Reformulations and solution algorithms for the maximum leaf spanning tree problem

Abstract

Given a graph G = (V, E), the maximum leaf spanning tree problem (MLSTP) is to find a spanning tree of G with as many leaves as possible. The problem is easy to solve when G is complete. However, for the general case, when the graph is sparse, it is proven to be NP-hard. In this paper, two reformulations are proposed for the problem. The first one is a reinforced directed graph version of a formulation found in the literature. The second recasts the problem as a Steiner arborescence problem over an associated directed graph. Branch-and-Cut algorithms are implemented for these two reformulations. Additionally, we also implemented an improved version of a MLSTP Branch-and-Bound algorithm, suggested in the literature. All of these algorithms benefit from pre-processing tests and a heuristic suggested in this paper. Computational comparisons between the three algorithms indicate that the one associated with the first reformulation is the overall best. It was shown to be faster than 123 290 A. Lucena et al. the other two algorithms and is capable of solving much larger MLSTP instances than previously attempted in the literature.

Regenerator Location Problem and survivable extensions: A hub covering location perspective

“…As networks get dense, the RLP solution times also tend to decrease dramatically. This observation is consistent with the findings in Chen et al (2009) and Lucena et al, 2010. In fact, both of these papers could report exact solutions for either very small dimensions (at most 100 nodes) or very dense networks (p values at least 0.1).…”

“…Chen et al (2009) and Lucena et al (2010) present the most efficient of those algorithms. To check the performances of our algorithms against this benchmark we emulated the random graph generation procedures commonly adopted in these two studies.…”

“…Using this equivalence, Sen et al (2008) provide another heuristic for the RLP. Lucena et al (2010) provide branch and cut solution methodologies for the MLSTP and MCDSP and ultimately for the RLP. In a recent study, Gendron et al (2014) propose efficient exact solution methodologies built on Benders decomposition and branch and cut techniques.…”

Matheuristics for Variants of the Dominating Set Problem

“…Exact methods and formulations have been proposed mainly for the MCDS problem. These approaches are unable to solve most instances in a reasonable time (van Rooij & Bodlaender, 2011;Lucena et al, 2010;Gendron et al, 2014).…”

A Simple Approach to Dynamic Optimisation of Flexible Optical Networks with Practical Application

“…For completeness, relevant optimisation problems and solutions from other fields (in networking and graph theory) include the Minimum Dominating Set (MDS) [14] and the Minimum Connected Dominating Set (MCDS) [15], Minimum Spanning Tree (MST) [16], Maximum Leaf Spanning Tree (MLSP) [17], and Shortest Path (SPP) [18] problems. We draw on components of all of these in this paper, described, where appropriate, in what follows.…”