Solving the Quadratic Minimum Spanning Tree Problem

Cordone, Roberto; Passeri, Gianluca

doi:10.1016/j.amc.2012.05.043

Cited by 24 publications

(50 citation statements)

References 17 publications

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“…Like the compact tree representation used in (Cordone & Passeri, 2012;Fu & Hao, 2014), we uniquely represent each feasible solution T as a rooted tree (with vertex 1 fixed as the root vertex, being different from (Cordone & Passeri, 2012) where the root changes dynamically during the search process), corresponding to a one-dimensional vector T = {t i , i ∈ V }, where t i denotes the parent vertex of vertex i only except the root vertex 1 (let t 1 = null). Inversely, given a vector T = {t i , i ∈ V }, the corresponding solution tree can be easily reconstructed.…”

Section: Solution Presentationmentioning

confidence: 99%

“…Very recently, Lozano et al (2013) propose an iterated greedy (IG) and a strategic oscillation (SO) heuristic, and combine them with the ITS (Palubeckis, Rubliauskas, & Targamadzè, 2010) algorithm to obtain a powerful hybrid algorithm named HSII. In addition to the standard QMSTP, for the variant only with adjacency costs, Maia, Goldbarg, & Goldbarg (2013) propose a Pareto local search algorithm and adapt the 108 instances in (Cordone & Passeri, 2012) as benchmarks to evaluate the proposed algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…The quadratic minimum spanning tree problem (QMSTP) requires to determine a spanning tree T = (V, X), so as to minimize its total cost F (T ), i.e., the sum of the linear costs plus the quadratic costs. Naturally, this problem can be formulated as follows (Cordone & Passeri, 2012):…”

Section: Introductionmentioning

confidence: 99%

“…Öncan and Punnen (2010) combine the Lagrangian relaxation scheme with an extended formulation of valid inequalities to obtain tighter bounds. Cordone and Passeri (2012) re-implement the Lagrangian branch-and-bound procedure in (Assad & Xu, 1992) with some improvements. Very recently, several exact algorithms are proposed for solving other closely related variants.…”

Section: Introductionmentioning

confidence: 99%

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A three-phase search approach for the quadratic minimum spanning tree problem

Hao

2015

Engineering Applications of Artificial Intelligence

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Given an undirected graph with costs associated with each edge as well as each pair of edges, the quadratic minimum spanning tree problem (QMSTP) consists of determining a spanning tree of minimum total cost. This problem can be used to model many real-life network design applications, in which both routing and interference costs should be considered. For this problem, we propose a three-phase search approach named TPS, which integrates 1) a descent-based neighborhood search phase using two different move operators to reach a local optimum from a given starting solution, 2) a local optima exploring phase to discover nearby local optima within a given regional search area, and 3) a perturbation-based diversification phase to jump out of the current regional search area. Additionally, we introduce dedicated techniques to reduce the neighborhood to explore and streamline the neighborhood evaluations. Computational experiments based on hundreds of representative benchmarks show that TPS produces highly competitive results with respect to the best performing approaches in the literature by improving the best known results for 31 instances and matching the best known results for the remaining instances only except two cases. Critical elements of the proposed algorithms are analyzed.

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Section: Solution Presentationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A three-phase search approach for the quadratic minimum spanning tree problem

Hao

2015

Engineering Applications of Artificial Intelligence

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“…The goal is then to find a spanning tree that minimizes the total interference. Formulations, exact, and heuristic solution algorithms for the QMSTP were studied in [2,3,13,19,21].…”

Section: Introductionmentioning

confidence: 99%

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

Pereira¹,

Cunha

2017

Networks

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Given a complete and undirected graph G, the adjacent only quadratic minimum spanning tree problem (AQM-STP) 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.

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

2015

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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

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Solving the Quadratic Minimum Spanning Tree Problem

Cited by 24 publications

References 17 publications

A three-phase search approach for the quadratic minimum spanning tree problem

A three-phase search approach for the quadratic minimum spanning tree problem

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

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

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