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Branch‐and‐cut and Branch‐and‐cut‐and‐price algorithms for the adjacent only quadratic minimum spanning tree problem
Author(s) -
Pereira Dilson Lucas,
Gendreau Michel,
Salles da Cunha Alexandre
Publication year - 2015
Publication title -
networks
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.977
H-Index - 64
eISSN - 1097-0037
pISSN - 0028-3045
DOI - 10.1002/net.21580
Subject(s) - spanning tree , branch and cut , maximum cut , mathematics , minimum spanning tree , linear programming relaxation , integer programming , linear programming , linearization , branch and bound , cutting plane method , quadratic equation , combinatorics , relaxation (psychology) , branch and price , mathematical optimization , algorithm , graph , nonlinear system , psychology , social psychology , physics , geometry , quantum mechanics
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