On minimizing vertex bisection using a memetic algorithm

Jain, Pallavi; Saran, Gur; Srivastava, Kamal

doi:10.1016/j.ins.2016.07.055

Cited by 9 publications

(12 citation statements)

References 11 publications

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“…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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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 has practical applications in fault tolerance [8]. It is also related to the complexity of sending messages to processors in interconnection networks via vertex disjoint paths [10].…”

Section: Introductionmentioning

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%

“…We used 13 different sets of graphs, totalizing 477 instances. It is important to mention that some of these graphs are considerably big and have been solved only by the memetic algorithm for VBP (MAVBMP) proposed in [8]. For the experiment, we use the well-known optimization engine CPLEX (v12.5).…”

Section: Introductionmentioning

confidence: 99%

“…The results of this research were also used to assess the performance of the best stochastic algorithm for VBP: MAVBMP [8]. Thus, we compare the results explicitly reported in [8] with the best results obtained by all the formulations. From 221 instances, MAVBMP found 178 optimal solutions, achieving an effectiveness of 80.5%.…”

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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On Roman Domination of Graphs Using a Genetic Algorithm

Khandelwal

Srivastava

Saran

2020

Computational Methods and Data Engineering

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A Roman dominating function (RDF) on a graph G is a labelling f : V → {0, 1, 2} such that every vertex labelled 0 has at least one neighbour with label 2. The weight of G is the sum of the labels assigned. Roman domination number (RDN) of G, denoted by γ R (G), is the minimum of the weights of G over all possible RDFs. Finding RDN for a graph is an NP-hard problem. Approximation algorithms and bounds have been identified for general graphs and exact results exist in the literature for some standard classes of graphs such as paths, cycles, star graphs and 2 × n grids, but no algorithm has been proposed for the problem for exact results on general graphs in the literature reviewed by us. In this paper, a genetic algorithm has been proposed for the Roman domination problem in which two construction heuristics have been designed to generate the initial population, a problem specific crossover operator has been developed, and a feasibility function has been employed to maintain the feasibility of solutions obtained from the crossover operator. Experiments have been conducted on different types of graphs with known optimal results and on 120 instances of Harwell-Boeing graphs for which bounds are known. The algorithm achieves the exact RDN for paths, cycles, star graphs and 2 × n grids. For Harwell-Boeing graphs, the results obtained lie well within bounds.

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On minimizing vertex bisection using a memetic algorithm

Cited by 9 publications

References 11 publications

Constructive heuristic for the vertex bisection problem

Constructive heuristic for the vertex bisection problem

Two new integer linear programming formulations for the vertex bisection problem

On Roman Domination of Graphs Using a Genetic Algorithm

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