A Probabilistic Tabu Search Algorithm for the Generalized Minimum Spanning Tree Problem

In this paper we present a probabilistic tabu search algorithm for the generalized minimum spanning tree problem. The basic idea behind the algorithm is to use preprocessing operations to arrive at a probability value for each vertex which roughly corresponds to its probability of being included in an optimal solution, and to use such probability values to shrink the size of the neighborhood of solutions to manageable proportions. We report results from computational experiments that demonstrate the superiority of this method over the generic tabu search method.

Selecting Optimal Minimum Spanning Trees that Share a Topological Correspondence with Phylogenetic Trees

Choi et. al (2011) introduced a minimum spanning tree (MST)-based method called CLGrouping, for constructing tree-structured probabilistic graphical models, a statistical framework that is commonly used for inferring phylogenetic trees. While CLGrouping works correctly if there is a unique MST, we observe an indeterminacy in the method in the case that there are multiple MSTs. In this work we remove this indeterminacy by introducing so-called vertex-ranked MSTs. We note that the effectiveness of CLGrouping is inversely related to the number of leaves in the MST. This motivates the problem of finding a vertex-ranked MST with the minimum number of leaves (MLVRMST). We provide a polynomial time algorithm for the MLVRMST problem, and prove its correctness for graphs whose edges are weighted with tree-additive distances

On the probabilistic min spanning tree Problem

We study a probabilistic optimization model for min spanning tree, where any vertex vi of the input-graph G(V,E) has some presence probability pi in the final instance G′ ⊂ G that will effectively be optimized. Suppose that when this “real” instance G′ becomes known, a spanning tree T, called anticipatory or a priori spanning tree, has already been computed in G and one can run a quick algorithm (quicker than one that recomputes from scratch), called modification strategy, that modifies the anticipatory tree T in order to fit G ′. The goal is to compute an anticipatory spanning tree of G such that, its modification for any G ′ ⊆ G is optimal for G ′. This is what we call probabilistic min spanning tree problem. In this paper we study complexity and approximation of probabilistic min spanning tree in complete graphs under two distinct modification strategies leading to different complexity results for the problem. For the first of the strategies developed, we also study two natural subproblems of probabilistic min spanning tree, namely, the probabilistic metric min spanning tree and the probabilistic min spanning tree 1,2 that deal with metric complete graphs and complete graphs with edge-weights either 1, or 2, respectively

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 effective heuristics (Modified 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

Heuristics based on 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-complete. A Greedy Randomized Adaptive Search Procedure (GRASP) and different versions of Variable Neighbourhood Search (VNS) are proposed. 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