Combining variable neighborhood search with integer linear programming for the generalized minimum spanning tree problem

We consider the generalized version of the classical Minimum Spanning Tree problem where the nodes of a graph are partitioned into clusters and exactly one node from each cluster must be connected. We present a Variable Neighborhood Search (VNS) approach which uses three different neighborhood types. Two of them work in complementary ways in order to maximize search effectivity. Both are large in the sense that they contain exponentially many candidate solutions, but efficient polynomial-time algorithms are used to identify best neighbors. For the third neighborhood type we apply Mixed Integer Programming to optimize local parts within candidate solution trees. Tests on Euclidean and random instances with up to 1280 nodes indicate especially on instances with many nodes per cluster significant advantages over previously published metaheuristic approaches. This work is supported by the RTN ADONET under grant 504438.     

A tabu search heuristic for the generalized minimum spanning tree problem

This paper describes an attribute based tabu search heuristic for the generalized minimum spanning tree problem (GMSTP) known to be NP-hard. Given a graph whose vertex set is partitioned into clusters, the GMSTP consists of designing a minimum cost tree spanning all clusters. An attribute based tabu search heuristic employing new neighborhoods is proposed. An extended set of TSPLIB test instances for the GMSTP is generated and the heuristic is compared with recently proposed genetic algorithms. The proposed heuristic yields the best results for all instances. Moreover, an adaptation of the tabu search algorithm is proposed for a variation of the GMSTP in which each cluster must be spanned at least once.     

A GRASP algorithm for the multi-criteria minimum spanning tree problem

This paper proposes a GRASP (Greedy Randomized Adaptive Search Procedure) algorithm for the multi-criteria minimum spanning tree problem, which is NP-hard. In this problem a vector of costs is defined for each edge of the graph and the problem is to find all Pareto optimal or efficient spanning trees (solutions). The algorithm is based on the optimization of different weighted utility functions. In each iteration, a weight vector is defined and a solution is built using a greedy randomized constructive procedure. The found solution is submitted to a local search trying to improve the value of the weighted utility function. We use a drop-and-add neighborhood where the spanning trees are represented by Prufer numbers. In order to find a variety of efficient solutions, we use different weight vectors, which are distributed uniformly on the Pareto frontier. The proposed algorithm is tested on problems with r=2 and 3 criteria. For non-complete graphs with n=10, 20 and 30 nodes, the performance of the algorithm is tested against a complete enumeration. For complete graphs with n=20, 30 and 50 nodes the performance of the algorithm is tested using two types of weighted utility functions. The algorithm is also compared with the multi-criteria version of the Kruskal’s algorithm, which generates supported efficient solutions. This work was funded by the Municipal Town Hall of Campos dos Goytacazes city. The used computer was acquired with resource of CNPq.     

On the prize-collecting generalized minimum spanning tree problem

The prize-collecting generalized minimum spanning tree problem (PC-GMSTP), is a generalization of the generalized minimum spanning tree problem (GMSTP) and belongs to the hard core of -hard problems. We describe an exact exponential time algorithm for the problem, as well we present several mixed integer and integer programming formulations of the PC-GMSTP. Moreover, we establish relationships between the polytopes corresponding to their linear relaxations and present an efficient solution procedure that finds the optimal solution of the PC-GMSTP for graphs with up 240 nodes.     

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.     

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.     

Expected runtimes of a simple evolutionary algorithm for the multi-objective minimum spanning tree problem

Evolutionary algorithms are applied to problems that are not well understood as well as to problems in combinatorial optimization. The analysis of these search heuristics has been started for some well-known polynomial solvable problems. Such analyses are starting points for the analysis of evolutionary algorithms on difficult problems. We present the first runtime analysis of a multi-objective evolutionary algorithm on a NP-hard problem. The subject of our analysis is the multi-objective minimum spanning tree problem for which we give upper bounds on the expected time until a simple evolutionary algorithm has produced a population including for each extremal point of the Pareto front a corresponding spanning tree. These points are of particular interest as they give a 2-approximation of the Pareto front. We show that in expected pseudopolynomial time a population is produced that includes for each extremal point a corresponding spanning tree.     

Simulated annealing algorithm for the robust spanning tree problem

This paper addresses the robust spanning tree problem with interval data, i.e. the case of classical minimum spanning tree problem when edge weights are not fixed but take their values from some intervals associated with edges. The problem consists of finding a spanning tree that minimizes so-called robust deviation, i.e. deviation from an optimal solution under the worst case realization of interval weights. As it was proven in Kouvelis and Yu (Robust Discrete Optimization and Its Applications, Kluwer Academic, Norwell, 1997), the problem is NP-hard, therefore it is of great interest to tackle it with some metaheuristic approach, namely simulated annealing, in order to calculate an approximate solution for large scale instances efficiently. We describe theoretical aspects and present the results of computational experiments. To the best of our knowledge, this is the first attempt to develop a metaheuristic approach for solving the robust spanning tree problem.     

单位无穷范数下边权有界的最小支撑树逆最优值问题

研究了单位$l_{\infty}$范数下边权有界的最小支撑树逆最优值问题。给定一个边赋权无向连通网络$G=(V, E, w)$, 支撑树$T^0$, 下界向量$\bm{l}$, 上界向量$\bm{u}$及数值$K$, 寻求一个新的边权向量$\bm{\bar{w}}$满足上下界约束$\bm{l}\le\bar{\bm w}\le {\bm u}$, 且$T^0$是在向量$\bm{\bar{w}}$下权值为$K$的一个最小支撑树, 目标是在单位$l_{\infty}$范数下使得修改成本$\|\bar{\bm w}-{\bm w}\|$最小。本文给出了该问题的数学模型, 分析了其最优性条件, 设计了求解该问题的时间复杂度为$O(|V||E|)$的强多项式时间算法。     

单位无穷范数下边权有界的最小支撑树逆最优值问题

研究了单位$l_{\infty}$范数下边权有界的最小支撑树逆最优值问题。给定一个边赋权无向连通网络$G=(V, E, w)$, 支撑树$T^0$, 下界向量$\bm{l}$, 上界向量$\bm{u}$及数值$K$, 寻求一个新的边权向量$\bm{\bar{w}}$满足上下界约束$\bm{l}\le\bar{\bm w}\le {\bm u}$, 且$T^0$是在向量$\bm{\bar{w}}$下权值为$K$的一个最小支撑树, 目标是在单位$l_{\infty}$范数下使得修改成本$\|\bar{\bm w}-{\bm w}\|$最小。本文给出了该问题的数学模型, 分析了其最优性条件, 设计了求解该问题的时间复杂度为$O(|V||E|)$的强多项式时间算法。     

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.     

Np-hardness proof and an approximation algorithm for the minimum vertex ranking spanning tree problem

The minimum vertex ranking spanning tree problem (MVRST) is to find a spanning tree of G whose vertex ranking is minimum. In this paper, we show that MVRST is NP-hard. To prove this, we polynomially reduce the 3-dimensional matching problem to MVRST. Moreover, we present a (⌈D_s/2⌉+1)/(⌊log₂(D_s+1)⌋+1)-approximation algorithm for MVRST where D_s is the minimum diameter of spanning trees of G.     

TAGE iterative algorithm and nonpolynomial spline basis for the solution of nonlinear singular second order ordinary differential equations

In the present paper, we discuss three point difference method based on nonpolynomial spline basis for the second order ordinary differential equation. Difference schemes are derived for linear and nonlinear case and are used to solve via two parameter alternating group explicit iterative algorithm. The schemes have a fourth and second order of uniform convergence for the choice of the parameters involved in the method. Computational results are presented comparing the two methods in terms of accuracy and execution times. The results indicate the advantage of using parallel implementation of the new method.