A comparative analysis of several formulations for the generalized minimum spanning tree problem

Abstract

This article describes eight formulations for the Generalized Minimum Spanning Tree Problem (GMSTP). Relationships between the polytopes of their linear relaxations are established. It is shown that four of these polytopes are strictly included in the remaining ones. This analysis suggests which formulations should be preferred for the construction of a branch-and-cut algorithm and for the evaluation of heuristics.

Generalized minimum spanning tree games

“…Among the four existing GMSTP formulations presented by Myung et al (1995), four formulations by Feremans et al (2002) and four others in Pop (2009), we consider a multi-commodity flow formulation defined on a directed graph D = (V, A) of the first paper. The motivation behind the choice of this particular model is its compact form (with a polynomial number of constraints) compared to other models (with an exponential number of constraints).…”

“…The authors also present four integer linear programming formulations and a branch-and-bound algorithm to solve instances of up to 100 vertices. Feremans et al (2002) and Pop (2009) describe twelve different formulations for GMSTP and study the relationships between the polytopes of their linear relaxations. This paper introduces the generalized minimum spanning tree game (GMSTG) and proposes the computational methods for calculating its cost allocation.…”

Minimum Spanning Tree Problem with Label Selection

“…It is known that GMSTP is NP-hard [1], and the problem is still NP-hard even on trees [2]. Thus, linear programming relaxations are considered [1]- [4]. Chang and Leu introduced the minimum labeling spanning tree problem (MLSTP) [5], where the edges of a graph are colored and the number of colors of a spanning tree should be minimized.…”

An ant colony system approach for solving the at-least version of the generalized minimum spanning tree problem

“…Conversely, if T cbest (t) is significantly worse than T gbest (t), its edges receive almost negligible reinforcement (attenuated as J gbest (t)/J 2 cbest (t)). All other edges receive only a negative reinforcement (11). Fig.…”

“…Unlike the MST, both versions of the GMST have been shown to be NP-hard [2,3]. Specifically for the L-GMST, it is shown in [3] that no constant factor polynomial time algorithm exists unless P = N P. While most research regarding the GMST has focused on the E-GMST [2,4,7,8,11], some solution methods have also been reported in the literature for solving the L-GMST problem. These include integer linear programming (ILP), local search and metaheuristics such as tabu search and genetic algorithms.…”