP-hard. In this paper, an approximation algorithm is derived for minimizing the maximum receiver interference of a dual power network by utilizing the approximation algorithm for Dual Power Management Problem. The proposed algorithm is supported by the simulation results. We term this problem as Dual Power Receiver Interference Problem and show that it is NP-complete using a polynomial time reduction from Degree Constrained Minimum Spanning Tree problem. We also prove the NP-completeness of Dual Power Management Problem by a polynomial reduction from Vertex Cover Problem.

Exact solution approaches for the Multi-period Degree Constrained Minimum Spanning Tree Problem

The Multi-period Degree Constrained Minimum Spanning Tree Problem (MP-DCMSTP) is defined in terms of a finite discretized planning horizon, an edge weighted undirected graph G, degree bounds and latest installation dates assigned to the vertices of G. Since vertices must be connected to a root node no later than their latest installation dates and edges’ weights are non-increasing over time, the problem asks for optimally choosing and scheduling edges’ installation over the planning horizon, enforcing connectivity of the solution at each time period, so that in the end of the planning horizon, a degree constrained spanning tree of G is found. We show that the decision version of a combinatorial relaxation for the problem, that of finding a Multi-period Minimum Spanning Tree Problem (MP-MSTP), is NP-Complete. We propose a new integer programming formulation for MP-DCMSTP that is at least as good as the multi-commodity flow formulation in the literature. We also introduce some new valid inequalities which allowed our strengthened formulation to produce the strongest known bounds to date. Two MP-DCMSTP exact algorithms exploring the strengthened formulation are introduced here. One of them, RCBC, is a hybrid method involving two phases, the first being a Lagrangian Relax-and-cut method that works as a pre-processor procedure to the second phase, a Branch-and-cut algorithm. The other approach, SRCBC, uses RCBC to solve a sequence of smaller MP-DCMSTP instances generated from the original one in the hope of solving the latter faster. Our computational results indicate that SRCBC solved more instances to proven optimality, generally in one fourth of the time taken by RCBC to solve similar instances. For those instances left unsolved by both, SRCBC also provided much better feasible solutions within the same CPU time limit.

DIFFERENT TIME INSTALLATION EFFECT ON THE QUALITY OF THE SOLUTION FOR THE MULTIPERIOD INSTALLATION PROBLEM USING MODIFIED PRIM’S ALGORITHM

Most in network design problems, The Minimum Spanning Tree (MST) is usually used as the backbone. If we add degree restriction on its vertices (can represent cities, stations, etc) of the graph (represents the network), the problem becomes the Degree Constrained Minimum Spanning Tree (DCMST) problem. However, to do the installation or connecting the network, it is possible that the process must be done into some stages or periods. That situation occurs because of the weather constraint, fund constraint, etc. By restricting and dividing the stages or periods of the network’s installation, the problem emerges as The Multi Period Degree Constrained Minimum Spanning Tree (MPDCMST) problem or Multiperiod Installation Problem. We develop two algorithms based on Modified Prim’s algorithms (WAC1 and WAC2) to solve the MPDCMST problem, show and compare the different time installation effect on quality of the solution by implementing and comparing those algorithms using 300 generate problems. Keywords and phrases minimum spanning tree; degree constrained; installation, period

Min-degree constrained minimum spanning tree problem with fixed centrals and terminals: Complexity, properties and formulations

We consider a variant of the Min-Degree Constrained Minimum Spanning Tree Problem where the central and terminal nodes are fixed a priori. We prove that the optimization problem is NP-Hard even for complete graphs and the feasibility problem is NP-Complete even if there is an edge between each central and each terminal in the input graph. Actually, this complexity result still holds when the minimum degree of each central node is restricted to be a same value d ≥ 2. We derive necessary and sufficient conditions for feasibility. We present several integer linear programming formulations – based on known formulations for the minimum spanning tree problem – along with a theoretical comparison among the lower bounds provided by their linear relaxations. We propose three Lagrangian heuristics. Computational experiments compare the performances of the heuristics and the formulations.

Genetic algorithms to balanced tree structures in graphs

Abstract Given an edge-weighted graph G = ( V , E ) with vertex set V and edge set E, we study in this paper the following related balanced tree structure problems in G. The first problem, called Constrained Minimum Spanning Tree Problem (CMST), asks for a rooted tree T in G that minimizes the total weight of T such that the distance between the root and any vertex v in T is at most a given constant C times the shortest distance between the two vertices in G. The Constrained Shortest Path Tree Problem (CSPT) requires a rooted tree T in G that minimizes the maximum distance between the root and all vertices in V such that the total weight of T is at most a given constant C times the minimum tree weight in G. The third problem, called Minimum Maximum Stretch Spanning Tree (MMST), looks for a tree T in G that minimize the maximum distance between all pairs of vertices in V. It is easy to conclude from the literatures that the above problems are NP-hard. We present efficient genetic algorithms that return (as shown by our experimental results) high quality solutions for these problems.

A strong symmetric formulation for the Min-degree Constrained Minimum Spanning Tree Problem

Given an integer K≥3 and an edge weighted undirected graph G, the Min-Degree Constrained Minimum Spanning Tree Problem (MDMST) asks for a minimum cost spanning tree of G, such that every non-leaf vertex has degree at least K. We introduce a new reformulation for MDMST that consists of splitting spanning trees into two parts: a backbone tree spanning only the non-leaf vertices and an assignment of the leaves to non-leaf vertices. The backbone tree is modelled with a lifted version of generalized subtour breaking constraints (GSECs). Our computational results show that we can go beyond the strongest MDMST Linear Programming lower bounds in the literature. In addition, a Branch-and-cut algorithm based on our new lower bounds seems to be competitive with the best MDMST exact approach, despite the lack of a primal heuristic. As a by product, new best lower bounds are provided for several unsolved MDMST instances.

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.

Polynomial Time Approximation Schemes for the Constrained MinimumSpanning Tree Problem

Let G = (V, E) be an undirected graph with a weight function and a cost function on edges. The constrained minimum spanning tree problem is to find a minimum cost spanning tree T in G such that the total weight in T is at most a given bound B. In this paper, we present two polynomial time approximation schemes (PTASs) for the constrained minimum spanning tree problem.

Min-degree constrained minimum spanning tree problem: complexity, properties, and formulations

This paper addresses a new combinatorial problem, the Min-Degree Constrained Minimum Spanning Tree (md-MST), that can be stated as: given a weighted undirected graph with positive costs on the edges and a node-degrees function , the md-MST is to find a minimum cost spanning tree T of G, where each node i of T either has at least a degree of or is a leaf node. This problem is closely related to the well-known Degree Constrained Minimum Spanning Tree (d-MST) problem, where the degree constraint is an upper bound instead. The general NP-hardness for the md-MST is established and some properties related to the feasibility of the solutions for this problem are presented, in particular we prove some bounds on the number of internal and leaf nodes. Flow-based formulations are proposed and computational experiments involving the associated Linear Programming (LP) relaxations are presented.

The min-degree constrained minimum spanning tree problem: Formulations and Branch-and-cut algorithm

Given an edge weighted undirected graph G and a positive integer d, the Min-Degree Constrained Minimum Spanning Tree Problem (MDMST) consists of finding a minimum cost spanning tree of G, such that each vertex is either a leaf or else has degree at least d in the tree. In this paper, we discuss two formulations for MDMST based on exponentially many undirected and directed subtour breaking constraints and compare the strength of their Linear Programming (LP) bounds with other bounds in the literature. Aiming to overcome the fact that the strongest of the two models, the directed one, is not symmetric with respect to the LP bounds, we also presented a symmetric compact reformulation, devised with the application of an Intersection Reformulation Technique to the directed model. The reformulation proved to be much stronger than the previous models, but evaluating its bounds is very time consuming. Thus, better computational results were obtained by a Branch-and-cut algorithm based on the original directed formulation. With the proposed method, several new optimality certificates and new best upper bounds for MDMST were provided.

Models and heuristics for the k -degree constrained minimum spanning tree problem with node-degree costs

The k -Degree constrained Minimum Spanning Tree Problem (k -DMSTP) consists in finding a minimal cost spanning tree satisfying the condition that every node has a degree no greater than a fixed value k. Here we consider an extension where besides the edge costs, a concave cost function is associated to the degree of each node. Several integer linear programming formulations based on reformulation techniques are presented and their linear programming relaxations are compared. A GRASP heuristic to obtain feasible solutions for the problem together with a Path Relinking strategy is also described. We include computational results using instances with up to 200 nodes that give an empirical assessment of the linear programming bounds of the different models as well as the ability of using these models to solve the problems by using an Integer Linear Programming package. The results also show that the proposed GRASP heuristic appears to perform rather well for this type of problems. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012

Modelling the Hop Constrained Connected Facility Location Problem on Layered Graphs

Abstract Gouveia et al. [Gouveia, L., L. Simonetti and E. Uchoa, Modelling hop-constrained and diameter-constrained minimum spanning tree problems as Steiner tree problems over layered graphs , Mathematical Programming (2010). URL http://dx.doi.org/10.1007/s10107-009-0297-2 ] show how to model the Hop Constrained Minimum Spanning tree problem as Steiner tree problem on a layered graph. Following their ideas, we provide three possibilities to model the Hop Constrained (HC) Connected Facility Location problem (ConFL) as ConFL on layered graphs. We show that on all three layered graphs the respective LP relaxations of two cut based models are of the same quality. In our computational study we compare a compact hop-indexed tree model against the two cut based models on the simplest layered graph. We provide results for instances with up to 1300 nodes and 115000 arcs.

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.

Finding min-degree constrained spanning trees faster with a Branch-and-cut algorithm

Abstract In this paper, two formulations for the Min-degree Constrained Minimum Spanning Tree Problem, one based on undirected Subtour Elimination Constraints and the other on Directed Cutset inequalities, are discussed. The quality of the Linear Programming bounds provided by them is addressed and a Branch-and-cut algorithm based on the strongest is investigated. Our computational experiments indicate that the method compares favorably with other exact and heuristic approaches in the literature, in terms of solution quality and execution times. Several new optimality certificates and new best upper bounds are provided here.