For a given graph G = (V, E) the maximum independent set problem is to find the&nbsp;largest subset of pairwise nonadjacent vertices. We propose a new model which is a reformulation&nbsp;of the maximum independent set problem as an unconstrained quadratic binary programming, and&nbsp;we resolve it afterward by means of a genetic algorithm. The efficiency of the approach is confirmed&nbsp;by results of numerical experiments on DIMACS benchmarks

Douiri, Sidi Mohamed

Elbernoussi, Souad

English

Douiri, Sidi Mohamed Mohamed

Nonlinear Analysis: Modelling and Control

410 Nonlinear Analysis: Modelling and Control, 2012, Vol. 17, No. 4, 410–417An unconstrained binary quadratic programmingfor the maximum independent set problemSidi Mohamed Douiri, Souad ElbernoussiResearch laboratory Mathematics Computing and ApplicationsFaculty of Sciences, University Mohammed V-Agdal4 Avenue Ibn Battouta, B.P. 1014 RP, Rabat, Moroccodouirisidimohamed@hotmail.fr; souad.elbernoussi@gmail.comReceived: 21 September 2011 / Revised: 16 August 2012 / Published online: 22 November 2012Abstract. For a given graph G = (V,E) the maximum independent set problem is to find thelargest subset of pairwise nonadjacent vertices. We propose a new model which is a reformulationof the maximum independent set problem as an unconstrained quadratic binary programming, andwe resolve it afterward by means of a genetic algorithm. The efficiency of the approach is confirmedby results of numerical experiments on DIMACS benchmarks.Keywords: maximum independent set, unconstrained binary quadratic programming, geneticalgorithm.1 IntroductionThe maximum independent set problem (MIS) is one of the central combinatorial op-timization problems and shown to be NP-hard [1]. This problem is relevant for manypractical applications in computer science and operation research, and engineering [2],such as register allocation in a compiler, assigning channels to radio station, schedulingexam, graph coloring, and the reader collision problem [3]. Some studies were madeon the basis of the greedy algorithms and tabu search [4, 5], or using the intersectiongraph of an axis-parallel rectangles [6]. In [7] the authors proposed a method based on animprovement of the maximum independent set algorithm given by F. Glover in [8]. Otherstudies based on a method that utilizes the polynomially solvable critical independent setproblem [9]. The unconstrained binary quadratic programming problem is to maximize(or minimize) the function:f(x) = xTQx,where Q = (qij) is an n× n matrix of constants and x is an n-vector of binary variables.This formulation has an ability to model a wide range of different problems on manyareas as traffic management [10], machine scheduling [11], UBQP gives evidence toits relevance and effectiveness in the face of known problems by their complexity suchc© Vilnius University, 2012An unconstrained binary quadratic programming for the maximum independent set problem 411as the set-partitioning problem [12], the set packing problem [13], the vertex coloringproblem [14], and the linear ordering problem [15]. Given its NP-hard nature [1], variousapproaches have been proposed for solving this model using exact methods [16, 17] andmetaheuristic methods as memetic algorithms [18, 19], scatter search [20], simulatedannealing [21], adaptive memory approaches based on tabu search [22–24], and recentlycombination of GRASP and tabu search [25].This paper presents an efficient method for solving the maximum independent setproblem (MIS) via its modeling as a unconstrained binary quadratic programming(UBQP).This paper is organized as follows: we define the maximum independent set prob-lem in Section 2. Section 3 presents the transformation of linear programming to un-constrained binary quadratic programming associated to the maximum independent setproblem. In Section 4 the ingredients of our algorithm are described, including an adaptedgenetic algorithm, and Section 5 draws some conclusions.2 Problem definitionThe maximum independent set problem (MIS) may be written in a form of a linearproblem with binary variables as follows:(LPMIS)maxx0 =∑ni=1 xi,xi + xj ≤ 1, {i, j} ∈ E,x binary,(1)where x2i = xi and xi ∈ (0, 1).The problem (LPMIS) is constituted of a linear objective function with one types ofconstraints xi+xj ≤ 1 and the number of these inequality constraints is equal to card(E).Our purpose is to reformulate the problem (LPMIS) in a binary quadratic problem withoutconstraints in the form:(UBQPMIS){maxx0 = xTQx,x binary,where Q is a square symmetric matrix of dimension card(V ). For this we apply a trans-formation on the constraints set.3 TransformationWe introduce the constraints in the objective function in the following way. The objectiveis transformed under shape xTDx by (1), then the constraints xi+ xj ≤ 1 are introducedby the product xixj . The problem (LPMIS) is thus replaced by the quadratic problemwithout constraint:(UBQPMIS){maxx0 =∑ni=1 xi −∑(i,j)∈E xixj ,x binary.Nonlinear Anal. Model. Control, 2012, Vol. 17, No. 4, 410–417412 S.M. Douiri, S. ElbernoussiExample. We consider the undirected graph in Fig. 1, on this graph we obtain threeindependent sets: C1 = (2, 4, 5, 7, 9), C2 = (1, 6, 8) and C3 = (3).Fig. 1. Independent sets of a graph.We reformulate it into an unconstrained binary quadratic programming problem. Theexample satisfies the following linear programming:(LPMIS)maxx0 =∑9i=1 xi, x3 + x6 ≤ 1,x1 + x2 ≤ 1, x3 + x7 ≤ 1,x1 + x9 ≤ 1, x3 + x8 ≤ 1,x2 + x3 ≤ 1, x5 + x6 ≤ 1,x3 + x4 ≤ 1, x8 + x9 ≤ 1,x3 + x5 ≤ 1, x binary.We apply the transformation and obtain:Q =1 −1 0 0 0 0 0 0 −1−1 1 −1 0 0 0 0 0 00 −1 1 −1 −1 −1 −1 −1 00 0 −1 1 0 0 0 0 00 0 −1 0 1 −1 0 0 00 0 −1 0 −1 1 0 0 00 0 −1 0 0 0 1 0 00 0 −1 0 0 0 0 1 −1−1 0 0 0 0 0 0 −1 1.The independent set C1 = (2, 4, 5, 7, 9), i.e., x = (010110101) gives an optimal solutionfor UQPMIS with x0 = 5, if we add or remove one or more vertex to C1 the value ofUQPMIS will be inferior, for example if we add to C1 the vertex 3, i.e., x = (011110101)we find x0 = −2.4 Solving UBQPMISAfter having transformed our MIS problem in an unconstrained binary quadratic form,we try to solve it using an adapted genetic algorithm well by choosing aptly the operatorswhich will be ideal to the MIS problem.www.mii.lt/NAAn unconstrained binary quadratic programming for the maximum independent set problem 4134.1 Genetic algorithmOur resolution approach of UBQPMIS problem is based on genetic algorithm (GA). TheGA is a research process based on the laws of natural selection and genetics. Generally,a GA consists of three simple operations. Their functioning is extremely simple, we startwith an initial population, we evaluate the performance of each individual, and we createa new population of potential solutions using evolutionary operators: selection, crossoverand mutation, GA cycle is repeated until a desired stopping criterion is reached.The individuals of initial population are generated randomly with equal probabilitysuch as the genes are assigned a value of 0 or 1. To build a diversified initial population,each individual added to the population must be different to all the existing solutions ofthe population. Using the best known lower bound based on degrees of vertices di givenby Caro and Tuza [26]: ∑i∈V1di + 1≤ card(MIS(G))During the initialization of the population we fix the size of the independent set to p =b∑i∈V 1/(di + 1)c that is, the number of 1 in each individual of the population.We consider the objective function xTQx as evaluation function (fitness) of eachindividual in the population.For the selection, first, we randomly choose two individuals, then we apply the tour-nament selection operator in order to keep the best individual. The comparison betweentwo individuals is carried out according to their fitness.Crossover is a recombination operator that combines parts of two individual parents toproduce offspring that contain some genetic information from both parents. A probabilityparameter pc, is set to determine the crossover rate. We opt for a special crossing, wechoose two random integers r, s in each parents P1 and P1 such as r, s ∈ [1, n] whorepresent points inter-genes. We opt for a crossing which aims to permute two blocks ofgenes of each parents pair. The two selected blocks have the same number of genes whichare worth one and the same size. By this crossing we always keep the number of 1 oneach new individual obtained.C1(i) ={P1(i) if i /∈ [r1, s1],P2(j) if i ∈ [r1, s1] and j ∈ [r2, s2],C2(i) ={P2(i) if i /∈ [r2, s2],P1(j) if i ∈ [r2, s2] and j ∈ [r1, s1].Table 1 presents a comparison between the results obtained using a 1-point crossover,a 2-point crossover and the proposed crossover for 25 tests. The tests were performedfor a crossover rate equal to pc = 0.8 and a mutation rate equal to pm = 0.2. Theaverage values (Xavg) solutions obtained with the proposed crossover are clearly superiorcompared to the two types of crossover.The mutation is used with the aim to further explore the search space and reachingsolutions that the crossing cannot touch them. We choose randomly two distinct genesthat we permute them after, usually with small probability pm.Nonlinear Anal. Model. Control, 2012, Vol. 17, No. 4, 410–417414 S.M. Douiri, S. ElbernoussiFigure 2: Used crossover operator.Table 1: Comparison of solution quality for different type of crossoveroperations.One point crossover Two point crossover Proposed crossoverGraph XAvg XAvg XAvganna 77.84 76.53 79.04david 33.71 33.60 34.18huck 27 26.14 27fpsol2.i.1 301.28 303.59 306.47inithx.i.1 563.06 561.25 564.80zeroin.i.3 118.72 120.55 122.36myciel4 11 11 11myciel5 20.61 18.27 23myciel6 47 45.93 47myciel7 93.65 93.12 94.07mug100-25 32.19 31.45 31.86mulsol.i.5 86.04 84.27 86.73Table 1 presents a comparison between the results obtained using a1-point crossover, a 2-point crossover and the proposed crossover for 256Figure 2: Used crossover p ator.Table 1: Comparison of solution quality for different type of cr ssoveroperations.One point crossover Two point crossover Proposed crossoverGraph XAvg XAvg XAvganna 77.84 76.53 79.04david 33.71 33.60 34.18huck 27 26.14 27fpsol2.i.1 301.28 303.59 306.47inithx.i.1 563.06 561.2 564.80zeroin.i.3 118.72 120.55 122.36myciel4 11 11 11myciel5 20.61 18.27 23myciel6 47 45.93 47myciel7 93.65 93.12 94.07mug100-25 32.19 31.45 31.86mulsol.i.5 86.04 84.27 86.73Table 1 presents a comparison between the results ob ained using a1-point crossover, a 2-point crossover and the proposed crossover for 256Figure 2: Used crossover operator.Table 1: Comparison of solution quality for different type of crossoveroperations.One point crossover Two point crossover Proposed crossoverGraph XAvg XAvg XAvganna 77.84 76.53 79.04david 33.71 33.60 34.18huck 27 26.14 27fpsol2.i.1 301.28 303.59 306.47inithx.i.1 563.06 561.25 564.80zeroin.i.3 118.72 120.55 122.36myciel4 11 11 11myciel5 20.61 18.27 23myciel6 47 45.93 47myciel7 93.65 93.12 94.07mug100-25 32.19 31.45 31.86mulsol.i.5 86.04 84.27 86.73Table 1 presents a comparison between the results obtained using a1-point crossover, a 2-point crossover and the proposed crossover for 256Figure 2: Used crossover p ator.Table 1: Comparison of solution quality for different type of cr ssoveroperations.One point crossover Two point crossover Proposed crossoverGraph XAvg XAvg XAvganna 77.84 76.53 79.04david 33.71 33.60 34.18huck 27 26.14 27fpsol2.i.1 301.28 303.59 306.47inithx.i.1 563.06 561.2 564.80zeroin.i.3 118.72 120.55 122.36myciel4 11 11 11myciel5 20.61 18.27 23myciel6 47 45.93 47myciel7 93.65 93.12 94.07mug100-25 32.19 31.45 31.86mulsol.i.5 86.04 84.27 86.73Table 1 presents a comparison between th results ob ained using a1-point crossover, a 2-point crossover and the proposed crossover for 256Fig. 2. Used crossover operator.Table 1. Comparison of solution quality for different type of crossover operations.One point crossover Two point crossover Proposed crossoverGr ph Xavg Xavg Xavganna 77.84 76.53 79.04david 33.71 33.60 34.18huck 27 26.14 27fpsol2.i.1 301.28 303.59 306.47inithx.i.1 563.06 561.25 564.80zeroin.i.3 118.72 120.55 122.36myciel4 11 11 11myciel5 20.61 18.27 23myciel6 47 45.93 47myciel7 93.65 93.12 94.07mug100-25 32.19 31.45 31.86mulsol.i.5 86.04 84.27 86.73Remark. We fix p = b∑i∈V 1/(di + 1)c, then we apply the transformation to obtainUBQPMIS problem, after we run the genetic algorithm, and each time we find a maximumindependent set we increment the number p by one in order to find an independent setlarger.Algorithm 1. UBQPMIS algorithm.1. Give the problem UBQPMIS.2. Put p = b∑i∈V 1/(di + 1)c.3. While T ≤ Tmax.4. Evaluate each individual by calculating its fitness x0 = xTQx.5. Make a selection by tournament.6. Apply the crossing.7. Apply the mutation with a very low probability.8. Evaluate the individuals.9. If we find a feasible solution x0 then p← p+ 1.10. End while.www.mii.lt/NAAn unconstrained binary quadratic programming for the maximum independent set problem 4154.2 Obtained resultsIn this section, we present the experimental results after running our algorithm on in-stances available in the literature (see the web page http://mat.gsia.cmu.edu/COLOR03/ for a complete description of the instances) and compare them with thosegiven by [7,9]. The platform of our experiments is a personal computer windows 7, AMDAthlon(tm) X2 dual-core QL-65 (2cpus) 2. GHz with 4 GB RAM. The results obtainedby our algorithm are reported in the Table 2. For each instance, we indicate the numberof vertices n, the number of edges d0, the values XS, XDBG and XUBQP respectivelydenote the cardinal of the maximal independent set given by S. Butenko et al. [9], DBGalgorithm [7], and UBQPMIS algorithm, the symbol ‘–’ means that the information is notavailable. The parameters of the GA for the UBQPMIS problem are: population size= 100,crossover rate pc = 0.8, mutation rate pm = 0.2, and the maximum iterations numberTmax = 200.Table 2. Computational results.Graph n d0 XS XDBG XUBQP Time UBQP (s)anna 138 493 80 80 80 0.87david 87 406 36 36 36 0.49fpsol2.i.1 496 11654 307 307 307 12.39fpsol2.i.2 451 8691 261 261 261 10.77fpsol2.i.3 425 8688 238 238 238 9.18huck 74 301 27 27 27 0.51inithx.i.1 864 18707 566 566 566 15.26inithx.i.2 645 13979 365 365 365 10.94inithx.i.3 621 13969 360 360 360 7.89jean 80 254 38 38 38 0.60zeroin.i.1 211 4100 120 120 120 4.22zeroin.i.2 211 3541 127 127 127 4.05zeroin.i.3 206 3540 123 123 123 5.131-FullIns5 282 3247 – 138 138 6.673-FullIns4 405 3524 – 193 193 8.422-Inser4 149 541 – 74 74 1.053-Inser3 56 110 – 27 27 0.384-Inser3 79 156 – 39 39 0.96games120 120 638 – 22 22 1.20mulsol.i.1 197 3925 – 100 100 5.26mulsol.i.5 186 3973 – 88 88 4.81mug88-1 88 146 – – 29 1.17mug88-25 88 146 – – 29 0.95mug100-1 100 166 – – 33 1.78mug100-25 100 166 – – 33 2.34myciel3 11 20 – – 5 0.21myciel4 23 71 – – 11 0.33myciel5 47 236 – – 23 0.57myciel6 95 755 – – 47 0.84Nonlinear Anal. Model. Control, 2012, Vol. 17, No. 4, 410–417416 S.M. Douiri, S. ElbernoussiGraph n d0 XS XDBG XUBQP Time UBQP (s)myciel7 191 2360 – – 95 3.96dsjc125.1 125 736 – – 32 2.03dsjc250.1 250 3218 – – 39 5.49le450_15a 450 8168 – – 71 9.41le450_15d 450 16750 – – 37 12.16le450_25a 450 8260 – – 85 11.07le450_25d 450 17425 – – 40 13.225 ConclusionIn this paper we have presented a method to reformulate the linear program of maximumindependent set problem as an unconstrained binary quadratic problem, the proposedalgorithm proves to be highly effective in solving a range of benchmark instances from theliterature. Most of these instances have been easily resolved with a reasonable executiontime. For the first 13 instances we obtained identical results than those in [9], our resultsare also identical to the first 21 instances given in [7], to show the effectiveness of ouralgorithm we tested 15 other instances.References1. M.R. Garey, D.S. 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An unconstrained binary quadratic programming for the maximum independent set problem

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An unconstrained binary quadratic programming for the maximum independent set problem

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Nonlinear Analysis: Modelling and Control