A new Integer Linear Programming and Quadratically Constrained Quadratic Programming Formulation for Vertex Bisection Minimization Problem

Jain, Pallavi; Saran, Gur; Srivastava, Kamal

doi:10.14313/jamris_1-2016/9

Cited by 4 publications

(8 citation statements)

References 7 publications

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“…In Sect. 4.1 we present the best integer linear programming formulation from the literature: ILPLIT [7]. Section 4.2 presents our mathematical formulation based on the redefinition of the objective function proposed in Sect.…”

Section: Mathematical Formulations For Vbpmentioning

confidence: 99%

“…Despite its applications, a small number of solution methods have been proposed for solving VBP. In the literature reviewed, we found three integer linear programming (ILP) formulations [3,7], two branch and bound algorithms [3,6] and one memetic algorithm [8].…”

Section: Introductionmentioning

confidence: 99%

“…Currently, the best ILP formulation for VBP is the one proposed by Jain et al [7]. We call this formulation ILPLIT since it is the best ILP formulation in the literature.…”

Section: Introductionmentioning

confidence: 99%

“…The strength of ILPLIT resides in the small number of variables and constraints generated. Specifically, ILPLIT generates 2(m + n) variables and 8m + n constraints [7]. Here, n and m stand for the number of vertices and edges of the graph, respectively.…”

Section: Introductionmentioning

confidence: 99%

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Two new integer linear programming formulations for the vertex bisection problem

Castillo-García

Hernández

2019

Comput Optim Appl

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The vertex bisection problem (VBP) is an NP-hard combinatorial optimization problem with important practical applications in the context of network communications. The problem consists in finding a partition of the set of vertices of a generic undirected graph into two subsets (A and B) of approximately the same cardinality in such a way that the number of vertices in A with at least one adjacent vertex in B is minimized. In this article, we propose two new integer linear programming (ILP) formulations for VBP. Our first formulation (ILPVBP) is based on the redefinition of the objective function of VBP. The redefinition consists in computing the objective value from the vertices in B rather than from the vertices in A. As far as we are aware, this is the first time that this representation is used to solve VBP. The second formulation (MILP) reformulates ILPVBP in such a way that the number of variables and constraints is reduced. In order to assess the performance of our formulations, we conducted a computational experiment and compare the results with the best ILP formulation available in the literature (ILPLIT). The experimental results clearly indicate that our formulations outperform ILPLIT in (i) average objective value, (ii) average computing time and (iii) number of optimal solutions found. We statistically validate the results of the experiment through the well-known Wilcoxon rank sum test for a confidence level of 99.9%. Additionally, we provide 404 new optimal solutions and 73 new upper and lower bounds for 477 instances from 13 different groups of graphs.

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Section: Mathematical Formulations For Vbpmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Currently, the best ILP formulation for VBP is the one proposed by Jain et al [7]. We call this formulation ILPLIT since it is the best ILP formulation in the literature.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Two new integer linear programming formulations for the vertex bisection problem

Castillo-García

Hernández

2019

Comput Optim Appl

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“…In the literature we can find both exact and approximate solution methods. Regarding the exact methods, seven approaches have been proposed for VBP, five Integer Linear Programming (ILP) formulations (Castillo-García & Hernández, 2019;Fraire et al, 2014;Jain, Saran & Srivastava, 2016a) and two branch and bound algorithms (Fraire et al, 2014;Jain, Saran, & Srivastava, 2016b). In addition, there are three metaheuristic algorithms (Herrán, Colmenar, & Duarte, 2019;Jain, Saran & 2016c;Terán-Villanueva et al, 2019) and one constructive algorithm for VBP (González et al, 2015), totalizing four approximate solutions.…”

Section: Introductionmentioning

confidence: 99%

Constructive heuristic for the vertex bisection problem

Castillo-García

Hernández-Hernández

2020

JART

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The Vertex Bisection Problem (VBP) consists in partitioning a generic graph into two equallysized subgraphs and such that the number of vertices in with at least one adjacent vertex in is minimized. This problem is NP-hard with practical applications in the telecommunication industry. In this article we propose a new constructive algorithm for VBP based on the Greedy Randomized Adaptive Search Procedure (GRASP) methodology. We call our algorithm CVBP. We compare CVBP with a previously published GRASP-based constructive algorithm (LIT) in order to assess the performance of our algorithm in practice. The results of the experiment showed that CVBP outperformed LIT by 75.83 % in solution quality. The validation of the experimental evidence was performed by the well-known Wilcoxon Signed Rank Sum Test. The test found statistical significance for a confidence level of 99.99 %. Therefore, we consider that our constructive heuristic is a good alternative to stochastically solve the Vertex Bisection Problem.

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