Branch‐and‐cut and Branch‐and‐cut‐and‐price algorithms for the adjacent only quadratic minimum spanning tree problem

Abstract

The quadratic minimum spanning tree problem (QMSTP) consists of finding a spanning tree of a graph G such that a quadratic cost function is minimized. In its adjacent only version (AQMSTP), interaction costs only apply for edges that share an endpoint. Motivated by the weak lower bounds provided by formulations in the literature, we present a new linear integer programming formulation for AQMSTP. In addition to decision variables assigned to the edges, it also makes use of variables assigned to the stars of G. In doing so, the model is naturally linear (integer), without the need of implementing usual linearization steps, and its linear programming relaxation better estimates the interaction costs between edges. We also study a reformulation derived from the new model, obtained by projecting out the decision variables associated with the stars. Two exact solution approaches are presented: a branch‐and‐cut‐and‐price algorithm, based on the first formulation, and a branch‐and‐cut algorithm, based on its projection. Our computational results indicate that the lower bounds introduced here are much stronger than previous bounds in the literature. Being designed for the adjacent only case, our duality gaps are one order of magnitude smaller than the Gilmore–Lawler lower bounds for AQMSTP. As a result, the two exact algorithms introduced here outperform the previous exact solution approaches in the literature. In particular, the branch‐and‐cut method we propose managed to solve AQMSTP instances with as many as 50 vertices to proven optimality. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(4), 367–379 2015

Polyhedral results, branch‐and‐cut and Lagrangian relaxation algorithms for the adjacent only quadratic minimum spanning tree problem

“…These extra costs account for the additional equipment that needs to be in place in order to convert data going from one edge to the other. The AQMSTP was investigated in [2,12,17,18].…”

“…The star reformulation introduced by Pereira et al [17,18] is the strongest known linear integer programming formulation for the AQMSTP. Besides binary variables assigned to the edges in E, it uses exponentially many decision variables assigned to the stars of G; a star being any subset of edges incident to a given vertex.…”

“…Besides binary variables assigned to the edges in E, it uses exponentially many decision variables assigned to the stars of G; a star being any subset of edges incident to a given vertex. Two AQMSTP exact solution approaches based on the star reformulation were introduced in [18]: a branch-and-cut-and-price (BCP) and a branch-andcut (BC) method. They differ in the way of handling these exponentially many variables.…”

“…Two complementary research paths are followed here, their common starting point being the star reformulation [18] for the problem. In the first research direction, we investigate the cone associated with the projection of the star reformulation onto a compact variable space.…”

“…Although the inequalities we characterized do not fully describe the projection cone, they help us to define a strong formulation on a compact variable space. As a result, three new BC algorithms emerge, two of which perform much better than the best exact algorithm in the AQMSTP literature, provided in [18]. In the second research direction, we introduce a Lagrangian relaxation scheme for the star reformulation in [18] and use the associated Lagrangian relaxation lower bounds as the main ingredient in a new AQMSTP branch-and-bound (BB) algorithm.…”