Degree Constrained Minimum Spanning Tree Research Articles

WAC4 algorithm to solve the multiperiod degree constrained minimum spanning tree problem

The Multiperiod Degree Constrained Minimum Spanning Tree (MPDCMST) is a problem of finding the smallest weight spanning tree while also maintaining the degree restriction in every vertex and satisfying the vertex installation requirement in every period. This problem arises in the networks installation problem where the degree restriction represents the reliability of each vertex and the vertex installation requirement represents the priority vertices that must be installed in the network on a certain period. The installation is divided into some periods because some conditions occur such as harsh weather, fund limitation, etc. In this paper, we propose a WAC4 Algorithm to solve the MPDCMST problem. The performance of the algorithm will be compared to the WAC1 algorithm already in the literature.

USING MODIFICATION OF PRIM’S ALGORITHM AND GNU OCTAVE AND TO SOLVE THE MULTIPERIODS INSTALLATION PROBLEM

The Minimum Spanning Tree (MST) is one of the famous problems that is used mostly as the backbone in many network design problems. 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 requirement on every period. Two algorithms (WWM1 and WWM2) are proposed for solving this problem. GNU OCTAVE is used for coding and visualization. GNU is a recursive acronym for "GNU's Not Unix!", and that name is chosen because it is like Unix but differs from Unix because it is free and contains no Unix code. Those algorithms were implemented using 300 randomly generated problems. Moreover, we compare WWM1 and WWM2 algorithms using existing data from the literature and the results show that WWM2 is the best.
 ABSTRAK: Minimum Spanning Tree (MST) merupakan salah satu masalah mahsyur yang banyak digunakan sebagai tulang belakang kepada masalah banyak rekaan jaringan. Menerusi graf G(V,E), di mana V adalah himpunan titik dan E adalah himpunan garis yang menghubungkan titik-titik dalam V, dan bagi setiap garis eij terdapat berat berhubung cij ?0, Multi-period Degree Constrained Minimum Spanning Tree (MPDCMST) merupakan masalah dalam menentukan MST, pada masa sama turut menimbangkan kekangan pada setiap titik vertek, dan memenuhi syarat keperluan pemasangan pada setiap detik. Dua algoritma (WWM1 dan WWM2) dicadangkan bagi menyelesaikan masalah ini. GNU OCTAVE digunakan bagi pengaturcaraan dan visualisasi. GNU merupakan suatu singkatan kepada “GNU's Not Unix”, dan nama tersebut dipilih karena ianya seperti Unix, tetapi berbeza dari Unix kerana ia percuma dan tidak mempunyai kod Unix. Algoritma tersebut dilaksana dengan menggunakan 300 masalah terhasil secara rawak. Tambahan, algoritma WWM1 dan WWM2 dibandingkan dengan kajian terdahulu dan hasil kajian menunjukkan WWM2 adalah terbaik.

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.

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.

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

Comparative analysis of some modified prim�s algorithms to solve the multiperiod degree constrained minimum spanning tree problem

Objectives: To compare the WAC1, WAC2, and WAC3 algorithms against WADR1, WADR2, WADR3, WADR4, and WADR5 algorithms to solve the Multiperiod Degree Constrained Minimum Spanning Tree. Methods/Statistical analysis: WAC1, WAC2, and WAC3 are algorithms developed by modifying Prim’s algorithm for the Minimum Spanning Tree (MST) and adopting the period of installation/connecting for vertices in the network. In WAC1, WAC2, and WAC3 algorithms we consider the HVTk , the set of vertices that must be connected on kth period, while in WADR5 is not. In WAC1 the vertices in HVTk must be installed as early as possible, while in WAC2 is not given priority to be connected as soon as possible, but can be any time as long as the connection still on that current period. In WAC3, we adopt the smallest value for 2-path for vertex under consideration to be connected. All algorithm proposed used the same data as used in WADR1, WADR2,WADR3, WADR4 and WADR5. Findings: Since WAC1, WAC2, and WAC3, WADR5 algorithms are based on modified Prim’s algorithm, then the connectivity property is maintained during the process of installation/connection not like WADR1, WADR2, WADR3, and WADR4 where based on kruskal’s. Based on the same data used and connectivity property, the result shows that the performance of WAC2 is the best among the other algorithms developed. Application/Improvements: Considering real life application in the network installation problem, the WAC2 algorithm is one of alternative solutions since it maintains connectivity property and performs best. Keywords: Comparative Analysis, Connectivity, Degree Constrained, Modified Prim, Multi Period

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.

K-opt를 적용한 차수 제약 최소신장트리 알고리즘

k-opt를 적용한 차수 제약 최소신장트리 알고리즘

A Simple Degree-Constrained Minimum Spanning Tree Algorithm

A Simple Degree-Constrained Minimum Spanning Tree Algorithm

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.

A Hybrid Algorithm for Degree-Constrained Minimum Spanning Tree

A Hybrid Algorithm for Degree-Constrained Minimum Spanning Tree

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.

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.

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.

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.