Constrained Minimum Spanning Tree Research Articles

Novel Degree Constrained Minimum Spanning Tree Algorithm Based on an Improved Multicolony Ant Algorithm

Degree constrained minimum spanning tree (DCMST) refers to constructing a spanning tree of minimum weight in a complete graph with weights on edges while the degree of each node in the spanning tree is no more thand(d≥ 2). The paper proposes an improved multicolony ant algorithm for degree constrained minimum spanning tree searching which enables independent search for optimal solutions among various colonies and achieving information exchanges between different colonies by information entropy. Local optimal algorithm is introduced to improve constructed spanning tree. Meanwhile, algorithm strategies in dynamic ant, random perturbations ant colony, and max-min ant system are adapted in this paper to optimize the proposed algorithm. Finally, multiple groups of experimental data show the superiority of the improved algorithm in solving the problems of degree constrained minimum spanning tree.

Exact solution for branch vertices constrained spanning problems

Given a connected graph G, a vertex v of G is said to be a branch vertex if its degree is strictly greater than 2. The Minimum Branch Vertices Spanning Tree problem (MBVST) consists in nding a spanning tree of G with the minimum number of branch vertices. This problem has been well studied in the literature and has several applications specially for routing in optical networks. However, this kind of applications do not explicitly impose a sub-graph as solution. A more flexible structure called hierarchy is proposed. Hierarchy, which can be seen as a generalization of trees, is de ned as a homomorphism of a tree in a graph. Since minimizing the number of branch vertices in a hierarchy does not make sense, we propose to search the minimum cost spanning hierarchy such that the number of branch vertices is less than or equal to an integer R. We introduce the Branch Vertices Constrained Minimum Spanning Hierarchy (BVCMSH) problem which is NP-hard. The Integer Linear Program (ILP) formulation of this new problem is given. To evaluate the di erence of cost between trees and hierarchies, we confront the BVCMSH problem to the Branch Vertices Constrained Minimum Spanning Tree (BVCMST) problem by comparing its exact solutions. It appears from this comparison that when R 2, the hierarchies improve the average cost from more than 8% when jVGj = 20 and from more than 12% when jVGj = 30.

A botnet-based command and control approach relying on swarm intelligence

Survivability and scalability are the main emerging challenges in command and control of ubiquitous networked entities operating in untrusted communication scenarios, due to the increasing sophistication of the detection and mitigation/defeating techniques together with the increasing number of elements to be controlled and their distribution over multiple heterogeneous communication infrastructures. Accordingly, this work focuses on a new more robust and scalable botnet-based command and control architecture, aiming at wiping off any rigid master-slave relationship and autonomizing the bot operating roles, with significant agility gains in the whole overlay communication infrastructure. It relies on swarm intelligence and in particular on stigmergic communication, ensuring spontaneous, implicit coordination and collaboration among the independent bot agents. The resulting architecture presents improved fault tolerance and dynamic adaptation to varying network conditions, by propagating control messages to any bot node through multiple short-range hops structured according to a dynamically built Degree Constrained Minimum Spanning Tree, whose distributed calculation is inspired to ant colony's foraging behavior. For this reason, it may constitute the basis for an evolutionary malware-based control and management scheme that can be used in several homeland security/defense scenarios where the botnet technology may be used as a support tool in strategic military or intelligence operations.

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.

An efficient method to solve least-cost minimum spanning tree (LC-MST) problem

In this paper, least-cost minimum spanning tree (LC-MST) problem is defined as a method to construct a minimum cost spanning tree that has the least-cost edges in the network by using the distance (cost) matrix. The paper presents a new algorithm based on the distance matrix to solve the LC-MST problem. The studied cases show that the presented algorithm is efficient to solve the LC-MST problem in less time. Also, the presented algorithm can be modified to solve the DC-MST (Delay Constrained-Minimum Spanning Tree) problem presented by Lee and Atiquzzaman (2007) and the MRCT (Minimum Routing Cost Tree) problem presented by Cambos and Ricardo (2008), given as the applications of the presented algorithm.

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.

Md-MST is NP-hard for

Abstract Let G = ( V , E ) be a connected weighted undirected graph. Given a positive integer d, the Min-Degree Constrained Minimum Spanning Tree (md-MST) problem consists in finding a spanning tree T for G having minimum total edge cost and such as each node i in the tree either has degree at least d or is a leaf node. In the present work we prove that this recently introduced combinatorial problem is NP-hard in general.

The subdivision-constrained minimum spanning tree problem

Motivated by the constrained minimum spanning tree (CST) problem in Hassin and Levin [R. Hassin, A. Levin, An efficient polynomial time approximation scheme for the constrained minimum spanning tree problem using matroid intersection, SIAM Journal on Computing 33 (2) (2004) 261–268], we study a new combinatorial optimization problem in this paper, called the general subdivision-constrained spanning tree problem (GSCST): given a graph G = ( V , E ; w , c ) with two nonnegative integers w ( e ) and c ( e ) for each edge e ∈ E , two positive integers B and d , the GSCST problem is to first find a spanning tree T = ( V , E T ) of G with weight ∑ e ∈ E T w ( e ) ≤ B and then to insert some new vertices on some suitable edges in T such that each edge in the subdivision tree T ′ of T has its weight not beyond d . The objective is to minimize the cost ∑ e ∈ E T i n s e r t ( e ) c ( e ) of such new vertices inserted on the suitable edges among all spanning trees of G subject to the two preceding constraints, where a subdivision tree T ′ of T is constructed by inserting some new vertices on the suitable edges in T , the value i n s e r t ( e ) = ⌈ w ( e ) d ⌉ − 1 is the least number of vertices inserted and c ( e ) is the cost of each vertex inserted on the edge e . We obtain the following main results: (1) the GSCST problem and its variant are still NP-hard, by a reduction from the 0–1 knapsack problem, respectively; (2) the GSCST problem as well as its variant is polynomially equivalent to the CST problem, which implies the existence of a polynomial time approximation scheme to solve the GSCST problem and its variant; (3) we finally design three strongly polynomial time algorithms to solve the special versions of the GSCST problem and its variant, respectively.

Constraint Programming for the Diameter Constrained Minimum Spanning Tree Problem

Abstract We propose a new formulation for the Diameter Constrained Minimum Spanning Tree Problem using constraint programming. Computational results have shown that this formulation combined with an appropriate search procedure solves larger instances and is faster than the other approaches in the literature.

SOLVING THE DEGREE CONSTRAINED MINIMUM SPANNING TREE PROBLEM USING TABU AND MODIFIED PENALTY SEARCH METHODS

In this paper we consider the Degree Constrained Minimum Spanning Tree Problem. This problem is concerned with finding, in a given edge weighted graph G (all weights are non-negative), 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 center. Because of its NP-completeness, a number of heuristics have been proposed. In this paper we propose two new search methods: one based on the method of Tabu search and the other based on a penalty function approach. For comparative analysis, we test our methods on some benchmark problems. The computational results support our methods.
 

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.