On the Biobjective Adjacent Only Quadratic Spanning Tree Problem

Maia, Sílvia M. D. M.; Goldbarg, Elizabeth Ferreira Gouvêa; Goldbarg, Marco César

doi:10.1016/j.endm.2013.05.135

Cited by 14 publications

(11 citation statements)

References 8 publications

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“…Constraints (13) are clearly implied by (4) and (10). To check that constraints (14) are also implied by P RLT , formulate (5) in terms of an edge f and S ¼ V=fvg, for any vertex v. Then subtract the resulting inequality from (4) (also formulated for f), to obtain X e A δðvÞ y fe Z x f ; which, together with (10), implies (14).□…”

Section: Rlt-1 Based Boundsmentioning

confidence: 95%

“…Öncan and Punnen [6] added the valid inequalities (14) to P AX92 and formulated QMSTP as where P OP10 ¼ P AX92 \ ðx; yÞ A R m þ m 2 : ðx; yÞ satisfies ð4Þ n o . The following results hold true.…”

Section: Rlt-1 Based Boundsmentioning

confidence: 99%

“…A bi-objective version of AQMSTP was investigated by Maia et al [14]. The linear ( P e A E q ee x e ) and the quadratic ( P e;f A E;e a f q ef x e x f ) parts of the AQMSTP cost function were considered as two separate objective functions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Lower bounds and exact algorithms for the quadratic minimum spanning tree problem

Pereira

Gendreau

Cunha

2015

Computers & Operations Research

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Section: Rlt-1 Based Boundsmentioning

confidence: 95%

Section: Rlt-1 Based Boundsmentioning

confidence: 99%

See 1 more Smart Citation

Lower bounds and exact algorithms for the quadratic minimum spanning tree problem

Pereira

Gendreau

Cunha

2015

Computers & Operations Research

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“…For example, in transportation, telecommunication or oil supply networks, the linear function represents the costs for building each road, communication link or pipe, and the quadratic function represents the extra costs needed for transferring from one road (link, pipe) to another one. Normally, the interference costs are limited to pairs of adjacent edges (Maia, Goldbarg, & Goldbarg, 2013;Pereira, Gendreau, & Cunha, 2013), but in some special cases, the interference costs also exist between any pair of edges, especially for situations where the topology has little relation to the physical layout. As discussed in (Assad & Xu, 1992;Öncan & Punnen, 2010; Palubeckis, Rubliauskas, & Targamadzè, 2010), the QMSTP has several equivalent formulations such as the stochastic minimum spanning tree problem (SMSTP), the quadratic assignment problem (QAP), and the unconstrained binary quadratic optimization problem (UBQP).…”

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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“…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].…”

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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On the Biobjective Adjacent Only Quadratic Spanning Tree Problem

Cited by 14 publications

References 8 publications

Lower bounds and exact algorithms for the quadratic minimum spanning tree problem

Lower bounds and exact algorithms 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

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