The quadratic minimum spanning tree problem: A lower bounding procedure and an efficient search algorithm

Öncan, Temel; Punnen, Abraham P.

doi:10.1016/j.cor.2010.01.004

Cited by 37 publications

(54 citation statements)

References 16 publications

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“…Once constraints (13) are relaxed in F RLT , the resulting problem is polynomially solvable by an adaptation of the Gilmore-Lawler procedure [7,10]. Pereira et al [19] used this fact to develop a Lagrangian relaxation scheme where constraints (13) are relaxed and dualized in a Lagrangian fashion and QMSTP lower bounds were obtained by subgradient optimization.…”

Section: Aqmstp Formulations and Solution Approachesmentioning

confidence: 99%

“…Constraints (13) are needed only if one chooses to explicitly distinguish y ef from y fe . The distinction is useful, for example, when applying the Lagrangian relaxation scheme in [19].…”

Section: Aqmstp Formulations and Solution Approachesmentioning

confidence: 99%

“…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%

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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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Section: Aqmstp Formulations and Solution Approachesmentioning

confidence: 99%

“…Constraints (13) are needed only if one chooses to explicitly distinguish y ef from y fe . The distinction is useful, for example, when applying the Lagrangian relaxation scheme in [19].…”

Section: Aqmstp Formulations and Solution Approachesmentioning

confidence: 99%

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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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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…In addition to these early methods, even more heuristics have been proposed in recent years, mostly based on local search. For example, the Tabu Thresholding algorithm (Öncan & Punnen, 2010) alternatively performs local search and random moves. In (Palubeckis, Rubliauskas, & Targamadzè, 2010), an iterated tabu search (ITS) is proposed and compared to a multi-start simulated annealing algorithm and a hybrid genetic algorithm, showing that ITS performs the best.…”

Section: Introductionmentioning

confidence: 99%

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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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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The quadratic minimum spanning tree problem: A lower bounding procedure and an efficient search algorithm

Cited by 37 publications

References 16 publications

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

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

A three-phase search approach for the 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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