Capacitated Minimum Spanning Tree Research Articles

Scatter search for capacitated minimum spanning tree

We develop a scatter search algorithm for the classical capacitated minimum spanning tree (CMST) problem. We address both the homogeneous and the more difficult heterogeneous variant of the problem. The CMST is central in a wide range of network design applications in industrial engineering, routing and logistics, and communication networks. Since the problem is NP-Complete, heuristic solution methods are often required to find high-quality solutions within practical time limits. Research efforts have been focused on complex algorithm designs combining advanced neighborhood search techniques with exact solution methods or hybrids of genetic algorithms with various alternative search techniques and likewise supplemented with optimization procedures. In this study we go back to the roots and offer an alternative algorithm that competes with the best algorithms in the literature and both avoids complicated artifacts and is simpler to implement. Key features of the proposed algorithm include an oscillation neighborhood search method characterized by two classes of neighborhoods and an evolutionary search procedure based on a scatter search framework.

A hybrid evolutionary algorithm for the capacitated minimum spanning tree problem

The capacitated minimum spanning tree (CMST) problem is a fundamental problem in telecommunication network design. Given a central node and a set of remote terminal nodes with specified demands for traffic, the goal of CMST is to find a minimum cost spanning tree (network) such that the traffic on any arc of the network satisfies the capacity constraint. To tackle this NP-hard problem, we propose an effective hybrid evolutionary algorithm that integrates a backbone-based crossover operator, a destroy-and-repair mutation operator to generate meaningful offspring solutions, and an intensification-driven adaptive tabu search procedure that considers both feasible and infeasible solutions in search of high-quality local optima. Extensive computational results on a set of 126 well-known CMST benchmark instances from the literature indicate that the proposed algorithm competes favorably with the state-of-the-art heuristics. For a selection of 25 most challenging CMST instances with unknown optimal solutions, the proposed algorithm reports new upper bounds (improved best-known solutions) in 9 cases, while reaching the best-known result for 12 instances. Furthermore, we provide experimental analyses to identify the key algorithmic features of the proposed approach.

The capacitated minimum spanning tree problem with arc time windows

We consider a new variant of the minimum spanning tree problem, where time windows are associated with the arcs of the underlying graph and capacities relate to the maximum number of vertices that subtrees may incorporate. The problem is referred to as the Capacitated Minimum Spanning Tree with Arc Time Windows (CMSTP_ATW) and emerges in routing situations with flow disruptions across road segments. We devise a Mixed Integer Programming (MIP) formulation to model the problem, which can be solved using CPLEX. To examine the quality of the solutions obtained, we convert the data sets of Solomon (1987) to appropriately capture CMSTP_ATW instances and provide results for the problems with 25, 50 and 100 vertices. Furthermore, we compare the CPLEX built-in heuristic that determines the initial integer solution for the CMSPT_ATW, vis-a-vis a greedy heuristic we have developed that offers high quality solutions in short computational times for the large size test problems. Experimental results show that there is a strong negative correlation between the GAP of CPLEX and the total number of iterations in one-hour performance time, against the no or positive correlation between the GAP of CPLEX and the initial iterations. Finally, we modify the MIP by adding parts of the solution derived, using the greedy heuristic, in the set of problem constraints, and observe that the CPLEX results for the CMSTP_ATW are in general improved, offering evidence that it is a promising solution approach.

A Branch-and-Price-and-Cut Algorithm for the Cable-Routing Problem in Solar Power Plants

A solar power plant is a large-scale photovoltaic (PV) system designed to supply usable solar power to the electricity grid. Building a solar power plant needs consideration of arrangements of several important components, such as PV arrays, solar inverters, combiner boxes, cables, and other electrical accessories. The design of solar power plants is very complex because of various optimization parameters and design regulations. In this study, we address the cable-routing problem arising in the planning of large-scale solar power plants, which aims to determine the partition of the PV arrays, the location of combiner boxes, and cable routing such that the installation cost of the cables connecting the components is minimized. We formulate the problem as a mathematical programming problem, which can be viewed as a generalized capacitated minimum spanning tree (CMST) problem, and then devise a branch-and-price-and-cut (BPC) algorithm to solve it. The BPC algorithm uses two important valid inequalities, namely the capacity inequalities and the subset-row inequalities, to tighten the lower bounds. We also adopt several acceleration strategies to speed up the algorithm. Using real-world data sets, we show by numerical experiments that our BPC algorithm is superior to the typical manual-based planning approach used by many electric power planning companies. In addition, when solving the CMST problem with unitary demands, our algorithm is highly competitive compared with the best exact algorithm in the literature.

Offshore wind farm electrical cable layout optimization

This article explores an automated approach for the efficient placement of substations and the design of an inter-array electrical collection network for an offshore wind farm through the minimization of the cost. To accomplish this, the problem is represented as a number of sub-problems that are solved in series using a combination of heuristic algorithms. The overall problem is first solved by clustering the turbines to generate valid substation positions. From this, a navigational mesh pathfinding algorithm based on Delaunay triangulation is applied to identify valid cable paths, which are then used in a mixed-integer linear programming problem to solve for a constrained capacitated minimum spanning tree considering all realistic constraints. The final tree that is produced represents the solution to the inter-array cable problem. This method is applied to a planned wind farm to illustrate the suitability of the approach and the resulting layout that is generated.

A biased random-key genetic algorithm for the capacitated minimum spanning tree problem

This paper focuses on the capacitated minimum spanning tree (CMST) problem. Given a central processor and a set of remote terminals with specified demands for traffic that must flow between the central processor and terminals, the goal is to design a minimum cost network to carry this demand. Potential links exist between any pair of terminals and between the central processor and the terminals. Each potential link can be included in the design at a given cost. The CMST problem is to design a minimum-cost network connecting the terminals with the central processor so that the flow on any arc of the network is at most Q. A biased random-key genetic algorithm (BRKGA) is a metaheuristic for combinatorial optimization which evolves a population of random vectors that encode solutions to the combinatorial optimization problem. This paper explores several solution encodings as well as different strategies for some steps of the algorithm and finally proposes a BRKGA heuristic for the CMST problem. Computational experiments are presented showing the effectiveness of the approach: Seven new best-known solutions are presented for the set of benchmark instances used in the experiments.

On the interdependency between multi-channel scheduling and tree-based routing for WSNs in smart grid environments

Field tests show that the link-quality of wireless links in different smart grid environments, such as outdoor substation, varies greatly both in space and time because of various factors, including multi-path, fading, node contentions, radio frequency (RF) interference, and noise. This leads to both time and location dependent capacity limitations of wireless links in smart grid environments. To improve network capacity in such environments, multichannel communication and the use of proper routing topologies emerge as efficient solutions to achieve simultaneous, interference-free transmissions over multiple channels. In this paper, we explore the impact of multi-channel communication and the selection of efficient routing topologies on the performance of wireless sensors networks in different smart grid spectrum environments. Particularly, we evaluate the network performance using a receiver-based channel selection method and using different routing trees, including routing trees constructed considering the link qualities, Capacitated Minimum Spanning Trees (CMSTs), capacitated minimum spanning tree considering link qualities and Minimum Hop Spanning Trees (MHSTs). We focus on performance measures such as delay and throughput that can benefit from the simultaneous parallel transmissions and show that the use multiple channels together with routing trees that consider network capacity and link quality, i.e., capacitated minimum spanning tree considering link qualities, substantially improve the network performance in harsh smart-grid environments compared to single-channel communication and minimum-hop routing trees.

Heuristics for the multi-level capacitated minimum spanning tree problem

The capacitated minimum spanning tree (CMST) problem is fundamental to the design of centralized communication networks. In this paper we consider the multi-level capacitated minimum spanning tree problem, a generalization of the well-known CMST problem. Based on work previously done in the field, three heuristics are presented, addressing unit and non-unit demand cases. The proposed heuristics have been also integrated into a mixed integer programming solver. Evaluation results are presented, for an extensive set of experiments, indicating the improvements that the heuristics bring to the particular problem.

A filter-and-fan algorithm for the capacitated minimum spanning tree problem

The capacitated minimum spanning tree (CMST) is a notoriously difficult problem in combinatorial optimization. Extensive investigation has been devoted to developing efficient algorithms to find optimal or near-optimal solutions. This paper proposes a new CMST heuristic algorithm that effectively combines the classical node-based and tree-based neighborhoods embodied in a filter-and-fan (F&F) approach, a local search procedure that generates compound moves in a tree search fashion. The overall algorithm is guided by a multi-level oscillation strategy used to trigger each type of neighborhood while allowing the search to cross feasibility boundaries. Computational results carried out on a standard set of 135 benchmark problems show that a simple F&F design competes effectively with prior CMST metaheuristics, rivaling the best methods, which are significantly more complex.

A node rooted flow-based model for the local access network expansion problem

In this paper, we present a new formulation for the local access network expansion problem. Previously, we have shown that this problem can be seen as an extension of the well-known Capacitated Minimum Spanning Tree Problem and have presented and tested two flow-based models. By including additional information on the definition of the variables, we propose a new flow-based model that permits us to use effectively variable eliminations tests as well as coefficient reduction on some of the constraints. We present computational results for instances with up to 500 nodes in order to show the advantages of the new model in comparison with the others.

Parametric enhancements of the Esau–Williams heuristic for the capacitated minimum spanning tree problem

The Capacitated Minimum Spanning Tree Problem is NP-hard and several heuristic solution methods have been proposed. They can be classified as classical ones and metaheuristics. Recent developments have shown that metaheuristics outperform classical heuristics. However, they require long computation times and there are difficulties in their parameter calibration and coding phases. This explains the popularity of the Esau–Williams (EW) heuristic in practice, and its use in many successful metaheuristics and second-order greedy methods. In this work, we are concerned with the EW heuristic and we propose simple new enhancements. Based on our computational experiments, we can say that they considerably improve its accuracy with minor increase in computation time, and without harming its simplicity.

A Modified Edge Removal Stiener Tree Heuristic for Global Routing in VLSI Design

Routing is a key stage for VLSI physical design. Steiner tree construction is a well studied topic in design automation. There have been a number of significant theoretical advances in the past few years. The focus of this paper is on combining the speed and solution quality of a high quality Steiner heuristic with the reality of modern routing. Practical designs contain routing congestion and blockages; routing is implemented across multiple layers. Each routing layer has preferred directions, and connecting vias have significant cost. In a modern design, many trees are in competition with each other for scarce routing resources. The objective is not to simply build trees with the lowest length; they must also be low cost. We present an approach that is as fast as spanning tree construction, while accurately modeling routing costs. Our work extends an earlier Steiner tree heuristic algorithm by adding the ability to minimize the routing congestion without altering the computational complexity of the underlying algorithm. We compare our CAST algorithm with the capacitated minimum spanning tree (CMST) and the ER algorithm finding that our approach offers impressive reduction in congestion cost in average about 23.1% and 63.4%, respectively.

A dandelion-encoded evolutionary algorithm for the delay-constrained capacitated minimum spanning tree problem

This paper proposes an evolutionary algorithm with Dandelion-encoding to tackle the Delay-Constrained Capacitated Minimum Spanning Tree (DC-CMST) problem. This problem has been recently proposed, and consists of finding several broadcast trees from a source node, jointly considering traffic and delay constraints in trees. A version of the problem in which the source node is also included in the optimization process is considered as well in the paper. The Dandelion code used in the proposed evolutionary algorithm has been recently proposed as an effective way of encoding trees in evolutionary algorithms. Good properties of locality has been reported on this encoding, which makes it very effective to solve problems in which the solutions can be expressed in form of trees. In the paper we describe the main characteristics of the algorithm, the implementation of the Dandelion-encoding to tackled the DC-CMST problem and a modification needed to include the source node in the optimization. In the experimental section of this article we compare the results obtained by our evolutionary with that of a recently proposed heuristic for the DC-CMST, the Least Cost (LC) algorithm. We show that our Dandelion-encoded evolutionary algorithm is able to obtain better results that the LC in all the instances tackled.

GRASP with hybrid heuristic-subproblem optimization for the multi-level capacitated minimum spanning tree problem

We propose a GRASP using an hybrid heuristic-subproblem optimization approach for the Multi-Level Capacitated Minimum Spanning Tree (MLCMST) problem. The motivation behind such approach is that to evaluate moves rearranging the configuration of a subset of nodes may require to solve a smaller-sized MLCMST instance. We thus use heuristic rules to define, in both the construction and the local search phases, subproblems which are in turn solved exactly by employing an integer programming model. We report numerical results obtained on benchmark instances from the literature, showing the approach to be competitive in terms of solution quality. The proposed GRASP have in fact improved the best known upper bounds for almost all of the considered instances.

Design of capacitated minimum spanning tree with uncertain cost and demand parameters

The classical Capacitated Minimum Spanning Tree Problem (CMSTP) deals with finding a minimum-cost spanning tree so that the total demand of the vertices in each subtree does not exceed the capacity limitation. In most of the CMSTP models, the edge costs and the demands of the vertices in the network are assumed to be known with certainty. This paper considers the CMSTP model, where the edge costs and/or the demands are only approximately known. A fast approximate reasoning algorithm, which is based on the Esau–Williams savings heuristic and fuzzy logic rules, is proposed. The computational results of the study based on the proposed approach are also reported.

Solving the variable size bin packing problem with discretized formulations

In this paper we study the use of a discretized formulation for solving the variable size bin packing problem (VSBPP). The VSBPP is a generalization of the bin packing problem where bins of different capacities (and different costs) are available for packing a set of items. The objective is to pack all the items minimizing the total cost associated with the bins. We start by presenting a straightforward integer programming formulation to the problem and later on, propose a less straightforward formulation obtained by using a so-called discretized model reformulation technique proposed for other problems (see [Gouveia L. A 2n constraint formulation for the capacitated minimal spanning tree problem. Operations Research 1995; 43:130–141; Gouveia L, Saldanha-da-Gama F. On the capacitated concentrator location problem: a reformulation by discretization. Computers and Operations Research 2006; 33:1242–1258]). New valid inequalities suggested by the variables of the discretized model are also proposed to strengthen the original linear relaxation bounds. Computational results (see Section 4) with up to 1000 items show that these valid inequalities not only enhance the linear programming relaxation bound but may also be extremely helpful when using a commercial package for solving optimally VSBPP.

Robust branch-cut-and-price for the Capacitated Minimum Spanning Tree problem over a large extended formulation

This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arborescence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms: powerful new cuts expressed over a very large set of variables are added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very significant improvements over previous algorithms. Several open instances could be solved to optimality.

The Multilevel Capacitated Minimum Spanning Tree Problem

In this paper, we consider the multilevel capacitated minimum spanning tree (MLCMST) problem, a generalization of the well-known capacitated minimum spanning tree (CMST) problem, that allows for multiple facility types in the design of the network. We develop two flow-based mixed integer programming formulations that can be used to find tight lower bounds for MLCMST problems with up to 150 nodes. We also develop several heuristic procedures for the MLCMST problem. First, we present a savings-based heuristic. Next, we develop local search algorithms that use exponential size, node-based, cyclic and path exchange neighborhoods. Finally, we develop a hybrid genetic algorithm for the MLCMST. Extensive computational results on a large set of test problems indicate that the genetic algorithm is robust and, among the heuristics, generates the best solutions. They are typically 6.09% from the lower bound and 0.25% from the optimal solution value.

Least cost heuristic for the delay-constrained capacitated minimum spanning tree problem

This study deals with the Delay-Constrained Capacitated Minimum Spanning Tree (DC-CMST) problem arising in topology discovery of local network. While the traditional Capacitated Minimum Spanning Tree (CMST) problem deals with only traffic capacity constraint of a port of a source node, and Delay-Constrained Minimum Spanning Tree (DCMST) considers only the maximum end-to-end delay constraint, the DC-CMST problem deals with mean network delay and traffic capacity constraints simultaneously. The DC-CMST problem consists of finding a set of minimum cost spanning trees to link end-nodes to a source node which satisfy the traffic requirements at end-nodes and the required mean delay of a network. We formulate the DC-CMST problem and present the Least-Cost DC-CMST Heuristic, which consists of node exchange algorithm, node shift algorithm, and mean delay link capacity algorithm to solve the DC-CMST problem. Results from performance analysis show that the proposed heuristic can produce, in a very short computation time, better results than the previous CMST algorithms modified to solve the DC-CMST problem. The proposed Least-Cost DC-CMST Heuristic can be applied to the topological design of large local networks and to the construction of least cost broadcast and multicast trees for efficient routing algorithms.

Savings based ant colony optimization for the capacitated minimum spanning tree problem

The problem of connecting a set of client nodes with known demands to a root node through a minimum cost tree network, subject to capacity constraints on all links is known as the capacitated minimum spanning tree (CMST) problem. As the problem is NP-hard, we propose a hybrid ant colony optimization (ACO) algorithm to tackle it heuristically. The algorithm exploits two important problem characteristics: (i) the CMST problem is closely related to the capacitated vehicle routing problem (CVRP), and (ii) given a clustering of client nodes that satisfies capacity constraints, the solution is to find a MST for each cluster, which can be done exactly in polynomial time. Our ACO exploits these two characteristics of the CMST by a solution construction originally developed for the CVRP. Given the CVRP solution, we then apply an implementation of Prim's algorithm to each cluster to obtain a feasible CMST solution. Results from a comprehensive computational study indicate the efficiency and effectiveness of the proposed approach.