the maximum edge weight clique problem:...

The Maximum Edge Weight Clique Problem:Formulations and Solution Approaches

Seyedmohammadhossein Hosseinian, Dalila B.M.M. Fontes, Sergiy Butenko,Marco Buongiorno Nardelli, Marco Fornari, and Stefano Curtarolo

Abstract Given an edge-weighted graph, the maximum edge weight clique(MEWC) problem is to find a clique that maximizes the sum of edge weightswithin the corresponding complete subgraph. This problem generalizes the classicalmaximum clique problem and finds many real-world applications in molecularbiology, broadband network design, pattern recognition and robotics, informationretrieval, marketing, and bioinformatics among other areas. The main goal ofthis chapter is to provide an up-to-date review of mathematical optimizationformulations and solution approaches for the MEWC problem. Information onstandard benchmark instances and state-of-the-art computational results is alsoincluded.

S. Hosseinian • S. Butenko (�)Texas A&M University, 2027 Emerging Technologies Building, Mail Stop 3131,College Station, TX 77843, USAe-mail: [email protected]; [email protected]

D.B.M.M. FontesFaculdade de Economia da Universidade do Porto, and LIAAD/INESC TEC, Rua Dr. RobertoFrias, 4200-464 Porto, Portugale-mail: [email protected]

M.B. NardelliPhysics, Chemistry and iARTA, Initiative for Advanced Research in Technology and the Arts,University of North Texas, Denton, TX 76203, USAe-mail: [email protected]

M. FornariDepartment of Physics, Central Michigan University, Mount Pleasant, MI 48859, USAe-mail: [email protected]

S. CurtaroloMaterials Science, Electrical Engineering, Physics and Chemistry, Duke University,144 Hudson Hall, Durham, NC 27708, USAe-mail: [email protected]

© Springer International Publishing AG 2017S. Butenko et al. (eds.), Optimization Methods and Applications,Springer Optimization and Its Applications 130,https://doi.org/10.1007/978-3-319-68640-0_10

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https://doi.org/10.1007/978-3-319-68640-0_10

218 S. Hosseinian et al.

1 Introduction

Network models are widely used to cast many different relationships and problemsthat are part of our everyday life. Networks can be used to represent relationsbetween entities in various complex systems, including biological, social, political,economic, and technological applications. A common problem arising in networkmodels is to find a part of an original network that satisfies some required structuralproperties and optimizes a given performance measure. In this work, we deal withone of such problems—the maximum edge weight clique (MEWC) problem.

The problem of interest generalizes the maximum clique problem (MCP), whichinvolves finding a complete subgraph of maximum cardinality in a given graph. TheMCP is a classical combinatorial optimization problem that was among the firstproblems shown to be NP-hard [31]. Over the years, many applications in numerousfields have been reported in the literature. For a recent comprehensive survey on theMCP, the reader is referred to [55].

Given a simple, undirected graph with vertex and/or edge weights, the followingweighted versions of the MCP have been considered in the literature:

• The maximum vertex weight clique (MVWC) problem,• The maximum edge weight clique (MEWC) problem,• And the maximum total weight clique (MTWC) problem.

In all three versions, a clique that induces a subgraph with maximum totalweight is searched for, the difference being that in the MVWC the weight ofthe clique is given by the sum of the weights of the vertices, in the MEWC itis given by the sum of the weights of the edges, and in the MTWC it is givenby the sum of the weights of all the vertices and the edges of the correspondingcomplete subgraph. Two common variants of each problem are considered—withand without a restriction on the cardinality of the cliques sought. The cardinality-restricted maximum weight clique asks for a maximum weight clique of cardinalityat most b and is typically considered in complete graphs. We will use the maximumallowed clique cardinality, b, in the parentheses to denote the cardinality-restrictedversion of each problem: MVWC.b/, MEWC.b/, and MTWC.b/, respectively. Theabbreviations used in the itemized list above will be used for the versions withoutthe cardinality restriction.

All six versions of the maximum weight clique problem described above areNP-hard. The versions without cardinality restrictions can be stated as the Booleanquadratic problem (BQP) [43], and the cardinality-restricted variants are specialcases of the quadratic knapsack problem [29, 47]. The focus of this paper is on theMEWC problem on simple, undirected, edge-weighted graphs; however, wheneverappropriate, we will be referring to the other maximum weight clique versions.

Several particular cases of the MEWC problem are known under different namesin the literature. For example, the heaviest k-subgraph problem, also known as

The Maximum Edge Weight Clique Problem 219

k-cluster and maximum edge-weighted subgraph [37], in which one wishes to finda subset of k vertices such that the sum of the weights of edges in the subgraphinduced by the subset is maximized. Also, dispersion and diversity problems can,in some specific cases, be seen as MEWC problems. These problems, which havebeen studied since the early 1990s, deal with selecting a subset of elements from agiven set in such a way that the diversity among the selected elements is maximized.Therefore, they also involve finding a subset of k vertices, which for complete graphswill be a clique. Several objective functions have been proposed and discussed (see,e.g., [48]), and one of the most common and well-studied versions, the max-sum,involves maximizing the sum of the weights of the edges in the induced graph.Therefore, if the distance is not 0 between every pair of elements, then a maximumedge weight clique is a solution to this problem. Recent comprehensive reviews canbe found in [34, 39].

Although most of the work has been developed considering complete graphs andsearching for weighted cliques of a fixed cardinality, many practical applications aredefined on sparse graphs. For example, in protein interaction networks, the proteinsare the vertices, and edges only exist when two proteins interact, while the weightsrepresent the strength (reliability) of protein interactions. Such networks tend to besparse. Other sparse networks arise if edges with weights below a certain thresholdare removed.

Numerous applications of the MEWC models are found in facility locationand dispersion problems, molecular biology [29], broadband network design [46],computer vision, pattern recognition and robotics [9], information retrieval, exper-imental design, signal transmission [8], and marketing [12]. More recently, theweighted cliques have been garnering significant attention in bioinformatics appli-cations, such as protein structure prediction, detection, and discovery of proteincomplexes in protein interaction networks; see, e.g., [11, 35, 54] and referencestherein. Finally, we will mention an emerging application of the MEWC problemin materials science. A system of materials can be represented as an edge-weightedgraph, where the individual materials are represented as vertices, and weights ofedges between pairs of vertices provide a quantitative description of the level ofsimilarity between the corresponding materials. Metrics of similarity between pairsof materials can take advantage of various characteristics of materials, such asdensity of states (DoS) functions or Hamiltonian materials fingerprinting basedon projection on localized basis sets [1–3]. Dropping the edges of low weight,cliques with high total edge weights are expected to correspond to groups of similarmaterials that can be mutually substitutable in certain technologies.

This chapter is organized as follows. In Section 2, we present existing formu-lations for the MEWC problem. We start by quadratic formulations, and then weprovide several integer linear programming models for this problem. Section 3discusses known solution approaches. Section 4 introduces the standard benchmarkinstances along with some computational results from the literature, and Section 5concludes the chapter.


2 Mathematical Optimization Formulations

Let G D .V; E/ be a simple, undirected, and edge-weighted graph, where V is theset of vertices and E is the set of edges. A clique is a subset of vertices C � V thatinduces a complete subgraph, i.e., for all i; j 2 C, fi; jg 2 E. The weight W.C/ ofa clique C is defined as the sum of the weights of all edges in the correspondinginduced subgraph GŒC�, that is, W.C/ D P

i;j2C wij, where wij denotes the weightassociated with an edge fi; jg 2 E. A maximum edge weight clique of G is a cliqueC in G which maximizes W.C/.

As mentioned above, this problem is NP-hard, as it generalizes the maximumclique problem (MCP) which is known to be NP-hard [31]. If all weights wij are setto 1, the problem of finding the MEWC is equivalent to the MCP. Most work on theedge-weighted cliques considers MEWC.b/, where one looks for a maximum edgeweight clique defined on a complete graph satisfying a vertex cardinality bound,although sometimes both vertex and edge-cardinality bounds are imposed. For moststudies, such a constraint is needed as complete networks and nonnegative weightsare, typically, considered. Note that the MEWC in a complete network having allweights nonnegative is the full set of vertices, i.e., V . In contrast, if all weights arenonpositive, any vertex on its own is a MEWC. To the best of our knowledge, theonly works dealing with the MEWC without cardinality constraints are [19, 25, 27,28, 50].

A straightforward nonlinear and binary formulation for the cardinality-constrained MEWC problem in complete, edge-weighted graphs is given by .Q1/

below [43]:

MaximizePn�1

iD1

PnjDiC1 wijxixj

subject toPn

iD1 xi � b;

xi 2 f0; 1g; i 2 V;

(Q1)

where b is a given upper bound on the number of vertices in a clique. A decisionvariable xi is set to 1 if the corresponding vertex i 2 V is in the clique and 0otherwise. This formulation, however, is not appropriate for non-complete graphs,as in this case the model requires a constraint preventing two nonadjacent verticesto be in the clique:

xi C xj � 1 , xixj D 0; fi; jg 62 E: (1)

Other formulations in the literature mainly focus on getting rid of the nonlinearityissue and resort to integer (linear) programming, which will be presented shortly.But, very recently, the MEWC problem has been approached using a quadratic pro-gramming formulation by scraping integrality of variables [27, 28]. The integralityconstraint of the quadratic binary formulation has been relaxed by modification ofthe objective function to include non-existing edges with sufficiently small (i.e.,negatively large) weights:


Maximize�P

.i;j/2E wijxixj � P.i;j/62E Nwijxixj

�

subject to xi 2 Œ0; 1�; i 2 V;(Q2)

where Nwij denotes the auxiliary weights assigned to non-existing edges. In particular,Hosseinian et al. [28] studied optimality characteristic of this formulation andshowed that there is a one-to-one correspondence between local optima of (Q2)and characteristic vectors of the maximal cliques, implying that an optimal solutionof (Q2) is the characteristic vector of a maximum edge weight clique in theunderlying graph.

While the quadratic models are strictly vertex-based, consisting of n D jVjvariables, integer (linear) programming models have variables corresponding toedges and hence tend to be much larger. Several integer programming formulationshave been proposed for the MEWC problem. They are based on either the so-callednatural edge formulation [14] or on the extended model, first proposed in [15],which includes not only edge variables but also vertex variables. The “natural”edge formulation due to [14] is denoted by .L1/, while the extended formulationdiscussed in [29, 38, 46, 52] is designated by .L2/. Note that both models includevertex cardinality constraints.

MaximizePn�1

iD1

PnjDiC1 wijyij

subject to yij C yik � yjk � 1; i; j; k 2 V W i < j < kP

j;j>i yij C Pj;j<i yji � b � 1; i 2 V

yij C yjl C ylm � yil � yjm � 1; i; j; l; m 2 V W i < j < l < myij 2 f0; 1g; i; j 2 V:

(L1)

The first three sets of constraints represent three different types of facet-defininginequalities. The first type, called triangle inequality, forces the third edge of atriangle to be in the solution, whenever the other two edges are. The second typeis called partition inequality, and it forces to consider only cliques that contain nomore than b vertices. Finally, the third type are the so-called Z-inequalities, whichensure that there is only one clique in the solution.

MaximizePn�1

iD1

PnjDiC1 wijyij

subject to yij � xi; yij � xj; i; j 2 V W i < jxi C xj � yij � 1; i; j 2 V W i < jP

i xi � b;

yij 2 f0; 1g; i; j; i < j 2 V W i < jxi 2 f0; 1g; i 2 V:

(L2)

The first set of constraints ensures that all edges incident to a vertex not in the cliqueare also not in the clique, while the second set of constraints forces an edge to bein the clique if both its end vertices are in the clique. The remaining constraints are


the cardinality and integrality constraints. Note that the y variables can be treatedas continuous between 0 and 1, since the first two sets of constraints force yij to beeither 0 or 1, depending on the values of xi and xj.

The number of constraints of the formulation .L2/ is O.n2/, while in formulation.L1/–O.n4/, due to the Z-inequalities. Furthermore, formulation (L2) implies thetriangle inequalities, whose number is O.n3/.

Faigle et al. [15] propose extensions to formulation .L2/. In one such extension,they also include edge- cardinality constraints, and since the problem being exam-ined involved MEWC with exactly b vertices, both types of cardinality constraintsare written as equalities:

Pi xi D b; (2)

Pi;j;i<j yij D b.b�1/

2: (3)

Another extension was proposed by substituting the edge-cardinality constraints (3)by constraints on the edges cardinality per vertex:

Pj;j>i yij C P

j;j<i yji D .b � 1/xi; i 2 V: (4)

Other similar formulations have been used over the years, for example, in [29, 46]the edge-vertex formulation does not include the edge-cardinality constraints (3). In[46], a comparison of the linear programming relaxations of the two models (edgevariables and edge and vertex variables) was performed. The authors were able toconclude that the linear programming relaxation of (L2) formulation is stronger thanthat of (L1). The formulations in [38] and [52] substitute the cardinality constraintson the vertices and on the edges (Equations (2) and (3), respectively) by the so-calledstar inequalities (5) and (6), given next. These inequalities, first proposed in [46],limit the number of edges incident to each node to either b � 1 or 0, depending onwhether the node is in the clique or not, respectively.

Pj;j¤i yij � .b � 1/xi � 0; i 2 V; (5)

Pfi;jg2ı.i/ yij � .b � 1/xi � 0; i 2 V; (6)

where ı.i/ denotes the set of edges incident to vertex i.As mentioned above, many practical applications are defined on sparse graphs;

however, the models just discussed do not take advantage of network’s sparsity.Actually, some authors deal with non-complete graphs by adding dummy edgeswith a sufficiently large negative weight [44, 45], which leads to a model muchlarger than needed as it builds up the number of constraints.

In a contemporary work [25], Gouveia and Martins discuss formulations tailoredfor sparse graphs by analyzing their magnitude (number of variables and constraints)and the strength of the linear programming relaxations. Their first proposed formu-


lation .L3/ is based on the model introduced in [40] for the maximum/minimumedge neighborhood density clique problem. In this formulation, the vertex variablesare discretized by including an extra index that identifies the size of the clique thevertex belongs to and uses additional variables for the clique size as follows: ifvertex i is in a clique of size q, then v

qi is 1, otherwise it is 0; and a variable wq

is 1 if the clique has size q and 0, otherwise. The possible values for q, i.e., cliquecardinality, are given by Q D f2; : : : ; qmaxg, where qmax is an upper bound on theclique number that can be found by any known method; see, e.g., [22].

MaximizePn�1

iD1

PnjDiC1 wijyij

subject to yij � Pq2Q v

qi ; yij � P

q2Q vqj ; fi; jg 2 E

Pq2Q v

qi C P

q2Q vqj � yij � 1; fi; jg 2 E and wij < 0

PjWfi;jg2ı.i/ v

qj � .q � 1/v

qi ; i 2 V; q 2 Q

Pi2V v

qi D qwq; q 2 Q

Pq2Q wq D 1;

vqi 2 f0; 1g; i 2 V; q 2 Q

wq 2 f0; 1g; q 2 Qyij 2 f0; 1g; fi; jg 2 E:

(L3)

The equivalence of this formulation to (L2) can be easily seen by substituting thevertex variables with

Pq2Q v

qi . Thus, the first two sets of constraints impose that

an edge is in the clique if and only if both its end vertices are in the clique, whilethe next two sets of constraints enforce edges and vertices cardinality, respectively.Finally, the fifth constraint ensures the existence of a solution. This formulationinvolves O.jEj/ variables and O.jEj/ constraints, the same as (L2). Also, it has lessconstraints but more variables than (L1). Note that for sparse graphs, jEj may bemuch smaller than n2.

The authors of [25] also propose two different ways of strengthening the model,leading to two different versions of the formulation, namely, .L3a/, by adding starmerging equalities (7), and .L3b/, by adding both the star merging equalities (7)and inequalities (8) that force the number of vertices in the clique not adjacent tovertex i to be 0 if vertex i is in the clique. The star merging equalities state thateach vertex i in a clique with cardinality q requires .q � 1/ edges incident to it, andnone—if it is not in any clique.

Pfi;jg2ı.i/ yij D P

q2Q.q � 1/vqi ; i 2 V; (7)

PjWfi;jg62ı.i/ v

qj � .jVj � jı.i/j � 1/.wq � v

qi /; i 2 V; q 2 Q: (8)

Although all these models have the size of the same magnitude, .L3b/ providesa stronger linear programming relaxation than that of .L3a/, which in turn providesa stronger linear programming relaxation than that of (L3).


The second formulation proposed by Gouveia and Martins [25] uses the ideathat a clique can be viewed as an adequate intersection of stars. This idea was firstintroduced in [26] for characterizing cliques in directed graphs. The correspondingmodel .L4/ is substantially more compact as it involves only O.n/ constraints.

MaximizePn�1

iD1

PnjDiC1 wijyij

subject toP

fi;jg2ı.i/ yij D Pq2Q.q � 1/v

qi ; i 2 V

Pi2V v

qi D qwq; q 2 Q

Pq2Q wq D 1;

vqi 2 f0; 1g; i 2 V; q 2 Q

wq 2 f0; 1g; q 2 Qyij 2 f0; 1g; fi; jg 2 E:

(L4)

Three strengthened versions of formulation (L4) were considered, namely:

• .L4a/ obtained by including constraints

PjWfi;jg2ı.i/ v

qj � .q � 1/v

qi ; i 2 V; q 2 Q; (9)

• .L4b/ by including constraints

yij � Pq2Q v

qi ; yij � P

q2Q vqj ; fi; jg 2 E; (10)

• and .L4c/ by including constraints (8) and (10).

The number of constraints is increased to O..qmax � 1/jVj/ for .L4a/ and to O.jEj/for the others. Regarding the LP relaxations, .L4c/ is tighter than .L4b/, and allthree are tighter than (L4).

3 Solution Approaches

The first attempt to address the MEWC.b/ is due to Späth [53]. In this work, theauthor considers a facility selection-location problem, in which b facilities out of then > b possible ones are to be selected such that their average distance is minimized.Finding a solution to this problem is equivalent to finding a clique of b vertices withmaximum weight in an edge-weighted complete graph if edge weights are givenby M � de, where M is a strict upper bound on the distance de; e 2 E. Späth [53]proposes two heuristics that iteratively improve some initial clique by successivelyexchanging two vertices (one in the clique with another not in the clique). Thefirst heuristic uses the best improvement strategy, while the second uses the firstimprovement. The heuristics were tested on 27 random complete graphs with 25vertices to find MEWCs with n D 2; 3; : : : 24 vertices. These instances became thefirst MEWC(b) benchmarks.


Most of other early work on MEWC(b) problem focused on studying thepolyhedral properties of the b-clique polytope, which is the convex hull of allincidence vectors of cliques in Kn (complete graph on n vertices) with at most bvertices, and algorithmic approaches based on cutting planes [14, 29, 38, 43, 46, 52].The availability of strong valid inequalities was the key to the success of thesealgorithms. Branch-and-cut algorithms dominated the literature on MEWCs untilrecently. One of the main drawbacks of this type of approach is that, usually, it isnot computationally “easy” to generate violated facet-defining cutting planes.

Dijkhuizen and Faigle [14] consider cliques with up to b vertices and unrestrictededge weights for which they propose a “natural” edge variable formulation .L1/.They investigate the problem from a polyhedral point of view and introduce severalfacet-defining inequalities. A cutting-plane approach was proposed, which at eachiteration extends the current linear programming (LP) relaxation by imposingfacet-defining inequalities that were violated. Despite the extensive use of cuttingplanes and the eight different types of facet-inducing inequalities considered, thecomputational experiments reported showed that the cutting-plane approach was notsuitable to solve the MEWC(b) problem, even for small-sized instances. The largestinstance they solve refers to a 25 vertices graph, but extremely poor performancesare reported for quite smaller instances. Faigle et al. [15] propose an extendedformulation .L2/, including binary variables for vertices in addition to those foredges. Although the authors did not investigate the new model from a polyhedralpoint of view, they were able to show that by lifting the problem into higherdimensions, it was possible to obtain linear programming relaxations of goodquality. In addition, they have also considered a Lagrangian relaxation that for eachchoice of parameters was solved by a direct greedy algorithm. The computationalexperiments involved finding the MEWC with exactly b vertices in complete graphswith up to 200 vertices. The bounds obtained were very tight and for larger b valueof the gap between the lower and upper bound tended to zero. However, they werenot able to improve on the solutions found by Späth [53], despite concluding thatlifting the problem into higher dimensions had a substantial positive effect on thequality of the LP relaxation.

Following on these works, Park et al. [46] investigate both models. The authorsstart by summarizing the previous results on the polyhedral structure of thepolytope associated with the “natural” formulation .L1/ and then study the polytopeassociated with the extended formulation .L2/. This was in fact the first polyhedralstudy of the extended formulation. Park et al. [46] establish some classes offacet-defining inequalities for the polytopes associated with both formulations. Byrealizing that ignoring the clique cardinality constraints the problem can be viewedas a linearized version of BQP and that by ignoring the constraints forcing anedge to be in the solution when the vertices associated with both end points were,the MEWC reduces to a special case of the quadratic knapsack, the authors wereable to use the results of [30, 45] to provide the first polyhedral results for the b-clique polytope. They demonstrated this way that the extended formulation givesa tighter LP relaxation than the natural formulation, although with a much largernumber of constraints. The authors propose a branch and cut that successfully solves


problem instances involving graphs with up to 30 vertices. Most of the probleminstances have been solved to optimality by pure cutting planes (no branching wasnecessary). The computational performance was compared to that of [14] using theSpäth’s instances [53] with ten vertices. All instances were solved optimally within2 s, while [14] could not solve 18 of the 60 instances considered, with the timerequirements varying between 95 and 380 s.

Macambira and de Souza [38] continue this line of research. They introducedthree new classes of facet-defining inequalities extending the previous works [30]and [45]. The first class generalizes the clique, the cut, and the .s; t/-cut inequalities[45], while the second and third classes of inequalities generalize inequalitiesoriginally introduced in [30] and further studied in [46]. The authors then proposea branch-and-cut algorithm that uses different cutting-plane strategies based onthe newly defined facet inequalities. Initial lower bounds are derived by a primalconstructive heuristic, which fits the framework of greedy randomized adaptivesearch procedures (GRASP) [16]. Violated inequalities were found by enumeration,except for tree types of inequalities, since in this case the complexity would becomeO.n4/ or even O.n5/. The most fractional variable is chosen for branching, and thenext node to explore is the one with the best bound. Computational experimentswere carried out on problem instances with up to 48 vertices, larger than previouslyconsidered instances. All problem instances used were solved to optimality.

Hunting et al. [29] also propose a branch-and-cut algorithm, in which Lagrangianrelaxation and cutting planes are combined. However, they use a different relaxationtechnique, previously suggested for the traveling salesman problem [7]. At eachnode of the search tree, subproblems are obtained by including some vertex i inthe clique (xi D 1) or excluding it from the clique (xi D 0). Upper boundswere obtained by solving the Lagrangian dual and then iteratively improved byadding violated inequalities. In each search tree node, inequalities associated withLagrangian multipliers having a value close to zero are removed. The idea has beensuggested in [7] for the traveling salesman problem and has been employed, e.g., in[36] for Steiner problems in graphs. Lower bounds were found by using the heuristicproposed in [53]. The computational experiments were carried out on the probleminstances of [38] and have shown the method to have a comparable performance tothat of [38] for most problems and to be 2–3 times faster for the instances involvingmixed (i.e., both positive and negative) weights.

Sørensen [52] follows the same line of work and also introduces new facet-defining inequalities. Then a branch-and-cut algorithm using these new inequalities,as well as some of the previously defined ones, is proposed to solve the MTWC.b/

problem. Four classes of inequalities are considered, the first is a generalization ofthe tree inequalities, first introduced in [30] for the quadratic knapsack polytope,termed cut-tree inequalities, while the other classes are associated with multistars(clique-star, cut-star, and T-star inequalities). Generation of violated inequalities, toimpose as cutting planes, is controlled in an iterative and hierarchical manner thatimplies very frequent use of the inequalities that are employed in level 1, frequentuse of the ones that are used in level 2, and sporadic use of the level 3 ones.The choice of which inequalities are employed at each level reflects the relative


importance of the inequalities, previously found empirically. Whenever no violatedinequalities can be found or successive LPs have very close objective function value,branching occurs. In order to keep the LPs within a reasonable size, constraintidentification is terminated whenever 40n inequalities have been found, of whichonly up to the most violated 400 are considered, and nonbinding inequalities areremoved. Branching is based on vertices and the most fractional variable is chosen,ties being broken by weight value. The next branch to process is selected by abest-bound search strategy. A lower bound is obtained by a two-phase heuristic,composed of a greedy construction heuristic and an improvement heuristic, similarto that of [53]. This lower bound is used in the pruning phase. Two sets of probleminstances ranging from 30 to 61 vertices were used, and the computation timesobtained indicate that the algorithm is more efficient than previously proposedalgorithms [29, 38]. The first set consists of the instances proposed by Macambiraand de Souza [38] and also used in [29], while the second set consists of instancesof a column generation problem in the context of graph partitioning. The latterhave been generated in [51], and only these involve node weights. Regarding thefirst set of problems, a direct comparison with the computational times reportedin [29, 38] was not possible as different computers have been used. Nonetheless,the authors stated that, after adjusting for the relative speed of computers, theiralgorithm compares favorably with that of [29], as it is about five times faster onaverage (ranging from 2 to 12 times) for instances with positive weights and tentimes faster (ranging from 3 to 26) for instances with mixed weights.

More recently, Alidaee et al. [5] approached the MEWC.b/ by adapting a tabusearch heuristic designed for the general unconstrained binary quadratic program.The method applies the framework discussed in [23] that is centered around the useof a parametric oscillation. The value of the parameter “spam,” which is updated,indicates the amplitude of the oscillation around local optima. When local optimaare found, information regarding the values of the variables is stored, which is laterused to penalize potentially attractive add moves and induce drop moves avoidinggetting trapped into the same local optima. For further details and other applicationsof this framework, see, e.g., [4, 6, 32, 33]. The computational tests were performedon the problems proposed by Macambira and de Souza [38], which were alsosolved by Hunting et al. [29], Sørensen [52]. The authors were able to find anoptimal solution for all problem instances (as was the case with the other threeworks) improving significantly on the computational time requirements. Althoughcomparing the computational times for the various methods is not easy, as theauthors used different machines, the reported computational times are quite small,always less than 1 s.

In all the above studies, MEWC(b) problem was considered on a complete graph.However, many practical applications are defined on sparse graphs, for example, inprotein interaction networks [11, 54], where the proteins are the vertices and edgesonly exist when two proteins interact, having weights that represent the reliability ofprotein interactions. Such networks tend to be sparse. Other sparse networks arisedue to cutoff criteria, e.g., edges with weights below a certain threshold are removed.In this version, usually, no cardinality constraint is imposed.


The first approach in which edge variables are only defined for edges ofthe graph with nonzero weights is due to Mehrotra [43]. This work considersthe cardinality-constrained Boolean quadratic polytope which is the same as theb-clique polytope. After establishing many facet-defining inequalities, a branch-and-cut methodology is proposed. Computational experiments are reported forproblem instances associated with up to 30 vertices graph.

In a recent work, Mascia et al. [42] address the problem of characterizingstructural and functional sites in proteins, which requires determining the numberof sites in the protein and the set of residues involved in each of them, by searchingfor maximum weight cliques of an appropriate size in a weighted graph. The graphrepresents the residues (vertices) and their similarity (weighted edges). Only pairs ofresidues that satisfy the similarity threshold, i.e., only vertex pairs whose similarityis above a prespecified threshold, are adjacent. Solutions for the problem are foundby using a stochastic local search algorithm, which is based on the one proposed in[10] for the maximum clique problem. To avoid local search entrapment, the authorsimplement a prohibition period in a tabu search fashion, during which verticescannot be added or removed from the current solution.

Only more recently the MEWC problem defined on non-complete graphs andwithout cardinality constraints has been considered [19, 25, 27, 28, 50]. Pullan[50] extended and adapted the phased local search proposed in [49] to address themaximum edge (or vertex) weighted clique problem. The local search proposedconstructs a feasible clique by adding one vertex at a time until none can be added.New and better solutions are searched for by swapping a vertex not in the clique withone in the clique followed by the restart of the constructive phase whenever possible,otherwise another swap is performed. This is repeated until a prespecified numberof iterations is performed. The performance of this type of algorithms, typically,depends on the criteria used for choosing the vertices to add or remove from thecurrent solution. In order to avoid local entrapment, [50] uses three criteria, oneat a time. For the initial 50 iterations, a vertex is uniformly and randomly chosenamong the ones which would produce the largest increase in the clique weightif added. Then, for another 50 iterations, after identifying the vertices with thesmallest penalty, one is uniformly and randomly chosen among the ones whichwould produce the largest increase in the clique weight if added. Finally, for the last100 iterations after identifying the vertices with the largest degree, one is uniformlyand randomly chosen among the ones which would produce the largest increase inthe clique weight if added. The penalties are associated to the vertices in order todiscourage a frequent use of the same vertices. In the initialization, the first vertexis chosen randomly and the same happens when starting the last 100 iterations.However, at iteration 51 the algorithm continues from the clique obtained at theprevious iteration. Since the test instances used for the MEWC(b) problem werenot suitable for testing algorithms for the MEWC problem, Pullan [50] constructed80 benchmarks by adapting the maximum clique (MC) instances from the SecondDIMACS Implementation Challenge (see Section 4 for details). He performed aseries of long running experiments to obtain the “best” clique weight for eachproblem instance. Then, the heuristic method was run 100 times, and the results


reported include, for each problem instance, the number of times the “best” valuewas obtained, as well as the average computational time needed to obtain them.For the runs where the “best” value was not obtained, no additional informationis provided. For most instances, the “best” value was always obtained withinmodest computational times. However, for six instances, the “best” value was neverobtained, and for other three, it was only obtained in some of the runs and with muchlarger computational times. The graphs associated with the instances for which themethod was not able to achieve good results or performance are among the mostdense ones.

In a more recent work, Gouveia and Martins [25] study existing compactformulations for the MEWC on complete graphs in order to adapt them for sparsegraphs and propose new formulations for these types of graphs. The authors alsodiscuss some strengthening strategies. Computational experiments were carriedout on some of the instances proposed by [50] and on the following two real-world instances: the Reuters terror news (RTN) network, which was proposedin [13], and networks of interacting pathways and interacting metabolites foundin the bioinformatics literature [21]. Computational results are reported for ninedifferent integer programming models. All linear programming relaxations produce,in general, very large gaps, and they still remain large even after the strengthenedprocesses. However, the authors were able to observe some correlation between theperformance of different models and the graph sparsity. They also propose a newversion of the MEWC, designated by maximum edge weight neighborhood clique(MEWNC), in which one wants to find a clique that maximizes the total weight ofthe edges connecting the clique to its complement. This problem is the weightedversion of the maximum edge neighborhood clique problem proposed in [41]. Thecomputational experiments led the authors to the conclusion that this problem seemsto require even more computational effort than the MEWC.

Most recently, new methods for solving the MEWC problem based on thecontinuous quadratic formulation (Q2) are proposed [27, 28]. In [27], a constructionheuristic method based on an approximation of (Q2) is presented. The quadraticprogramming formulation is approximated by substituting the unit hypercubeconstraint by a unit hypersphere. A heuristic solution to the MEWC is then obtainedby considering all stationary points of the approximation problem. The stationarypoints are given by the eigenvectors of the objective function coefficients matrix.The algorithm extracts the clique corresponding to each eigenvector and outputs theone with the maximum total sum of edge weights. Every eigenvector is treated asan approximation of the characteristic vector of a maximal clique in the graph. Inorder to extract the corresponding clique, the eigenvector is sorted in a nonincreasingorder of its components implying an ordered list of vertices. The clique is initiatedby including the first vertex in the ordered list and expanded by appending theother vertices one-by-one, according to their orders, if connected to all vertices thatare already in the incumbent clique. A slightly modified version of this heuristicalgorithm has been used as a part of an exact method for the MEWC in [28].The exact algorithm is a combinatorial branch-and-bound method that uses thespectral heuristic to draw an initial lower bound, while it employs an upper bound


based on a quadratic relaxation of (Q2) at every node of the search tree. Thequadratic relaxation concerns optimization of the objective function of (Q2) subjectto a single hypersphere constraint. It has been shown that, under some regulatoryconditions, finding such upper bounds is not harder than finding the root of amonotonically decreasing function in a known interval. Computational experimentson some benchmark instances proposed in [50] have been carried out. The resultsreported in [28] show significant improvement in solution time for most of theinvestigated instances compared to the best exact results due to integer programmingformulations.

Another very recent approach has been proposed in [19], where a biasedrandom key genetic algorithm (BRKGA) is used to find good quality solutions withmodest time requirements. The BRKGA developed uses the framework proposed byGonçalves and Resende [24], which has been shown to have very good performancefor many different problems, including problems defined on graphs; see, e.g.,[17, 18]. The GA is responsible for evolving the chromosomes, which representthe vertex priorities used to construct solutions. The first gene is used to determinethe initial vertex in the clique construction. Then, the remaining vertices are sortedin increasing order of the corresponding gene values, i.e., random keys. The sortedvertices correspond to the sequence in which they will be added to the incumbentclique as long as they are adjacent to all vertices in the partial clique. Once a cliqueis obtained, its weight is computed. To obtain new solutions, two individuals arerandomly chosen to act as parents. One of them is chosen among the best solutions,while the other is chosen from the remainder of the population. Genes are chosenby using a biased uniform crossover, that is, for each gene, a biased coin is tossed todecide on which parent the gene is taken from. This way, the child inherits the genesfrom the top parent with higher probability. A new population is obtained by joiningthree subsets of solutions as follows: the first subset is obtained by copying the bestsolutions of the current population; the second subset is obtained by using a (biased)parameterized uniform crossover; and the remaining solutions, termed mutants, arerandomly generated as is the case for the initial population. The algorithm was testedon the instances proposed by Pullan [50], and the computational experiments showthat for most of the problems, the GA is able to obtain a solution matching the bestknown/optimal one within the time limit imposed (60 s). Actually, for most of theseinstances, the GA converged quite quickly, and the best solution found was obtainedearly on. However, this was not the case for less sparse graphs. For some of theinstances for which Pullan was able to quickly obtain a solution matching the bestknown one, the GA was not capable of doing so within the 60 s allowed. However,for one instance that Pullan was never able to obtain a solution as good as the bestknown/optimal one, after running his method 100 times each with a time limit,1 theGA obtained such a solution within the 60 s allowed. For another instance, Pullandid so only in 7 out of the 100 trials (with an average time of over 860 s), whereas

1Pullan [50] does not specify any time limit used; however for some instances, the average timereported is over 3300 s.


the GA was able to obtain it within the allowed 60 s. In [20] the GA reported in[19] is hybridized with a local search mechanism that allows for improved results.Furthermore, in the computational tests, the other problem instances used in [25]were also used, and the comparison with known results is favorable.

4 Benchmark Instances and Sample Computational Results

This section provides a description of the benchmark instances used to evaluate theperformance of algorithms for the MEWC problem. Sample computational resultsreported in the literature are also included.

Pullan [50] constructed benchmarks by adapting the maximum clique (MC)instances from the Second DIMACS Implementation Challenge (1992–1993).2

These problem instances range in size from less than 50 vertices and 1000 edgesto more than 3300 vertices and 5,000,000 edges. There are a total of 80 instancesthat have been generated from problems arising in coding theory, fault diagnosis,Keller’s conjecture on tilings using hypercubes, and Steiner triple, as well asrandomly generated instances.

The MC instances are converted into MEWC instances (termed DIMACS-EW)by associating a weight with each (existing) edge, which is calculated as follows:

wij D .i C j/ mod 200 C 1: (11)

Table 1 provides a description of the instances characteristics: the number of verticesjVj, the number of edges jEj, the edge density �.G/, as well as the best knownsolution W.C�/, i.e., the largest value for a clique weight reported in the literature.The values of W.C�/ that were proved to be optimal are shown in bold.

Next, we present some sample computational results reported by Pullan [50],Gouveia and Martins [25], Hosseinian et al. [28], and Fontes et al. [19]. Amongthese, [50] and [19] present heuristic results, [25] reports exact results, and [28]contains both heuristic and exact results. Since not all works attempt to solve all80 problem instances, we only summarize the results for 18 instances reportedin all of the four aforementioned papers in Table 2. One must, however, bear inmind that a more fair performance comparison would need to take into accountmany factors, such as the programming language, the data structures used, thecomputational equipment and protocols, and the algorithm parameters (stoppingcondition, parameter tuning, etc.) not mentioned here.

Pullan [50] ran his method 100 times for each problem instance with anundisclosed time limit and reported the number of times (“#” in the table) the bestknown solution was found, as well as the average time needed to find such solutions.For the sake of comparison, we report the time taken for all 100 trials. The GA

2http://dimacs.rutgers.edu/Challenges/.

http://dimacs.rutgers.edu/Challenges/


Tabl

e1

DIM

AC

Spr

oble

min

stan

ces:

char

acte

rist

ics

and

best

know

nso

lutio

nva

lues

(val

ues

know

nto

beop

timal

are

show

nin

bold

font

)

Inst

ance

jVjjEj

�.G

/W

.C�

/In

stan

cejVj

jEj�.G

/W

.C�

/

broc

k200

-120

014

,834

0.74

521

,230

john

son8

-4-4

7018

550.

768

6552

broc

k200

-220

098

760.

496

6542

john

son1

6-2-

412

054

600.

765

3808

broc

k200

-320

012

,048

0.60

510

,303

john

son3

2-2-

449

610

7,88

00.

879

16,2

05

broc

k200

-420

013

,089

0.65

813

,967

kelle

r417

194

350.

649

6745

broc

k400

-140

059

,723

0.74

835

,257

kelle

r577

622

5,99

00.

752

38,9

01

broc

k400

-240

059

,786

0.74

940

,738

kelle

r633

614,

619,

898

0.81

838

,901

broc

k400

-340

059

,681

0.74

846

,785

MA

NN

-a9

4591

80.

927

5460

broc

k400

-440

059

,765

0.74

954

,304

MA

NN

-a27

378

70,5

510.

990

799,

525

broc

k800

-180

020

7,50

50.

649

25,0

50M

AN

N-a

4510

3553

3,11

50.

996

5,93

0,06

4

broc

k800

-280

020

8,16

60.

651

27,9

32M

AN

N-a

8133

215,

506,

380

0.99

960

,504

,100

broc

k800

-380

020

7,33

30.

649

30,9

72p-

hat3

00-1

300

10,9

330.

244

3321

broc

k800

-480

020

7,64

30.

650

30,9

50p-

hat3

00-2

300

21,9

280.

489

31,5

64c-

fat2

00-1

200

1534

0.07

777

34p-

hat3

00-3

300

33,3

900.

744

63,3

90

c-fa

t200

-220

032

350.

163

26,3

89p-

hat5

00-1

500

31,5

690.

253

4764

c-fa

t200

-520

084

730.

426

168,

200

p-ha

t500

-250

062

,946

0.50

563

,870

c-fa

t500

-150

044

590.

036

10,7

38p-

hat5

00-3

500

93,8

000.

752

122,

302

c-fa

t500

-250

091

390.

073

38,3

50p-

hat7

00-1

700

60,9

990.

249

5185

c-fa

t500

-550

023

,191

0.18

620

5,86

4p-

hat7

00-2

700

121,

728

0.49

896

,380

c-fa

t500

-10

500

46,6

270.

374

804,

000

p-ha

t700

-370

018

3,01

00.

748

191,

148


C12

5.9

125

6963

0.89

866

,248

p-ha

t100

0-1

1000

122,

253

0.24

554

36

C25

0.9

250

27,9

840.

899

96,6

92p-

hat1

000-

210

0024

4,79

90.

490

107,

870

C50

0.9

500

112,

332

0.90

016

4,95

3p-

hat1

000-

310

003,

711,

746

7.43

122

9,50

0

C10

00.9

1000

450,

079

0.90

123

4,01

3p-

hat1

500-

115

0028

4,92

30.

253

7135

C20

00.5

2000

999,

836

0.50

014

,927

p-ha

t150

0-2

1500

568,

960

0.50

621

1,06

9

C20

00.9

2000

1,79

9,53

20.

900

316,

135

p-ha

t150

0-3

1500

847,

244

0.75

444

1,99

8

C40

00.5

4000

4,00

0,26

80.

500

19,3

04sa

n200

-0.7

-120

013

,930

0.70

045

,295

DSJ

C50

0-5

500

62,6

240.

502

9626

san2

00-0

.7-2

200

13,9

300.

700

15,0

73

DSJ

C10

00-5

1000

249,

826

0.50

012

,054

san2

00-0

.9-1

200

17,9

100.

900

242,

710

gen2

00-p

0.9-

4420

017

,910

0.90

094

,362

san2

00-0

.9-2

200

17,9

100.

900

178,

468

gen2

00-p

0.9-

5520

017

,910

0.90

015

0,83

9sa

n200

-0.9

-320

017

,910

0.90

069

,764

gen4

00-p

0.9-

5540

071

,820

0.90

015

0,98

1sa

n400

-0.5

-140

039

,900

0.50

074

42

gen4

00-p

0.9-

6540

071

,820

0.90

020

8,62

4sa

n400

-0.7

-140

055

,860

0.70

077

,719

gen4

00-p

0.9-

7540

071

,820

0.90

027

9,06

9sa

n400

-0.7

-240

055

,860

0.70

044

,155

ham

min

g6-2

6418

240.

905

32,7

36sa

n400

-0.7

-340

055

,860

0.70

024

,727

ham

min

g6-4

6470

40.

349

396

san4

00-0

.9-1

400

71,8

200.

900

496,

874

ham

min

g8-2

256

31,6

160.

969

800,

624

san1

000

1000

249,

750

0.50

010

,661

ham

min

g8-4

256

20,8

640.

639

12,3

60sa

nr20

0-0.

720

013

,930

0.70

016

,398

ham

min

g10-

210

0025

9,32

80.

519

13,1

40,8

16sa

nr20

0-0.

920

017

,910

0.90

085

,920

ham

min

g10-

410

2443

4,17

60.

829

83,2

80sa

nr40

0-0.

540

039

,900

0.50

082

98

john

son8

-2-4

2821

00.

556

192

sanr

400-

0.7

400

55,8

600.

700

22,7

91


Table 2 Summary of the sample results

Heuristic Exact

[50] [19] [28] [25] [28]

Instance # Time Time Gap(%) Time Time Time

brock200-1 100 4:00 2:85 0:00 1:98 >10;800:00 3047:57

brock200-2 100 2:00 1:61 0:00 0:75 9464:24 7:44

brock200-3 100 1:00 2:06 0:00 1:25 >10;800:00 55:91

brock200-4 100 2:00 2:34 1:65 1:56 >10;800:00 188:03

c-fat200-1 100 <1:00 0:46 0:00 0:05 3:87 0:48

c-fat200-2 100 <1:00 0:94 0:00 0:06 33:26 0:89

c-fat200-5 100 <1:00 60:00 0:00 0:63 155:30 >10;800:00

C125.9 100 7:00 1:81 1:26 0:73 >10;800:00 4558:17

hamming6-2 100 <1:00 0:27 0:00 0:05 0:30 4:44

hamming6-4 100 <1:00 0:09 0:00 <0:01 1:97 0:03

hamming8-4 100 1:00 3:00 0:00 2:97 >10;800:00 439:44

johnson8-2-4 100 <1:00 0:03 0:00 <0:01 0:14 <0:01

johnson8-4-4 100 <1:00 0:22 0:00 0:05 2:34 0:69

johnson16-2-4 100 <1:00 0:66 0:00 0:33 >10;800:00 84:69

keller4 100 1:00 1:42 0:00 0:80 >10;800:00 42:22

MANN-a9 100 <1:00 0:10 0:00 0:05 9:39 1:91

p-hat300-1 100 3:00 1:29 0:00 0:58 1273:05 3:28

p-hat300-2 100 <1:00 3:42 0:00 3:33 >10;800:00 171:28

All reported times are in CPU seconds

proposed in [19] was run once for each problem instance using three seeds with atime limit of 60 s (20 s per seed). The population size was set to 150 � jVj, basedon the performance achieved considering a pilot subset of instances consisting ofbrock200-1 and 400-1, c-fat200-1 and 200-2, and gen200p0.9-44 and 200p0.9-55.Literature recommendations were used to set the population parameters, as well asthe crossover probability. The computational time reported is the one required to findan optimal solution. For the construction heuristic algorithm proposed in [28], wereport the percentage optimality gap (i.e., W.C�/�W.C/

W.C�/� 100) and the computational

time of finding the solution.Regarding the exact methods, Gouveia and Martins [25] solved nine different

integer programming models. Here we report only the best solution time amongthem, due to different formulations for different instances. In this work, the solutiontime was limited to 3 h (10,800 s) for each model on each instance. The same timeconstraint has been considered in computational results of the combinatorial branch-and-bound algorithm reported in [28].


5 Conclusion

We surveyed the literature dealing with the MEWC problem. Our interest inthis problem is motivated by potential applications in network-based analysis ofsystems of materials in materials science. Despite numerous important applications,this problem is not nearly as well studied as its close relatives, the MVWC andMEWC.b/ problems. In particular, only a fraction of the standard benchmarksinstances has been solved to provable optimality. We hope that this review willmotivate future developments that will lead to theoretical characterizations of theMEWC problem, as well as exact and heuristic methods that would advance thecurrent state of the art.

Acknowledgments This work was carried out, while the second author was a visiting scholarat Texas A&M University, College Station, TX, USA, and is partially supported by scholarshipSFRH/BSAB/113662/2015. Partial support by DOD-ONR (N00014-13-1-0635) NSF (CMMI-1538493) grants is also gratefully acknowledged.

References

1. Agapito, L.A., Ferretti, A., Calzolari, A., Curtarolo, S., Buongiorno Nardelli, M.: Effectiveand accurate representation of extended Bloch states on finite Hilbert spaces. Phys. Rev. B 88,165127 (2013)

2. Agapito, L.A., Fornari, M., Ceresoli, D., Ferretti, A., Curtarolo, S., Buongiorno Nardelli, M.:Accurate tight-binding Hamiltonians for two-dimensional and layered materials. Phys. Rev. B93, 125137 (2016)

3. Agapito, L.A., Ismail-Beigi, S., Curtarolo, S., Fornari, M., Buongiorno Nardelli, M.: Accuratetight-binding Hamiltonian matrices from ab initio calculations: minimal basis sets. Phys. Rev.B 93, 035104 (2016)

4. Alidaee, B., Glover, F., Kochenberger, G.A., Rego, C.: A new modeling and solution approachfor the number partitioning problem. Adv. Decis. Sci. 2005(2), 113–121 (2005)

5. Alidaee, B., Glover, F., Kochenberger, G., Wang, H.: Solving the maximum edge weight cliqueproblem via unconstrained quadratic programming. Eur. J. Oper. Res. 181(2), 592–597 (2007)

6. Alidaee, B., Kochenberger, G., Lewis, K., Lewis, M., Wang, H.: A new approach for modelingand solving set packing problems. Eur. J. Oper. Res. 186(2), 504–512 (2008)

7. Balas, E., Christofides, N.: A restricted Lagrangean approach to the traveling salesmanproblem. Math. Program. 21(1), 19–46 (1981)

8. Balas, E., Yu, C.S.: Finding a maximum clique in an arbitrary graph. SIAM J. Comput. 15(4),1054–1068 (1986)

9. Ballard, D.H., Brown, C.M.: Computer Vision, 1982. Prenice-Hall, Englewood Cliffs, NJ(1982)

10. Battiti, R., Protasi, M.: Reactive local search for the maximum clique problem 1. Algorithmica29(4), 610–637 (2001)

11. Butenko, S., Wilhelm, W.E.: Clique-detection models in computational biochemistry andgenomics. Eur. J. Oper. Res. 173(1), 1–17 (2006)

12. Cavique, L.: A scalable algorithm for the market basket analysis. J. Retail. Consum. Serv.14(6), 400–407 (2007)

13. Corman, S.R., Kuhn, T., McPhee, R.D., Dooley, K.J.: Studying complex discursive systems.Hum. Commun. Res. 28(2), 157–206 (2002)


14. Dijkhuizen, G., Faigle, U.: A cutting-plane approach to the edge-weighted maximal cliqueproblem. Eur. J. Oper. Res. 69(1), 121–130 (1993)

15. Faigle, U., Garbe, R., Heerink, K., Spieker, B.: Lp-relaxations for the edge-weighted subcliqueproblem. In: Operations Research ’93, pp. 157–160. Springer, Berlin (1994)

16. Feo, T.A., Resende, M.G.C.: Greedy randomized adaptive search procedures. J. Glob. Optim.6, 109–133 (1995)

17. Fontes, D.B.M.M., Gonçalves, J.F.: Heuristic solutions for general concave minimum costnetwork flow problems. Networks 50(1), 67–76 (2007)

18. Fontes, D.B.M.M., Gonçalves, J.F.: A multi-population hybrid biased random key geneticalgorithm for hop-constrained trees in nonlinear cost flow networks. Optim. Lett. 7(6), 1303–1324 (2013)

19. Fontes, D.B.M.M., Gonçalves, J.F., Fontes, F.F.: A GA approach to the maximum edgeweight clique problem. In: CECNet 2016 – 6th International Conference on Electronics,Communications and Networks, Macau University of Science and Technology (2016)

20. Fontes, D.B.M.M., Gonçalves, J.F., Fontes, F.F.: An evolutionary approach to the maximumedge weight clique problem (2017, submitted)

21. Förster, J., Famili, I., Fu, P., Palsson, B.Ø., Nielsen, J.: Genome-scale reconstruction of thesaccharomyces cerevisiae metabolic network. Genome Res. 13(2), 244–253 (2003)

22. Gendron, B., Hertz, A., St-Louis, P.: A sequential elimination algorithm for computing boundson the clique number of a graph. Discret. Optim. 5(3), 615–628 (2008)

23. Glover, F., Kochenberger, G., Alidaee, B., Amini, M.: Tabu search with critical event memory:an enhanced application for binary quadratic programs. In: Meta-Heuristics, pp. 93–109.Springer, Berlin (1999)

24. Gonçalves, J.F., Resende, M.G.: Biased random-key genetic algorithms for combinatorialoptimization. J. Heuristics 17(5), 487–525 (2011)

25. Gouveia, L., Martins, P.: Solving the maximum edge-weight clique problem in sparse graphswith compact formulations. EURO J. Comput. Optim. 3(1), 1–30 (2015)

26. Gouveia, L., Moura, P.: Enhancing discretized formulations: the knapsack reformulation andthe star reformulation. Top 20(1), 52–74 (2012)

27. Hosseinian, S., Fontes, D.B.M.M., Butenko, S.: A quadratic approach to the maximum edgeweight clique problem. In: XIII Global Optimization Workshop (2016)

28. Hosseinian, S., Fontes, D.B.M.M., Butenko, S.: A nonconvex quadratic optimization approachto the maximum edge weight clique problem (2017, submitted)

29. Hunting, M., Faigle, U., Kern, W.: A Lagrangian relaxation approach to the edge-weightedclique problem. Eur. J. Oper. Res. 131(1), 119–131 (2001)

30. Johnson, E.L., Mehrotra, A., Nemhauser, G.L.: Min-cut clustering. Math. Program. 62(1–3),133–151 (1993)

31. Karp, R.M.: Reducibility Among Combinatorial Problems. Springer, Berlin (1972)32. Kochenberger, G., Glover, F., Alidaee, B., Lewis, K.: Using the unconstrained quadratic

program to model and solve max 2-sat problems. Int. J. Oper. Res. 1(1–2), 89–100 (2005)33. Kochenberger, G.A., Glover, F., Alidaee, B., Rego, C.: An unconstrained quadratic binary

programming approach to the vertex coloring problem. Ann. Oper. Res. 139(1), 229–241(2005)

34. Lei, T.L., Church, R.L.: On the unified dispersion problem: efficient formulations and exactalgorithms. Eur. J. Oper. Res. 241(3), 622–630 (2015)

35. Li, X., Wu, M., Kwoh, C.K., Ng, S.K.: Computational approaches for detecting proteincomplexes from protein interaction networks: a survey. BMC Genomics 11(1), 1 (2010)

36. Lucena, A.: Steiner problem in graphs: Lagrangean relaxation and cutting planes. COAL Bull.21(2), 2–8 (1992)

37. Macambira, E.M.: An application of tabu search heuristic for the maximum edge-weightedsubgraph problem. Ann. Oper. Res. 117, 175–190 (2002)

38. Macambira, E.M., de Souza, C.C.: The edge-weighted clique problem: valid inequalities, facetsand polyhedral computations. Eur. J. Oper. Res. 123, 346–371 (2000)


39. Martí, R., Gallego, M., Duarte, A., Pardo, E.G.: Heuristics and metaheuristics for the maximumdiversity problem. J. Heuristics 19(4), 591–615 (2013)

40. Martins, P.: Extended and discretized formulations for the maximum clique problem. Comput.Oper. Res. 37(7), 1348–1358 (2010)

41. Martins, P.: Cliques with maximum/minimum edge neighborhood and neighborhood density.Comput. Oper. Res. 39(3), 594–608 (2012)

42. Mascia, F., Cilia, E., Brunato, M., Passerini, A.: Predicting structural and functional sites inproteins by searching for maximum-weight cliques. In: AAAI. Citeseer (2010)

43. Mehrotra, A.: Cardinality constrained boolean quadratic polytope. Discret. Appl. Math. 79,137–154 (1997)

44. Mehrotra, A., Trick, M.A.: Cliques and clustering: a combinatorial approach. Oper. Res. Lett.22(1), 1–12 (1998)

45. Padberg, M.: The boolean quadric polytope: some characteristics, facets and relatives. Math.Program. 45(1–3), 139–172 (1989)

46. Park, K., Lee, K., Park, S.: An extended formulation approach to the edge-weighted maximalclique problem. Eur. J. Oper. Res. 95(3), 671–682 (1996)

47. Pisinger, D.: The quadratic knapsack problem – a survey. Discret Appl. Math. 155, 623–648(2007)

48. Prokopyev, O.A., Kong, N., Martinez-Torres, D.L.: The equitable dispersion problem. Eur. J.Oper. Res. 197(1), 59–67 (2009)

49. Pullan, W.: Phased local search for the maximum clique problem. J. Comb. Optim. 12(3),303–323 (2006)

50. Pullan, W.: Approximating the maximum vertex/edge weighted clique using local search. J.Heuristics 14(2), 117–134 (2008)

51. Sørensen, M.M.: New facets and a branch-and-cut algorithm for the weighted clique problem.Working paper 01–2, Department of Management Science and Logistics, The Aarhus Schoolof Business (2001)

52. Sørensen, M.M.: New facets and a branch-and-cut algorithm for the weighted clique problem.Eur. J. Oper. Res. 154, 57–70 (2004)

53. Späth, H.: Heuristically determining cliques of given cardinality and with minimal cost withinweighted complete graphs. Z. Oper. Res. 29(3), 125–131 (1985)

54. Tomita, E., Akutsu, T., Matsunaga, T.: Efficient algorithms for finding maximum and maximalcliques: effective tools for bioinformatics. INTECH Open Access Publisher (2011)

55. Wu, Q., Hao, J.K.: A review on algorithms for maximum clique problems. Eur. J. Oper. Res.242(3), 693–709 (2015)

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