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

作     者：Pereira, Dilson Lucas da Cunha, Alexandre Salles 

作者机构：Univ Fed Lavras Dept Ciencia Comp Lavras Brazil Univ Fed Minas Gerais Dept Ciencia Comp Belo Horizonte MG Brazil 

出 版 物：《NETWORKS》 (网络)

年 卷 期：2018年第71卷第1期

页      面：31-50页

核心收录：

学科分类：1201[管理学-管理科学与工程(可授管理学、工学学位)] 08[工学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 

基　　金：CNPq [408868/2016-3, 303677/2015-5, 471464/2013-9, 200493/2014-0] FAPEMIG [CEX - PPM-00187-15] 

主　　题：adjacent only quadratic minimum spanning tree problem projection branch-and-cut algorithms Lagrangian relaxation 

摘      要：Given a complete and undirected graph G, the adjacent only quadratic minimum spanning tree problem (AQMSTP) consists of finding a spanning tree that minimizes a quadratic function of its adjacent edges. The strongest AQMSTP linear integer programming formulation in the literature works on an extended variable space, using exponentially many decision variables assigned to the stars of G. In this article, we characterize three families of facet defining inequalities by investigating the projection of that formulation onto the space of the canonical linearization variables. On the algorithmic side, we introduce four new branch-and-bound algorithms. Three of them are branch-and-cut algorithms based on the inequalities characterized by projection. The fourth is based on a Lagrangian relaxation scheme, also devised for the star reformulation. Two of the branch-and-cut algorithms provide very good results, almost always dominating the previously best algorithm for the problem. The Lagrangian relaxation based branch-and-bound algorithm provides even better results. It manages to solve all previously solved AQMSTP instances in the literature in about one tenth of the time needed by its competitors. (c) 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 71(1), 31-50 2018