Constrained Minimum Spanning Tree Research Articles

Image Insertion Using Depth- and Topology- Constrained Minimum Spanning Tree for Compressed Image Sets

Image Insertion Using Depth- and Topology- Constrained Minimum Spanning Tree for Compressed Image Sets

CMSTR: A Constrained Minimum Spanning Tree Based Routing Protocol for Wireless Sensor Networks

How to extend the network lifetime with given limited energy budget is always one of the main concerns in Wireless Sensor Networks (WSNs). However, imbalanced energy consumption and overlong intra-cluster communication paths are prevalent in the hierarchical routing protocols, which shortens the network lifetime inevitably. To this end, an energy-efficient routing Protocol based on Constrained Minimum Spanning Tree (CMSTR) is proposed in this paper. To be specific, a new multichain routing scheme to balance the energy consumption for intra-cluster communications is presented. Based on the multichain routing scheme, the problem of establishing intra-cluster routing is transformed into a shortest Hamiltonian path problem on the basis of a graph-theoretic analysis model, which is solved through a Constrained Minimum Spanning Tree (CMST) algorithm proposed in this paper, with the aim to obtain the initial path for intra-cluster communications. In order to shorten the initial path length to obtain higher energy-efficient chain routes, a Neighbor Node Replacement (NNR) algorithm and a Link Intersection Detection and Elimination (LIDE) algorithm are proposed, in which the problem of potential long links and intersections is to be effectively alleviated. With shorter chain routes, unnecessary intra-cluster communication energy depletion can be reduced accordingly. In order to evaluate the performance of CMSTR, extensive simulation experiments are conducted. The results show that CMSTR can greatly prolong the network lifetime with regard to the metrics of FND and HND. To be specific, compared with LEACH, R-LEACH, and DCMSTR, the value of FND increased by 800%, 540% and 57%, that of HND increased by 322%, 286% and 22%, and overall network lifetime (AND) increased by 29%, 10% and 5%, respectively. Besides, CMSTR has a stable and lowest packet loss percentage (0.4%). In summary, CMSTR has excellent performance in terms of energy efficiency and network stability.

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.

Exact algorithms for finding constrained minimum spanning trees

For a given undirected graph with each edge associated with a weight and a length, the constrained minimum spanning tree (CMST) problem aims to compute a minimum weight spanning tree with total length bounded by a given fixed integer $$L\in {\mathbb {Z}}^{+}$$. In the paper, we first present an exact algorithm with a runtime $$O(mn^{2})$$ for CMST when the edge length is restricted to 0 and 1 based on combining the local search method and our developed bicameral edge replacement approach. Then we extend the algorithm to solve a more general case when the edge length is restricted to 0, 1 and 2 via iteratively improving a feasible solution of CMST towards an optimum solution. At last, numerical experiments are carried out to validate the practical performance of the proposed algorithms by comparing with previous algorithms as baselines.

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.

Solve Constrained Minimum Spanning Tree By cross-entropy (CE) method

Given an undirected graph and nonnegative numbers as weights for each edge, we consider the problem of finding a spanning tree that has lowest total cost with respect to cost weight and has constrained budget with respect to constrains weight [1]. Different from the traditional Lagrangean relaxation method, we proposed a new randomized method for solving the constrained spanning tree problem in this paper. For this newly proposed method every solution found is in the original problem’s feasible region, there is no need to do the hard work to close the gap between the Lagrangean relaxation and the original problem. And the proposed algorithm is very easy to parallelize, can take full advantage of multi-core processors to improve problem solving efficiency. For most optimization algorithms, properly selecting the super parameters have big impact on the algorithm’s practical performance. Through computer numerical simulation, we can see that our algorithm is robust to super parameters.

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.

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

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.

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.