A BRKGA-BASED MATHEURISTIC FOR THE MAXIMUM QUASI …celso/artigos/BRKGA_ExactQClique.pdf · imum...

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A BRKGA-BASED MATHEURISTIC FOR THE MAXIMUMQUASI-CLIQUE PROBLEM WITH AN EXACT LOCAL

SEARCH STRATEGY

Bruno Q. Pinto1, Celso C. Ribeiro2, José A. Riveaux3 andIsabel Rosseti4

Abstract. Given a graph G = (V,E) and a threshold γ ∈ (0, 1], themaximum cardinality quasi-clique problem consists in finding a maxi-mum cardinality subset C∗ of the vertices in V such that the densityof the graph induced in G by C∗ is greater than or equal to the thresh-old γ. This problem has a number of applications in data mining, e.g.in social networks or phone call graphs. We propose a matheuristicfor solving the maximum cardinality quasi-clique problem, based onthe hybridization of a biased random-key genetic algorithm (BRKGA)with an exact local search strategy. The newly proposed approach iscompared with a pure biased random-key genetic algorithm, which wasthe best heuristic in the literature at the time of writing. Compu-tational results show that the hybrid BRKGA outperforms the pureBRKGA.

Keywords: maximum cardinality quasi-clique problem, maximumclique problem, biased random-key genetic algorithm, metaheuristics,matheuristic, graph density.

June 7, 2019.1 Instituto Federal de Educação, Ciência e Tecnologia do Triângulo Mineiro, Uberlândia, MG38411-104, Brazil. e-mail: [email protected] Institute of Computing, Universidade Federal Fluminense, Niterói, RJ 24210-346, Brazil.e-mail: [email protected] Institute of Computing, Universidade Federal Fluminense, Niterói, RJ 24210-346, Brazil.e-mail: [email protected] Institute of Computing, Universidade Federal Fluminense, Niterói, RJ 24210-346, Brazil.e-mail: [email protected]

c© EDP Sciences 2019

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1. Introduction

Let G = (V,E) be a graph defined by a vertex set V and an edge set E ⊆ V ×V .G is a complete graph if there is an edge in E connecting every two different verticesin V . A graph G′ = (V ′, E′) is a subgraph of G if V ′ ⊆ V and E′ ⊆ E, whichis denoted by G′ ⊆ G. The graph G(V ′) induced in G by V ′ ⊆ V is that withvertex set V ′ and edge set formed by all edges of E with both ends in V ′. For anyV ′ ⊆ V , the subset E(V ′) ⊆ E is formed by all edges of E with both ends in V ′.In other words, E(V ′) is the edge set of the graph induced in G by V ′.

The density of graph G is given by dens(G) = |E|/(|V | × (|V | − 1)/2). For anyv ∈ V , the degree degG(v) denotes the number of vertices in G that are adjacentto v. In addition, for any v ∈ V and any V ′ ⊆ V , we denote by degG(v, V ′) thenumber of vertices of V ′ that are adjacent to v in G.

A subset C ⊆ V is a clique ofG if the graphG(C) induced inG by C is complete.Given a graph G = (V,E), the maximum clique problem (MCP) consists in findinga maximum cardinality clique of G. It was proved to be NP-hard by [15].

Given a graph G = (V,E) and a threshold γ ∈ (0, 1], a γ-clique is any subsetC ⊆ V such that the density of the subgraph G(C) is greater than or equal toγ. A γ-clique C is maximal if there is no other γ-clique C ′ that strictly containsC. The maximum quasi-clique problem (MQCP) amounts to finding a maximumcardinality subset C∗ of the vertices in V such that the density of the graphinduced in G by C∗ is greater than or equal to the threshold γ. This problem wasproved to be NP-hard in [20]. The problem has many applications and relatedclustering approaches include classifying molecular sequences in genome projectsby using a linkage graph of their pairwise similarities [8] and the analysis of massivetelecommunication data sets obtained from social networks or phone call graphs [1],as well as various other data mining and graph mining applications.

A few heuristics for MQCP exist in the literature, based on well known ap-proaches such as greedy randomized algorithms and their iterated extensions [18],stochastic local search [8], and GRASP [1]. Pinto et al. [22] proposed a biasedrandom-key genetic algorithm for finding approximate solutions to the maximumcardinality quasi-clique problem, using two different decoders. They showed thatthe decoder based on an optimized iterated greedy constructive heuristic led tothe best numerical results. They also showed that the use of a restart strat-egy significantly contributed to improve the robustness and the efficiency of thealgorithm. The resulting BRKGA-IG∗ heuristic strategy achieved the best perfor-mance and outperformed the restarted optimized iterated greedy (RIG∗) construc-tion/destruction heuristic of Oliveira et al. [18]. BRKGA-IG∗ was also comparedwith the exact algorithms AlgF3 and AlgF4 of Veremyev et al. [29] used as heuris-tics with time limits on their running times. BRKGA-IG∗ applied to sparse graphsalso outperformed the mixed integer programming approaches, finding target so-lution values in much smaller running times.

Ribeiro and Riveaux [24] proposed an exact enumeration algorithm to solve themaximum quasi-clique problem, based on a quasi-hereditary property. They alsoproposed a new upper bound that is used for pruning the search tree. Numerical

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results showed that their approach is competitive with the best integer program-ming formulations in [20, 29] solved by CPLEX and with the branch-and-boundalgorithm proposed by [19], in terms of both solution quality and running time.

We show that the exact enumeration algorithm QClique proposed by Ribeiroand Riveaux [24] can be hybridized with the biased random-key genetic algorithmBRKGA-IG∗ developed by Pinto et al. [22] as a local search strategy to improvethe quality of the solutions created by the decoder. This paper is organized asfollows. Section 2 presents the problem formulation. Section 3 introduces biasedrandom-key genetic algorithms and describes their customization to the maximumquasi-clique problem. Section 4 describes in detail the decoder DECODER-IG∗previously used used in the implementation of the biased random-key genetic al-gorithm for the maximum quasi-clique problem. The new decoder DECODER-LSQClique using an exact local search strategy is presented in Section 5, givingrise to a matheuristic for solving the maximum cardinality quasi-clique problembased on a biased random-key genetic algorithm. The exact local search algorithmLSQClique used by the new decoder is described in detail in Section 6. Numer-ical results are reported in Section 7. Concluding remarks are drawn in the lastsection.

2. Problem formulation and related work

The maximum quasi-clique problem can be formulated by associating a binaryvariable xi to each vertex of the graph [20]:

xi =

{1, if vertex vi ∈ V belongs to the solution,0, otherwise.

This formulation also considers a variable yij = xi · xj associated to each pairof vertices i, j ∈ V , with i < j, which is linearized as follows:

max∑i∈V

xi (1)

subject to: ∑(i,j)∈E:i


ensures that the density of the solution is greater than or equal to γ. Constraints(3) and (4) ensure that any edge may contribute to the density of a solution onlyif both of its ends are chosen to belong to this solution. Constraints (5) ensurethat any existing edge (i, j) ∈ E will contribute to the solution if both of itsends are chosen. Constraints (6) and (7) impose the binary and non-negativityrequirements on the problem variables, respectively.

Veremyev et al. [29] reported and compared four mixed integer programmingformulations for the maximum quasi-clique problem in sparse graphs. Two algo-rithms based on the best formulations led to better results than the mixed integerprogramming formulation proposed in [20], with all mixed integer programs solvedusing FICO Xpress-Optimizer [10] with the time limit of 3600 seconds. Ribeiroand Riveaux [24] developed an exact algorithm based on a quasi-hereditary prop-erty and proposed a new upper bound that is used for pruning the search tree.Numerical results showed that their approach is competitive and outperforms thebest integer programming approaches in the literature. The new upper bound isconsistently tighter than previously existing bounds.

3. Biased random-key genetic algorithms for maximumquasi-clique

Genetic algorithms with random keys, also called random-key genetic algo-rithms (RKGA), were first introduced by Bean [4] for combinatorial optimizationproblems whose solutions may be represented by permutation vectors. Solutionsare represented as vectors of randomly generated real numbers called keys. Adeterministic algorithm, called a decoder, takes as input a solution vector andassociates with it a feasible solution of the combinatorial optimization problem,for which an objective value or fitness can be computed. Two parents are selectedat random from the entire population to implement the crossover operation in theimplementation of an RKGA. Parents are allowed to be selected for mating morethan once in the same generation.

A biased random-key genetic algorithm (BRKGA) differs from an RKGA inthe way parents are selected for crossover, see Gonçalves and Resende [11] andResende and Ribeiro [23] for reviews. In a BRKGA, each element is generatedcombining one element selected at random from the elite solutions in the currentpopulation, while the other is a non-elite solution. The selection is said to bebiased because one parent is always an elite solution and has a higher probabilityof passing its genes to the new generation.

In the following, we summarize two variants of a biased random-key geneticalgorithm for MQCP, each of them using a different decoder. Both of them evolvea population of chromosomes that consists of vectors of real numbers. Each chro-mosome is represented by a vector of |V | components, in which each key is a realnumber in the range [0, 1) associated with one of the vertices of the graph G. Eachchromosome is decoded by an algorithm that receives the vector of keys and buildsa feasible solution for MQCP, i.e., the decoder returns a γ-clique as its output.


The two decoders DECODER-HCB and DECODER-IG∗ are described in the nextsection and have been originally presented in [22].

The parametric uniform crossover scheme proposed by [27] is used to combinetwo parent solutions and to produce an offspring. In this scheme, the offspringinherits each of its keys from the best fit of the two parents with a higher proba-bility. The biased random-key genetic algorithm developed in this work does notmake use of the standard mutation operator, where parts of the chromosomes arechanged with small probability. Instead, the concept of mutants is used: mutantsolutions are introduced in the population in each generation, randomly generatedin the same way as in the initial population. Mutants play the same role of themutation operator in traditional genetic algorithms, diversifying the search andhelping the procedure to escape from locally optimal solutions ([6, 7, 17]).

The |V | keys in the chromosome are randomly generated in the initial popula-tion. At each generation, the population is partitioned into two sets: TOP andREST . The size of the population is |TOP |+ |REST |. Subset TOP contains thebest solutions in the population. Subset REST is formed by two disjoint subsets:MID and BOT , with subset BOT being formed by the worst elements in the cur-rent population. As illustrated in Figure 1, the chromosomes in TOP are simplycopied to the population of the next generation. The elements in BOT are replacedby newly created mutants that are placed in the new set BOT . The remaining ele-ments of the new population are obtained by crossover, with one parent randomlychosen from TOP and the other from REST . This distinguishes a biased random-key genetic algorithm from the random-key genetic algorithm of [4], where bothparents are selected at random from the entire population. Since a parent solutioncan be chosen for crossover more than once in any given generation, elite solutionshave a higher probability of passing their random keys to the next generation. Inthis way, |MID | = |REST | − |BOT | offspring solutions are created.

The implementations of the biased random-key genetic algorithms for the max-imum quasi-clique problem make use of the C++ library brkgaAPI developedby [28], which is a framework for the development of biased random-key geneticalgorithms. It can also be used in parallel architectures running OpenMP.

The instantiation of the framework shown in Figure 2 to some specific opti-mization problem requires exclusively the development of a class implementingthe decoder for this problem. This is the only problem-dependent part of the tool.

According to [12], the BRKGA framework requires the following parameters:(a) the population size (p = |TOP | + |REST |); (b) the fraction pe of the popu-lation corresponding to the elite set TOP ; (c) the fraction pm of the populationcorresponding to the mutant set BOT ; (d) the probability rhoe that the offspringinherits each of its keys from the best fit of the two parents; and (e) the num-ber k of generations without improvement in the best solution until a restart isperformed.

In the remainder of this work, we consider and compare two variants of a bi-ased random-key genetic algorithm for solving MQCP, each of them based on adifferent decoder. Decoder DECODER-IG∗ was originally proposed by [22] andwill be summarized in the next section. The new decoder DECODER-LSQClique


TOP TOP

REST

BOT

XCrossover

Copy best solutions

Select one parentfrom TOP

Select otherparent from REST

Randomly generatedsolutions

most fit

least fit

MID

Figure 1. Population evolution between consecutive generationsof a BRKGA.

begingenerate P vectors

of random keys

stopping rulesatisfied?

decode each vectorof random keys

sort solutions bytheir fitness

classify solutions as

elite or non-elite

copy elite solutionsto next population

generate mutants innext population

combine selected parentsand add offspringto next population

update best

result

restart rulesatisfied?

yes

no

endyes

no

Figure 2. BRKGA framework.

proposed in this work is based on the exact enumeration algorithm proposed by[24] and will be presented in Section 5.

4. Decoder DECODER-IG∗

We first review decoder DECODER-HCB, as presented in [22]. Each solutionis associated with a set of |V | random keys. The decoder receives as parameters


the random keys rj ∈ [0, 1), j = 1, . . . , |V |. Each random key is a real numberin the range [0, 1) and corresponds to a vertex of the graph. Each chromosomerepresented by a set of random keys is decoded by an algorithm that receives thekeys and builds a feasible solution to MQCP. In other words, the decoder returnsa γ-clique associated with the set of random keys. Its pseudo-code is describedin Algorithm 1. The roles of parameters minsize and α are the same explainedin [22].

It may be used in two situations. First, to build a solution from scratch. Second,to complete (i.e., to reconstruct) a partially destroyed solution. For the secondcase, the decoder receives as an additional parameter a partial solution formed bya non-empty list of vertices.

Algorithm 1 DECODER-HCB(G, γ, α,minsize, S, r)

1: CL← V \ S2: if S = ∅ then3: RCL← {v ∈ CL : |{v′ ∈ CL : degG(v′) ≥ degG(v)}| ≤ max{li , α · |CL|}}4: x← argmin{rj : j ∈ RCL}5: S ← {x}6: end if7: while CL 6= ∅ do8: CL← ∅9: for all v ∈ V \ S do

10: if 2·(|E(S)|+degG(v,S))|S|(|S|+1) ≥ γ then11: CL← CL ∪ {v}12: end if13: end for14: if CL 6= ∅ then15: for all v ∈ CL do16: dif (v)← degG(v,CL) + |CL| · (degG(v, S)− γ · (|S|+ 1))17: end for18: RCL ← {v ∈ CL : |{v′ ∈ CL : dif (v′) ≥ dif (v)}| ≤ max{minsize, α ·|CL|}}

19: x← argmin{rj : j ∈ RCL}20: S ← S ∪ {x}21: end if22: end while

The second decoder DECODER-IG∗ is an extension of DECODER-HCB pre-sented above. It is based on the constructive heuristic HCB and on the optimizediterated, but decodes a population formed by longer vector of 2 · |V | random keysRj ∈ [0, 1), j = 1, . . . , |V |, |V | + 1, . . . , 2 · |V | each. The first |V | positions of eachvector of random keys are used in the construction of the initial solution and inthe reconstruction phase, while the last |V | positions are used in the destructionphase. The roles of parameters minsize, α, δ, and β are explained in [22].


The pseudo-code of Algorithm 2 starts by creating an initial solution S′ in line 1,using the decoder DECODER-HCB and the first random keys Rj , j = 1, . . . , |V |.The loop in lines 2 to 10 repeats the partial destruction (vertex eliminations)followed by the reconstruction (vertex insertions) of the current solution, until nofurther improvements can be obtained. The current solution S′ is copied to S inline 2. The current solution S′ is copied to S in line 3. The loop in lines 4 to 8removes one by one the δ · |S′| vertices that should be eliminated from the currentsolution. A restricted candidate RCL of size max{minsize, β · |CL|} is createdin line 5, containing the vertices with the smallest degrees in G(S′). The vertexwith the smallest random key R|V |+j , j ∈ RCL, is selected from the restrictedcandidate list in line 6 and eliminated from the current solution in line 7. Thereconstruction phase is performed in line 9, where the current partial solution S′is rebuilt by decoder DECODER-HCB, once again using the first random keysRj ,∈ RCL. The loop is interrupted in line 10 when the new solution S′ obtainedby destruction-reconstruction does not improve the incumbent S or the graphG(S′) is not connected; otherwise a new iteration resumes.

Algorithm 2 DECODER-IG∗(G, γ, α, δ, β,minsize, S,R)

1: S′ ← DECODER-HCB(G, γ, α,minsize, ∅, Rj : j = 1, . . . , |V |)2: repeat3: S ← S′4: for k = 1 to δ · |S′| do5: RCL ← {v ∈ S′ : |{v′ ∈ S′ : degG(v′, S′) ≤ degG(v, S′)}| ≤

max{minsize, β · |S ′|}}6: x← argmin{R|V |+j : j ∈ RCL}7: S′ ← S′ \ {x}8: end for9: S′ ← DECODER-HCB(G, γ, α,minsize, S′, Rj : j = 1, . . . , |V |)

10: until |S′| ≤ |S| or graph G(S′) is not connected

5. New decoder based on exact local search

The principle of the new decoder proposed in this section consists in replacingthe reconstruction phase of decoder DECODER-IG∗ (i.e., line 9 of Algorithm 2) byan exact algorithm that finds the maximum quasi-clique that can be reconstructedby DECODER-HCB starting from the remaining set of vertices S′ with α = 1. Aslightly modified version of the QClique algorithm of [24] is used with this goal.

Algorithm 3 describes the pseudo-code of decoder DECODER-LSQClique. Itrequires the same parameters as DECODER-IG∗, with the exception of an ex-tended vector with 2 · |V | + 1 random keys R+j , j = 1, . . . , 2 · |V |, 2 · |V | + 1 anda frequency parameter ρ: the exact search algorithm will be applied wheneverR+2·|V |+1 ≤ ρ, otherwise decoder DECODER-HCB will be used to reconstruct the


solution. We note that, if ρ = 1 the exact search algorithm will always be exe-cuted. If ρ = 0, it will never be executed and DECODER-LSQClique will behaveexactly as DECODER-IG∗. We observe that by appropriately tuning and settingthe best value for ρ one can optimize the performance of this decoder.

DECODER-LSQClique starts by creating an initial solution S′ in line 1, usingDECODER-HCB and the random keys Rj , j = 1, . . . , |V |. The loop in lines 2 to 15repeats the partial destruction (vertex eliminations) followed by the reconstruction(vertex insertions) of the current solution, until no further improvements can beobtained. The current solution S′ is copied to S in line 3. The loop in lines4 to 8 removes one by one the δ · |S′| vertices that should be eliminated fromthe current solution. A restricted candidate RCL of size max{minsize, β · |CL|}is created in line 5, containing the vertices with the smallest degrees in G(S′).The vertex with the smallest random key R|V |+j , j ∈ RCL, is selected from therestricted candidate list in line 6 and eliminated from the current solution inline 7. The reconstruction phase starts in line 9. If the random key R+2·|V |+1 issmaller than or equal to parameter ρ and G(S′) is a γ-clique, then the partialsolution S′ is extended in line 10 by an exact local search algorithm and a newsolution S∗ is obtained. Since algorithm LSQClique might stop before an optimalsolution is obtained due to its stopping criterion, then solution S∗ is completed byDECODER-HCB in line 11 (see Section 7.1 for more details). Otherwise, solutionS′ is rebuilt by DECODER-HCB in line 13, once again using the first randomkeys R+j , j = 1, . . . , |V |. The loop is interrupted in line 14 when the new solutionS′ obtained by destruction-reconstruction does not improve the incumbent S orthe graph G(S′) is not connected; otherwise a new iteration resumes. The bestsolution is returned in S.

Algorithm 3 DECODER-LSQClique(G, γ, α, δ, β,minsize, S,R+, ρ)

1: S′ ← DECODER-HCB(G, γ, α,minsize, ∅, R+j , j = 1, . . . , |V |)2: repeat3: S ← S′4: for k = 1 to δ · |S′| do5: RCL ← {v ∈ S′ : |{v′ ∈ S′ : degG(v′, S′) ≤ degG(v, S′)}| ≤

max{minsize, β · |S ′|}}6: x← argmin{R|V |+j : j ∈ RCL}7: S′ ← S′ \ {x}8: end for9: if R+2·|V |+1 ≤ ρ and dens(G(S

′)) ≥ γ then10: LSQClique(G,S′, γ, |S|, S∗)11: S′ ← DECODER-HCB(G, γ, α,minsize, S∗, R+j : j = 1, . . . , |V |)12: else13: S′ ← DECODER-HCB(G, γ, α,minsize, S′, R+j : j = 1, . . . , |V |)14: end if15: until |S′| ≤ |S| or graph G(S′) is not connected


6. Exact algorithm for new decoder

The exact algorithm QClique of Ribeiro and Riveaux [24] is mainly based on aquasi-heredity proposition that states that for any γ-clique C of cardinality k ofG, there exists a permutation Π of its vertices such that, for every i = 1, . . . , k,the last vertex v of a prefix Πi of size i of Π has minimum degree in the inducedsubgraphG(Πi). In case more than one vertex has minimum degree, and supposingthe vertices are labeled 1, . . . , |V |, then one with maximum label must be the lastin the permutation. Under these two assumptions, every solution is visited onlyonce.

However, we observe that if the vertices of S′ are assumed to be those in Πi, inthis case algorithm QClique may fail to enumerate all solutions that can be built byDECODER-HCB. Consider e.g. the graph in Figure 3(a) with the current solutionS = {1, 2, 4, 7, 8} and γ = 0.7. In Figure 3(b), S is partially destroyed and thepartial solution S′ = {1, 2} is obtained. We observe that solution {1, 2, 3, 4, 5, 6}showed in Figure 3(c), can be reconstructed by DECODER-HCB, but not byalgorithm QClique starting with Πi =< 1, 2 >, since vertex 2 of minimum degreein G(S′) would not be the last one in all possible permutations with prefix Πi.

5

6

3

4

1

2

7

8

5

6

3

4

1

2

7

8

(a) (b)

5

6

3

4

1

2

7

8

(c)

Figure 3. Maximum solution {1, 2, 3, 4, 5, 6} can not be recon-structed by QClique from the initial solution S′ = {1, 2} after thepartial destruction of S = {1, 2, 4, 7, 8}

Algorithm LSQClique is used as a variant of algorithm QClique in line 10 ofAlgorithm 3 to perform the exact local search. It does not make use of the quasi-heredity proposition [24]. It may investigate multiple permutations of the samesolution along the search tree, contrarily to QClique that visits only one.

Instead of working with permutations like in the original algorithm QClique,algorithm LSQclique works directly with the initial solution represented by theset of vertices S′. The partial solution S′ and the initial lower bound LB = |S|


Algorithm 4 LSQClique(G,S′, γ,LB , S∗)

1: S∗ ← S′2: i← |S′|3: if a solution of size k containing all vertices of S′ does not exist for somek = i+ 1, . . . ,LB + 1 then return

4: CL← {v ∈ V \ S′ : dens(G(S′ ∪ {v})) ≥ γ}5: if CL = ∅ then return6: for all j ∈ CL do7: LSQClique(G,S′ ∪ {j}, γ,max{LB, |S′|+ 1}, S′′)8: if |S′′| > |S∗| then9: S∗ ← S′′

10: LB ← max{LB, |S∗|}11: end if12: end for

are passed to Algorithm 4 as parameters. Line 1 of the pseudo-code copies to S∗the current partial solution S′. Line 2 sets i with the size of the current solutionS′. If a solution of size k containing all vertices in S′ does not exist for some valueof k = i+ 1, . . . , LB + 1, then the algorithm returns in line 3 (pruning).

The candidate list CL created in line 4 is formed by every vertex v ∈ V \ S′ :dens(G(S′ ∪ {v})) ≥ γ, as for DECODER-HCB. If the candidate list is empty,then the algorithm returns in line 5.

The loop in lines 6 to 12 attempts to extend the current solution S′ by eachvertex j in the candidate list CL. The vertices in the candidate list are takenin non increasing order of their degrees degG(j, S′). For each of them, line 7recursively invokes the algorithm for a new, extended partial solution S′∪{j} anda new, updated lower bound max{LB, |S′|+1}, returning a new solution S′′. Line8 checks if the new solution S′′ improves then current best S∗. If this is the case,the current best and the lower bound are updated in lines 9 and 10, respectively.When the recursion terminates, S∗ contains the best solution obtained by thealgorithm.

The pruning strategy discards the current node of the enumeration tree in line3 if a solution S of size k containing all vertices of S′ can not exist for somek = i+ 1, . . . , LB + 1. This is done by calculating an upper bound to the numberof edges in G(S) and checking if the number of edges in a γ-clique is greater thanthis upper bound. We follow a similar idea to that in [24].

Let E(S′, S \ S′) denote the set of edges in G between the vertices of S′ andS \ S′. If S is a γ-clique, then the following condition holds and gives a lowerbound to |E(S)|:

|E(S)| = |E(S′) ∪ E(S′, S \ S′) ∪ E(S \ S′)| ≥ γ ·(k

2

). (8)


Since the sets in Equation (8) are disjoint,

|E(S)| = |E(S′)|+ |E(S′, S \ S′)|+ |E(S \ S′)|. (9)

Since |E(S′)| is already known, in order to calculate an upper bound to |E(S)|we need to calculate an upper bound to |E(S′, S \ S′)|+ |E(S \ S′)|.

Let V ′ be formed by the k − i vertices of V \ S′ with the maximum numberof adjacent vertices in G that belong to S′. V ′ can be determined in polynomialtime O(|V |): first sort the vertices in the non increasing order of their degreesdegG(v, S), then select the k− i first vertices. Then, an upper bound to |E(S′, S \S′)| is given by |E(S′, V ′)|.

Now, let us define V ′′ as the set formed by the k − i vertices of V \ S′ withmaximum degree in G. Then, the following condition holds:∑

v∈S\S′degG(v) ≤

∑v∈V ′′

degG(v). (10)

Following a reasoning similar to that in [24], we obtain:

|E(S′, S \ S′)|+ |E(S \ S′)| ≤∑v∈V ′

degG(v, S′) + min

{∑v∈V ′′ degG(v)−

∑v∈V ′ degG(v, S

′)

2,

(k − i

2

)}.

(11)

Therefore, if the inequality below is not satisfied for at least one value of k =i+ 1, . . . , LB + 1, then a solution of size LB + 1 does not exist and we can prunethe current node of the search tree:

|E(S′)|+∑v∈V ′

degG(v, S′) + min

{∑v∈V ′′ degG(v)−

∑v∈V ′ degG(v, S

′)

2,

(k − i

2

)}≥ γ ·

(k

2

).

(12)

7. Computational results

7.1. Experimental setting

All algorithms were implemented using version 19.14.26429.4 of the Microsoft(R) C/C++ Optimizing Compiler for x64. The computational experiments havebeen performed on an Intel Core i5-5200 processor with 2.20 GHz and 8 GB ofRAM running under Windows 10.


Table 1. Parameters settings.

parameter BRKGA-IG∗ BRKGA-LSQCliquep 91 91pe 0.13 0.13pm 0.22 0.22rhoe 0.78 0.78α 0.01 0.01

δ 0.40 1− dens(G); if dens(G) < 0.82 · (1− dens(G)); if dens(G) ≥ 0.8

β 0.02 0.02maxnodes - 220ρ - max{1− dens(G), 0.80}

The newly proposed algorithm BRKGA-LSQClique was compared with theoriginal BRKGA-IG∗ heuristic of [22], which was the best heuristic for the maxi-mum quasi-clique problem at the time of writing. The best parameters for algo-rithm BRKGA-IG∗ were determined using the automatic tuning tool IRACE [16,21]. The same parameter settings were used for algorithm BRKGA-LSQClique,except for the parameter δ that controls the number of vertices to be destroyed. Inorder to enforce that the fraction of nodes eliminated in the destruction phase besmaller for dense graphs, DECODER-LSQClique empirically sets δ = 1−dens(G),if dens(G) < 0.8; δ = 2 · (1− dens(G)), otherwise.

DECODER-LSQClique makes use of an additional parameter ρ to establishwhether the exact local search should be applied or not to some solution. Again,we empirically set the probability that exact local search is applied at ρ = max{1−dens(G), 0.80}.

An additional parameter maxnodes is used to enforce a maximum limit to thenumber of nodes of the search tree generated by the exact algorithm LSQClique.It was set at 220 using the IRACE tool. In case the search tree generated byLSQClique reaches this maximum limit of nodes, then the algorithm returns theincumbent solution. For this reason, an additional application of the reconstruc-tion procedure DECODER-HCB was enforced in line 11 of Algorithm 3 to furtherimprove the current solution.

All parameter settings are shown in Table 1.We considered 27 DIMACS instances [14], 33 maximum clique instances of the

Benchmarks with Hidden Optimum Solutions for Graph Problems (BHOSLIB) [5,25, 26], and ten sparse instances from the University of Florida Sparse MatrixCollection [9] in the computational experiments.

7.2. Numerical results

In the first experiment, for each instance and each value of the threshold γ, weperformed 30 runs of each algorithm BRKGA-IG∗ and BRKGA-LSQClique until atarget solution value was found. Each run was limited to ten minutes of execution,


except for the instances on Table 7, whose runs were limited to 30 minutes ofexecution. We discarded the instances for which both algorithm obtained thetarget solution value for every value of γ in less than three generations on averageover the 30 runs.

Tables 2 to 7 show the results for 13 of the 27 DIMACS instances, all 33BHOSLIB instances, and three of the ten sparse instances that have not beendiscarded. For each instance, the tables shows its density, the threshold γ, the tar-get look4 considered in the experiment, and the results obtained by each algorithmBRKGA-IG∗ and BRKGA-LSQClique. For each algorithm, the table presents theaverage and best sizes of the solution found over the 30 runs, the average runningtime in seconds and the average generation in which the target was found over theruns that reached the target value (#look4). Whenever any of the two methodsfailed to find a solution as good as the target value look4 within the time limitfor all instances in the group of same density, we indicate it by “> 600 (0)” or“> 1800 (0)”.

The rows highlighted in dark gray ( ) correspond to the cases where BRKGA-IG∗ reached the target look4 more often or, when both of them reached the targetthe same number of times, it was faster than BRKGA-LSQClique in terms of theaverage time to target. Rows in light gray ( ) are those for which BRKGA-LSQClique performed better in terms of the same criteria. For all other cases,both algorithms found the target in the first generation in all runs.

These results show that BRKGA-IG∗ performed better for 43 out of the 46instances with density greater than 0.5 and tested with γ = 0.999, while BRKGAperfomed better for the remaining three (DSJC1000.5, frb30-15-1, and C4000.5).However, for smaller values of the threhold γ, algorithm BRKGA-LSQClique con-siderably outperformed BRKGA-IG∗ for 47 instances (i.e., for all but instancesbrock400_2 and PGPgiantcompo), showing that the exact local search is capa-ble to considerably improve the performance of the biased random key geneticalgorithm when the target is smaller than 0.999.

In the next experiment, we evaluate and compare the run time distributions(or time-to-target plots) of algorithms BRKGA-IG∗ and BRKGA-LSQClique forsome instances. Time-to-target plots display on the ordinate axis the probabilitythat an algorithm will find a solution at least as good as a given target valuewithin a given running time, shown on the abscissa axis. Run time distributionshave also been advocated by Hoos and Stützle [13] as a way to characterize therunning times of stochastic local search algorithms for combinatorial optimizationproblems. In this experiment, the two algorithms were made to stop whenever asolution with cost greater than or equal to a given target value was found. Thetargets are the same used in the first experiment. Each heuristic was run 200 timesfor each value of the threshold γ. Next, the empirical probability distributions ofthe time taken by each heuristic to find the target solution value are plotted. Toplot the empirical distribution for each heuristic, we followed the methodologyproposed by Aiex et al. [2, 3]. We associate a probability pi = (i − 12 )/200 withthe i-th smallest running time ti and plot the points (ti, pi), for i = 1, . . . , 200.The more to the left is a plot, the better is the algorithm corresponding to it.


BRKGA-IG*

BRKGA-LSQ

Cliq

ueInstan

ceDens.

γlo

ok4

Avg

.Best

Tim

e(#

look

4)Gen.

Avg

.Best

Tim

e(#

look

4)Gen.

C250.9

0.899

0.999

4444.00

440.18

(30)

2.77

44.00

440.40

(30)

4.37

0.95

105

105.00

105

0.52

(30)

2.40

105.03

106

0.43

(30)

1.07

C500.9

0.901

0.999

5757.00

5710.79(30)

49.40

57.00

5712.15(30)

55.10

0.95

157

157.00

157

8.72

(30)

10.17

157.00

157

4.19

(30)

4.83

C1000.9

0.901

0.999

6767.10

68126.51

(30)

60.97

67.00

68170.53

(27)

103.26

0.95

217

217.30

219

128.51

(30)

27.30

217.43

219

36.15(27)

6.37

DSJ

C500.5

0.500

0.999

1313.00

130.38

(30)

1.67

13.00

130.61

(30)

1.53

0.9

2121.00

213.04

(30)

4.87

21.00

212.89

(30)

3.10

0.8

3434.00

3420.66(30)

21.70

34.00

3410.85(30)

10.13

DSJ

C1000.5

0.502

0.999

1514.67

15265.43

(20)

349.20

14.73

15180.49

(22)

180.36

0.9

2423.30

24254.91

(9)

82.22

23.63

24203.97

(19)

71.05

0.8

3938.87

40242.46

(27)

31.59

39.03

4041.22(30)

7.03

brock4

00_1

0.748

0.999

2525.00

25101.51

(30)

852.43

24.83

25294.50

(25)

1504.56

0.9

6060.00

6023.60(30)

47.30

60.00

6010.78(30)

14.37

0.8

189

189.00

189

2.62

(30)

3.10

189.00

189

1.84

(30)

1.27

brock4

00_2

0.749

0.999

2525.00

251.57

(30)

13.10

25.00

254.81

(30)

22.43

0.9

6161.00

611.88

(30)

3.67

61.00

612.30

(30)

3.23

0.8

186

186.00

186

4.50

(30)

6.00

186.00

186

1.74

(30)

1.63

brock4

00_3

0.748

0.999

2525.00

251.25

(30)

10.47

25.00

251.57

(30)

9.10

0.9

6161.00

6129.52(30)

56.03

61.00

6114.58(30)

20.93

0.8

187

187.00

187

29.114

(30)

39.60

187.00

187

6.09

(30)

2.47

brock8

00_1

0.649

0.999

2121.00

2150.07(30)

108.77

21.00

2191.67(30)

135.43

0.9

4241.87

42143.39

(26)

54.88

42.00

4268.18(30)

26.67

0.8

9695.77

96254.22

(23)

80.22

96.00

9658.27(30)

19.03

brock8

00_2

0.651

0.999

2121.00

2154.91(30)

106.90

21.00

21131.81

(30)

189.13

0.9

4241.93

42182.78

(28)

70.07

42.00

4398.38(29)

39.59

0.8

9494.03

9594.59(30)

30.87

94.00

9433.99(30)

11.27

Tabl

e2.

First

expe

riment:

30runs

ofeach

instan

cean

dalgorithm

limited

to600second

sof

runn

ingtimeeach.


BRKGA-IG*

BRKGA-LSQ

Cliq

ueInstan

ceDens.

γlo

ok4

Avg

.Best

Tim

e(#

look

4)Gen.

Avg

.Best

Tim

e(#

look

4)Gen.

brock8

00_3

0.649

0.999

2121.00

2111.83(30)

18.67

21.00

2115.97(30)

24.17

0.9

4241.97

42190.77

(29)

79.76

42.00

4228.47(30)

12.30

0.8

9392.73

93237.31

(22)

78.91

93.00

9359.67(30)

20.27

frb3

0-15-1

0.824

0.999

2828.13

291.95

(30)

13.26

28.10

291.88

(30)

9.43

0.95

5959.00

597.31

(30)

13.07

59.00

597.21

(30)

9.80

0.90

139

139.00

139

25.30(30)

31.47

139.00

138

12.02(30)

11.36

frb3

0-15-2

0.823

0.999

2828.03

290.82

(30)

4.96

28.03

291.20

(30)

5.06

0.95

5858.00

5811.38(30)

19.40

58.00

588.03

(30)

10.60

0.90

137

137.40

138

281.27

(30)

24.60

312.58

138

220.50

(30)

195.72

frb3

0-15-4

0.824

0.999

2929.00

2958.10(30)

394.97

29.00

2972.97(30)

332.63

0.95

6160.80

61243.11

(24)

364.58

60.80

61169.83

(24)

183.75

0.90

138

137.70

138

255.28

(21)

303.10

137.97

138

163.85

(29)

150.90

frb3

0-15-5

0.824

0.999

2929.00

2955.16(30)

375.47

29.00

29137.72

(30)

694.60

0.95

6060.00

6099.01(30)

157.43

60.00

6042.16(30)

46.63

0.90

134

134.00

134

19.61(30)

24.60

134.00

134

11.10(30)

11.57

frb3

5-17-1

0.842

0.999

3333.00

3341.66(30)

168.80

33.00

33108.88

(30)

372.70

0.95

7877.97

78112.96

(29)

92.83

78.00

7867.17(30)

47.77

0.90

207

215.20

216

1.81

(30)

1.00

215.47

216

1.89

(30)

1.00

frb3

5-17-2

0.842

0.999

3333.00

3321.08(30)

85.87

33.00

3332.79(30)

115.53

0.95

7575.10

7681.34(30)

63.47

75.10

7633.02(30)

23.03

0.90

207

207.00

207

100.84

(30)

60.60

207.00

207

43.63(30)

23.40

frb3

5-17-4

0.843

0.999

3333.00

3347.44(30)

187.57

33.00

3377.14(30)

247.93

0.95

7979.07

8023.49(30)

18.93

79.17

8113.27(30)

9.27

0.90

238

237.77

238

258.20

(23)

170.00

238.00

238

155.42

(30)

98.00

frb3

5-17-5

0.841

0.999

3333.03

3415.21(30)

60.90

33.00

3324.78(30)

87.80

0.95

7979.00

7986.15(30)

69.50

79.03

8033.77(30)

23.07

0.90

225

224.53

225

389.49

(16)

252.63

225.00

225

109.48

(30)

71.33

Tabl

e3.

First

expe

riment:

30runs

ofeach

instan

cean

dalgorithm

limited

to600second

sof

runn

ingtimeeach.


BRKGA-IG*

BRKGA-LSQ

Cliq

ueInstan

ceDens.

γlo

ok4

Avg

.Best

Tim

e(#

look

4)Gen.

Avg

.Best

Tim

e(#

look

4)Gen.

frb4

0-19-1

0.857

0.999

3737.00

3727.10(30)

68.90

37.03

3832.11(30)

73.73

0.95

111

111.03

112

53.17(30)

22.30

111.20

113

27.26(30)

9.77

0.90

435

435.00

435

2.23

(30)

1.03

435.00

435

2.03

(30)

1.00

frb4

0-19-2

0.857

0.999

3737.00

3715.78(30)

40.13

37.00

3723.98(30)

55.23

0.95

103

102.90

104

178.76

(26)

82.04

103.07

104

71.23(30)

30.57

0.90

363

363.00

363

16.89(30)

6.80

363.00

363

6.77

(30)

2.83

frb4

0-19-4

0.856

0.999

3837.83

39145.57

(24)

370.67

37.60

38252.25

(18)

573.72

0.95

9595.00

9553.74(30)

24.57

95.20

9622.56(30)

9.33

0.90

292

291.10

292

395.01

(4)

160.50

291.73

292

229.11

(22)

90.50

frb4

0-19-5

0.856

0.999

3838.00

3899.55(30)

254.43

37.97

38120.16

(29)

282.14

0.95

100

99.50

100

264.97

(15)

115.07

99.97

100

123.23

(29)

51.97

0.90

333

332.80

333

219.13

(24)

87.38

333.00

333

32.61(30)

13.17

frb4

5-21-1

0.857

0.999

4141.07

4212.20(30)

18.97

41.03

4226.03(30)

37.90

0.95

121

121.33

123

43.20(30)

10.37

121.57

124

15.65(30)

3.63

0.90

508

508.00

508

35.54(30)

10.27

508.00

508

7.10

(30)

2.20

frb4

5-21-2

0.869

0.999

4141.07

4215.26(30)

23.63

41.03

4241.90(30)

62.47

0.95

120

120.23

122

71.53(30)

16.93

120.33

122

39.44(30)

9.83

0.90

491

490.17

491

291.03

(5)

73.80

491.00

491

151.37

(30)

42.80

frb4

5-21-4

0.869

0.999

4242.00

42125.88

(30)

189.07

41.93

42183.97

(28)

260.39

0.95

128

128.00

128

126.04

(30)

30.83

128.33

130

43.94(30)

9.67

0.90

554

554.00

554

17.76(30)

5.00

554.00

554

3.44

(30)

1.00

frb4

5-21-5

0.869

0.999

4241.90

43195.14

(26)

302.19

41.57

42186.09

(17)

274.47

0.95

121

120.93

123

177.89

(27)

44.82

121.20

122

96.31(30)

23.33

0.90

508

507.13

508

176.41

(4)

43.75

508.00

508

61.36(30)

16.97

frb5

0-23-1

0.879

0.999

4646.03

4729.85(30)

15.97

46.03

4770.75(30)

38.87

0.95

157

157.30

159

295.15

(28)

28.46

157.53

160

54.86(30)

6.63

0.90

797

796.03

797

87.44(1)

11.00

797.00

797

52.13(30)

8.13

Tabl

e4.

First

expe

riment:

30runs

ofeach

instan

cean

dalgorithm

limited

to600second

sof

runn

ingtimeeach.


BRKGA-IG*

BRKGA-LSQ

Cliq

ueInstan

ceDens.

γlo

ok4

Avg

.Best

Tim

e(#

look

4)Gen.

Avg

.Best

Tim

e(#

look

4)Gen.

frb5

0-23-2

0.878

0.999

4646.03

4727.28(30)

14.53

46.00

4661.16(30)

33.60

0.95

158

156.13

158

397.67

(6)

40.17

158.13

159

193.64

(30)

24.40

0.90

780

779.93

780

154.18

(28)

20.07

780.00

780

18.71(30)

2.80

frb5

0-23-4

0.879

0.999

4747.00

47159.94

(30)

85.43

46.77

47185.04

(23)

99.87

0.95

157

156.43

158

350.15

(19)

36.11

157.40

160

154.65

(29)

20.97

0.90

772

771.00

771

>600(0)

-771.60

772

217.30

(18)

33.61

frb5

0-23-5

0.879

0.999

4646.10

4726.44(30)

13.60

46.13

4754.29(30)

29.43

0.95

160

158.53

162

356.84

(6)

33.50

160.43

162

201.76

(30)

26.43

0.90

788

788.00

788

77.23(30)

10.00

788.00

788

8.17

(30)

1.17

frb5

3-24-1

0.883

0.999

5049.10

50366.36

(6)

123.83

48.77

49>600(0)

-0.95

199

198.57

200

336.91

(17)

20.88

199.50

201

142.60

(30)

10.93

0.90

978

977.43

978

272.78

(13)

27.46

978.00

978

23.79(30)

2.63

frb5

3-24-2

0.883

0.999

5049.43

51189.84

(13)

64.38

49.30

50216.89

(12)

77.08

0.95

172

169.80

172

382.90

(5)

26.20

172.07

174

202.27

(29)

18.00

0.90

926

925.07

926

285.68

(2)

25.00

925.90

926

123.69

(27)

13.00

frb5

3-24-4

0.883

0.999

5049.00

50283.98

(5)

91.40

48.70

50317.79

(4)

107.75

0.95

178

177.83

179

298.29

(22)

19.18

178.40

180

78.62(30)

5.80

0.90

932

932.00

932

13.05(30)

1.00

932.00

932

10.51(30)

1.00

frb5

3-24-5

0.883

0.999

4948.97

49165.88

(29)

54.83

48.80

50194.90

(23)

65.26

0.95

166

166.50

168

302.86

(26)

20.58

167.13

171

69.86(30)

5.50

0.90

894

893.00

893

>600(0)

-894.00

894

125.81

(30)

11.70

frb5

6-25-1

0.888

0.999

5351.63

53236.63

(4)

66.00

51.20

53188.83

(1)

56.00

0.95

225

224.30

225

242.23

(15)

10.80

225.33

227

69.21(30)

3.27

0.90

1161

1161.00

1161

52.68(30)

4.27

1161.00

1161

11.05(30)

1.00

frb5

6-25-2

0.888

0.999

5251.43

53280.08

(12)

74.17

51.03

52152.36

(4)

42.75

0.95

213

210.63

212

>600(0)

-213.80

217

198.41

(29)

12.00

0.90

1175

1175.00

1175

12.94(30)

1.00

1175.00

1175

10.68(30)

1.00

Tabl

e5.

First

expe

riment:

30runs

ofeach

instan

cean

dalgorithm

limited

to600second

sof

runn

ingtimeeach.


BRKGA-IG*

BRKGA-LSQ

Cliq

ueInstan

ceDens.

γlo

ok4

Avg

.Best

Tim

e(#

look

4)Gen.

Avg

.Best

Tim

e(#

look

4)Gen.

frb5

6-25-4

0.888

0.999

5151.07

5292.58(30)

25.20

51.00

52131.59

(29)

37.00

0.95

195

193.33

196

221.82

(1)

10.00

195.33

197

128.24

(30)

7.47

0.90

1125

1125.00

1125

16.50(30)

1.10

1125.00

1125

12.32(30)

1.00

frb5

6-25-5

0.888

0.999

5251.53

53240.08

(15)

66.40

51.23

52165.91

(8)

45.50

0.95

201

199.10

202

515.94

(6)

23.33

201.50

203

188.53

(30)

11.41

0.90

1138

1138.00

1138

15.19(30)

1.03

1138.00

1138

12.30(30)

1.00

frb5

9-26-1

0.892

0.999

5554.17

56346.35

(7)

81.00

53.57

55242.13

(2)

55.50

0.95

240

239.03

241

318.16

(10)

11.30

240.90

244

156.25

(30)

6.83

0.90

1388

1388.00

1388

15.50(30)

1.00

1388.00

1388

11.28(30)

1.00

frb5

9-26-2

0.893

0.999

5453.83

54199.91

(25)

46.44

53.60

54201.10

(18)

49.06

0.95

235

234.57

237

351.97

(16)

13.25

235.80

238

102.22

(30)

4.367

0.90

1388

1388.00

1388

13.60(30)

1.00

1388.00

1388

11.10(30)

1.00

frb5

9-26-4

0.892

0.999

5554.33

55228.03

(11)

52.45

53.73

55322.46

(4)

78.50

0.95

230

228.27

232

340.83

(6)

13.50

230.70

233

203.91

(30)

10.87

0.90

1373

1373.00

1373

14.22(30)

1.00

1373.00

1373

11.42(30)

1.00

frb5

9-26-5

0.893

0.999

5453.80

55198.84

(22)

46.32

53.50

54282.46

(15)

68.40

0.95

221

220.83

223

297.89

(21)

13.91

221.90

224

89.80(30)

3.73

0.90

1360

1360.00

1360

15.62(30)

1.00

1360.00

1360

12.46(30)

1.00

Tabl

e6.

First

expe

riment:

30runs

ofeach

instan

cean

dalgorithm

limited

to600second

sof

runn

ingtimeeach.


BRKGA-IG*

BRKGA-LSQ

Cliq

ueInstan

ceDens.

γlo

ok4

Avg

.Best

Tim

e(#

look

4)Gen.

Avg

.Best

Tim

e(#

look

4)Gen.

C2000.9

0.90

0.999

7473.70

7525.85(20)

247.92

73.47

7425.70(15)

305.73

0.95

276

270.50

275

>1800

(0)

-276.30

278

349.44

(25)

10.56

C4000.5

0.50

0.999

1716.93

17461.87

(28)

19.00

16.97

18353.46

(28)

14.79

0.8

4843.60

47>1800

(0)

-49.76

49449.38

(30)

5.35

0.90

2726.83

27761.12

(25)

6.76

27.07

28115.73

(30)

1.20

frb1

00-40

0.93

0.999

8685.90

87779.23

(23)

13.04

85.03

86771.79

(8)

16.00

0.95

1835

1834.97

1835

885.90

(28)

5.286

1836.00

1837

322.2(30)

1.53

EVA

0.00019

0.90

44.00

41.67

(30)

6.30

4.00

41.82

(30)

2.73

0.80

44.57

50.32

(30)

1.00

4.97

50.61

(30)

1.00

0.70

55.00

50.34

(30)

1.00

5.00

50.65

(30)

1.00

0.60

77.00

73.52

(30)

3.17

7.00

72.21

(30)

1.10

0.50

99.00

92.31

(30)

1.57

9.00

92.28

(30)

1.00

Geom

0.00044

0.90

2323.00

233.99

(30)

8.17

23.00

231.65

(30)

1.00

0.80

2325.90

262.32

(30)

1.00

26.00

263.38

(30)

1.00

0.70

3030.00

3038.06(30)

3.53

30.00

307.25

(30)

1.13

0.60

3635.53

36607.54

(16)

32.00

36.00

36420.33

(30)

28.70

0.50

4443.80

44486.85

(24)

17.79

44.00

4418.50(30)

1.00

PGPgian

t-0.00043

0.90

4242.00

423.77

(30)

1.10

42.00

425.07

(30)

1.10

compo

0.80

4848.00

4825.23(30)

2.40

48.00

489.31

(30)

1.03

0.70

5252.00

5212.63(30)

1.10

52.00

5214.14(30)

1.03

0.60

6060.00

6050.92(30)

2.23

60.00

6021.59(30)

1.07

0.50

7373.00

73173.76

(30)

4.33

73.00

7326.58(30)

1.00

Tabl

e7.

First

expe

riment:

30runs

ofeach

instan

cean

dalgorithm

limited

to1800

second

sof

runn

ingtimeeach.


Figures 4 and 5 illustrate the time-to-target plots for instance brock400_2 withγ = 0.999, 0.90, and 0.80 and for instance PGPgiantcompo with γ = 0.90, 0.80,and 0.50, respectively. We recall from Tables 2 and 7 that BRKGA-IG∗ obtainedbetter results for both instances with γ = 0.90 and for PGPgiantcompo withγ = 0.70. The plots in these figures show that BRKGA-IG∗ performs better forthe highest values of γ, while BRKGA-LSQClique becomes progressively better asγ decreases.

8. Concluding remarks

In this work, we showed that the exact enumeration algorithmQClique proposedby Ribeiro and Riveaux [24] can be hybridized with the biased random-key geneticalgorithm BRKGA-IG∗ developed by Pinto et al. [22] as a local search strategyto improve the quality of the solutions created by the decoder. The new decoderusing an exact local search strategy gives rise to a matheuristic for solving themaximum cardinality quasi-clique problem based on a biased random-key geneticalgorithm.

Computational experiments were performed on a set of 70 test instances con-sidering different thresholds.

The results showed that although BRKGA-IG∗ performed better for 43 out ofthe 46 instances with density greater than 0.5 and tested with γ = 0.999, algorithmBRKGA-LSQClique becomes progressively better as the threshold γ decreases.For smaller values of γ, algorithm BRKGA-LSQClique considerably outperformedBRKGA-IG∗ for 47 instances, showing that the exact local search is capable toconsiderably improve the performance of the biased random key genetic algorithmwhen the target is smaller than 0.999.

References

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Figure 4. Time-to-target plot for instance brock400_2.

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

cu

mu

lative

pro

ba

bili

ty

time to target solution

BRKGA-LSQCliqueBRKGA-IG*

(a) γ = 0.999

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16

cu

mu

lative

pro

ba

bili

ty



(b) γ = 0.90

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

cu

mu

lative

pro

ba

bili

ty



(c) γ = 0.80


Figure 5. Time-to-target plot for instance PGPgiantcompo.

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35

cu

mu

lative

pro

ba

bili

ty



(a) γ = 0.90

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300

cu

mu

lative

pro

ba

bili

ty



(b) γ = 0.80

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600

cu

mu

lative

pro

ba

bili

ty



(c) γ = 0.50

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