Minimum Label Spanning Tree Problem Research Articles

Community-based zigzag piloting algorithm for the strong generalized minimum label spanning tree problem

Community-based zigzag piloting algorithm for the strong generalized minimum label spanning tree problem

Finding minimum label spanning trees using cross‐entropy method

AbstractObtaining high‐quality solutions to the minimum label spanning tree problem is of crucial importance to efficient communication network design, since such solutions can reduce both the construction cost and the complexity of the ultimate architecture. However, the corresponding optimization task was shown to be hard even for complete graphs. As a consequence, no computationally efficient method for solving this problem exactly in a reasonable time is known to exist and one has to rely on approximation techniques such as heuristic and evolutionary algorithms. In this study, we investigate the performance of a different method called the Cross‐Entropy algorithm which relies on rigorous developments in the fields of information theory and stochastic simulation. Our findings indicate that the mathematical soundness of the Cross‐Entropy method makes it very reliable and robust as compared to its counterparts. In particular, the obtained results suggest that the Cross‐Entropy method is not sensitive to different graph models and that the proposed algorithm can obtain optimal or near‐optimal solutions while using a reasonable computational effort.

A Novel Binary Firey Algorithm for the Minimum Labeling Spanning Tree Problem

Given a connected undirected graph G whose edges are labeled, the minimum labeling spanning tree (MLST) problem is to find a spanning tree of G with the smallest number of different labels. The MLST is an NP-hard combinatorial optimization problem, which is widely applied in communication networks, multimodal transportation networks, and data compression. Some approximation algorithms and heuristics algorithms have been proposed for the problem. Firey algorithm is a new meta-heuristic algorithm. Because of its simplicity and easy implementation, it has been successfully applied in various fields. However, the basic firefly algorithm for the MLST problem is proposed in this paper. A binary operation method to update firefly positions and a local feasible handling method are introduced, which correct unfeasible solutions, eliminate redundant labels, and make the algorithm more suitable for discrete problems. Computational results show that the algorithm has good performance. The algorithm can be extended to solve other discrete optimization problems.

Heuristics for the strong generalized minimum label spanning tree problem

AbstractIn this work we introduce and study the strong generalized minimum label spanning tree (GMLST), a novel optimization problem defined on edge‐labeled graphs. Given a label set associated to each edge of the input graph, the aim is to look for the spanning tree using the minimum number of labels. Differently from the previously introduced GMLST problem, including a given edge in the solution means that all its labels are used. We present a mathematical formulation, as well as three heuristic approaches to solve the problem. Computational results compare the performances of the proposed algorithms.

Multi-objective evolutionary algorithm on simplified bi-objective minimum weight minimum label spanning tree problems

As general purpose optimisation methods, evolutionary algorithms have been efficiently used to solve multi-objective combinatorial optimisation problems. However, few theoretical investigations have been conducted to understand the efficiency of evolutionary algorithms on such problems, and even fewer theoretical investigations have been conducted on multi-objective combinatorial optimisation problems coming from real world. In this paper, we analyse the performance of a simple multi-objective evolutionary algorithm on two simplified instances of the bi-objective minimum weight minimum label spanning tree problem, which comes from real world. Though these two instances are similar, the analysis results show that the simple multi-objective evolutionary algorithm is efficient for one instance, but it may be inefficient for the other. According to the analysis on the second instance, we think that the restart strategy may be useful in making the multi-objective evolutionary algorithm more efficient for the bi-objective problem.

A hybrid metaheuristic for the minimum labeling spanning tree problem

An edge-labeled graph (ELG) is a graph in which each edge has a label associated. Given G, an ELG, the minimum labeling spanning tree problem (MLSTP) is an NP-hard problem that consists in finding a spanning tree in G by using a minimum number of labels. The MLSTP has applications in areas such as computer networks, multimodal transportation networks, and data compression. This paper introduces new concepts, presents a revised version of the maximum vertex covering algorithm and provides a tighter bound to its time complexity. Further, a new MIP-based metaheuristic is proposed for solving the MLSTP, the multi-start local branching (MSLB). It combines the efficiency of the proposed constructive heuristics with the capacity of exploration of a new local search method based on MIP. The computational experiments performed show that the MSLB is superior to the current state-of-the-art metaheuristics in respect to quality of the solutions and processing times.

A polyhedral approach to the generalized minimum labeling spanning tree problem

The minimum labeling spanning tree problem (MLSTP) is a combinatorial optimization problem that consists in finding a spanning tree in a simple graph G, in which each edge has one label, by using a minimum number of labels. It is an NP-hard problem that was introduced by Chang and Leu (Inf Process Lett 63(5):277–282, 1997. https://doi.org/10.1016/S0020-0190(97)00127-0 ). Chen et al. (Comparison of heuristics for solving the gmlst problem, in: Raghavan, Golden, Wasil (eds) Telecommunications modeling, policy, and technology, Springer, Boston, pp 191–217, 2008) subsequently proposed a generalization of the MLSTP, called the generalized minimum labeling spanning tree problem (GMLSTP), that allows a situation in which multiple labels can be assigned to an edge. Here, we show how the GMLSTP can be expressed as an MLSTP in a multigraph. Both problems have applications in various areas such as computer network design, multimodal transportation network design, and data compression. We propose a new compact binary integer programming model to solve exactly the GMLSTP and analyze the polytope associated with the formulation. The paper introduces new concepts, gives the polytope dimension, and describes five new facet families. The polyhedral comparison results for the studied polytope show that the new model is theoretically superior to current state-of-the-art formulations.

Carousel greedy: A generalized greedy algorithm with applications in optimization

In this paper, we introduce carousel greedy, an enhanced greedy algorithm which seeks to overcome the traditional weaknesses of greedy approaches. We have applied carousel greedy to a variety of well-known problems in combinatorial optimization such as the minimum label spanning tree problem, the minimum vertex cover problem, the maximum independent set problem, and the minimum weight vertex cover problem. In all cases, the results are very promising. Since carousel greedy is very fast, it can be used to solve very large problems. In addition, it can be combined with other approaches to create a powerful, new metaheuristic. Our goal in this paper is to motivate and explain the new approach and present extensive computational results.

Comparison of metaheuristics for thek‐labeled spanning forest problem

AbstractIn this paper, we study thek‐labeled spanning forest (kLSF) problem in which an undirected graph whose edges are labeled and an integer‐positive valueare given; the aim is to find a spanning forest of the input graph with the minimum number of connected components and the upper boundon the number of labels. The problem is related to the minimum labeling spanning tree problem and has several applications in the real world. In this paper, we compare several metaheuristics to solve this NP‐hard problem. In particular, the proposed intelligent variable neighborhood search (VNS) shows excellent performance, obtaining high‐quality solutions in short computational running time. This approach integrates VNS with other complementary approaches from machine learning, statistics, and experimental algorithmics, in order to produce high‐quality performance and completely automate the resulting optimization strategy.

Variable neighbourhood search for the k-labelled spanning forest problem

In this paper we present an efficient Variable Neighbourhood Search for the k-labelled spanning forest (kLSF) problem, an NP-hard graph problem related to the minimum labelling spanning tree problem, and with strong implications in multimodal transportation networks. Given an undirected input graph whose edges are labelled, and an integer positive value k¯, the aim is to find a spanning forest of the input graph having the minimum number of connected components and the upper bound k¯ on the number of labels to use.

OMEGA one multi ethnic genetic approach

The genetic algorithm (GA) is a quite efficient paradigm to solve several optimization problems. It is substantially a search technique that uses an ever-changing neighborhood structure related to a population which evolves according to a number of genetic operators. In the GA framework many techniques have been devised to escape from a local optimum when the algorithm fails in locating the global one. To this aim we present a variant of the GA which we call OMEGA (One Multi Ethnic Genetic Approach). The main difference is that, starting from an initial population, \(k\) different sub-populations are produced at each iteration and they independently evolve in \(k\) different environments. The resulting sub–populations are then recombined and the process is iterated. Our basic algorithmic scheme is tested on a recent and well-studied variant of the classic problem of the minimum spanning tree: the Minimum Labeling Spanning Tree problem. We compare our algorithm with several approaches drawn from the literature. The results are encouraging in view of future application of OMEGA to other classes of problems.

Solving the minimum labelling spanning tree problem by intelligent optimization

Research on intelligent optimization is concerned with developing algorithms in which the optimization process is guided by an “intelligent agent”, whose role is to deal with algorithmic issues such as parameters tuning, adaptation, and combination of different existing optimization techniques, with the aim of improving the efficiency and robustness of the optimization process. This paper proposes an intelligent optimization approach to solve the minimum labelling spanning tree (MLST) problem. The MLST problem is a combinatorial optimization problem where, given a connected, undirected graph whose edges are labelled (or coloured), the aim is to find a spanning tree whose edges have the smallest number of distinct labels (or colours). In recent work, the MLST problem has been shown to be NP-hard and some effective metaheuristics have been proposed and analysed. The intelligent optimization algorithm proposed here integrates the basic variable neighbourhood search heuristic with other complementary approaches from machine learning, statistics and experimental soft computing, in order to produce high-quality performance and to completely automate the resulting optimization strategy. We present experimental results on randomly generated graphs with different statistical properties, and demonstrate the implementation, the robustness, and the empirical scalability of our intelligent local search. Our computational experiments show that the proposed strategy outperforms heuristics recommended in the literature and is able to obtain high quality solutions quickly.

Performance Analysis of Evolutionary Algorithms for the Minimum Label Spanning Tree Problem

Some experimental investigations have shown that evolutionary algorithms (EAs) are efficient for the minimum label spanning tree (MLST) problem. However, we know little about that in theory. As one step towards this issue, we theoretically analyze the performances of the (1+1) EA, a simple version of EAs, and a multi-objective evolutionary algorithm called GSEMO on the MLST problem. We reveal that for the MLST$_{b}$ problem the (1+1) EA and GSEMO achieve a $\frac{b+1}{2}$-approximation ratio in expected polynomial times of $n$ the number of nodes and $k$ the number of labels. We also show that GSEMO achieves a $(2ln(n))$-approximation ratio for the MLST problem in expected polynomial time of $n$ and $k$. At the same time, we show that the (1+1) EA and GSEMO outperform local search algorithms on three instances of the MLST problem. We also construct an instance on which GSEMO outperforms the (1+1) EA.

The k-labeled Spanning Forest Problem

In the k-labeled Spanning Forest Problem (kLSF), given a graph G with a label (color) assigned to each edge, and an integer positive value kmax we look for the minimum number of connected components that can be obtained by using at most kmax different labels. The problem is strictly related to the Minimum Labelling Spanning Tree Problem (MLST), since a spanning tree of the graph (i.e. a single connected component) would obviously be an optimal solution for the kLSF, if it can be obtained without violating the bound on kmax. In this work we present heuristic and exact approaches to solve this new problem.

Intelligent variable neighbourhood search for the minimum labelling spanning tree problem

Given a connected, undirected graph whose edges are labelled (or coloured), the minimum labelling spanning tree (MLST) problem seeks a spanning tree whose edges have the smallest number of distinct labels (or colours). In recent work, the MLST problem has been shown to be NP-hard and some effective heuristics have been proposed and analyzed. In a currently ongoing project, we investigate an intelligent optimization algorithm to solve the problem. It is obtained by the basic Variable Neighbourhood Search heuristic with the integration of other complements from machine learning, statistics and experimental algorithmics, in order to produce high-quality performance and to completely automate the resulting optimization strategy. Computational experiments show that the proposed metaheuristic has high-quality performance for the MLST problem and it is able to obtain optimal or near-optimal solutions in short computational running time.

Solving the minimum labelling spanning tree problem using hybrid local search

Given a connected, undirected graph whose edges are labelled (or coloured), the minimum labelling spanning tree (MLST) problem seeks a spanning tree whose edges have the smallest number of distinct labels (or colours). In recent work, the MLST problem has been shown to be NP-hard and some efiective heuristics (Modifled Genetic Algorithm (MGA) and Pilot Method (PILOT)) have been proposed and analyzed. A hybrid local search method, that we call Group-Swap Variable Neighbourhood Search (GS-VNS), is proposed in this paper. It is obtained by combining two classic metaheuristics: Variable Neighbourhood Search (VNS) and Simulated Annealing (SA). Computational experiments show that GS-VNS outperforms MGA and PILOT. Furthermore, a comparison with the results provided by an exact approach shows that we may quickly obtain optimal or near-optimal solutions with the proposed heuristic.

Solving the Minimum Label Spanning Tree Problem by Mathematical Programming Techniques

We present exact mixed integer programming approaches including branch-and-cut and branch-and-cut-and-price for the minimum label spanning tree problem as well as a variant of it having multiple labels assigned to each edge. We compare formulations based on network flows and directed connectivity cuts. Further, we show how to use odd-hole inequalities and additional inequalities to strengthen the formulation. Label variables can be added dynamically to the model in the pricing step. Primal heuristics are incorporated into the framework to speed up the overall solution process. After a polyhedral comparison of the involved formulations, comprehensive computational experiments are presented in order to compare and evaluate the underlying formulations and the particular algorithmic building blocks of the overall branch-and-cut- (and-price) framework.

Greedy Randomized Adaptive Search and Variable Neighbourhood Search for the minimum labelling spanning tree problem

This paper studies heuristics for the minimum labelling spanning tree (MLST) problem. The purpose is to find a spanning tree using edges that are as similar as possible. Given an undirected labelled connected graph, the minimum labelling spanning tree problem seeks a spanning tree whose edges have the smallest number of distinct labels. This problem has been shown to be NP-hard. A Greedy Randomized Adaptive Search Procedure (GRASP) and a Variable Neighbourhood Search (VNS) are proposed in this paper. They are compared with other algorithms recommended in the literature: the Modified Genetic Algorithm and the Pilot Method. Nonparametric statistical tests show that the heuristics based on GRASP and VNS outperform the other algorithms tested. Furthermore, a comparison with the results provided by an exact approach shows that we may quickly obtain optimal or near-optimal solutions with the proposed heuristics.

A mixed integer linear formulation for the minimum label spanning tree problem

In this paper we deal with the minimum label spanning tree problem. This is a relevant problem with applications such as telecommunication networks or electric networks, where each edge is assigned with a label (such as a color) and it is intended to determine a spanning tree with the minimum number of different labels. We introduce some mixed integer formulations for this problem and prove that one of their relaxations always gives the optimal value. Finally we present and discuss the results of computational experiments.

Variable neighbourhood search for the minimum labelling Steiner tree problem

We present a study on heuristic solution approaches to the minimum labelling Steiner tree problem, an NP-hard graph problem related to the minimum labelling spanning tree problem. Given an undirected labelled connected graph, the aim is to find a spanning tree covering a given subset of nodes of the graph, whose edges have the smallest number of distinct labels. Such a model may be used to represent many real world problems in telecommunications and multimodal transportation networks. Several metaheuristics are proposed and evaluated. The approaches are compared to the widely adopted Pilot Method and it is shown that the Variable Neighbourhood Search that we propose is the most effective metaheuristic for the problem, obtaining high quality solutions in short computational running times.