## The Maximum Clique Problem (1999)

Citations: | 160 - 20 self |

### BibTeX

@MISC{Bomze99themaximum,

author = {Immanuel M. Bomze and Marco Budinich and Panos M. Pardalos and Marcello Pelillo},

title = {The Maximum Clique Problem},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computational Complexity 12 4 Bounds and Estimates 15 5 Exact Algorithms 19 5.1 Enumerative Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Exact Algorithms for the Unweighted Case . . . . . . . . . . . . . . 21 5.3 Exact Algorithms for the Weighted Case . . . . . . . . . . . . . . . . 25 6 Heuristics 27 6.1 Sequential Greedy Heuristics . . . . . . . . . . . . . . . . . . . . . . 28 6.2 Local Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.3 Advanced Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.1 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.2 Neural networks . . . . . . . . . . . . . . . . . . . . . . . .

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Citation Context ...ithm, unless P = NP . They also showed that the maximum clique problem does not belong to MAX NP but is in the lowest subclass of MAX \Pi 1 , see also [314]. In 1991, Crescenzi, Fiorini and Silvestri =-=[98]-=- proved that all problems in MAX NP are strongly reducible to the maximum clique problem. As a consequence, if the maximum clique is approximable within a constant factor, then all problems in MAX NP ... |

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Citation Context ...heory. Computer vision and pattern recognition problems for which the maximum clique formulation has proven to be effective include, for example, stereo correspondence [176], object recognition [51], =-=[52]-=-, [156], point pattern matching [248], and motion analysis [283], [280]. The framework has recently been extended to handle the problem of matching hierarchical relational structures, such as trees, a... |

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Citation Context ...hich approximating maximum clique becomes intractable [40, 42, 114, 41, 161]. The best polynomial-time approximation algorithm for the maximum clique problem was developed by Boppana and Halld'orsson =-=[66]-=-, and achieves an approximation ratio of n 1\Gammao(1) . In [162], Hastad shows that this is actually the best we can achieve, by proving that, unless NP = coR, the maximum clique problem cannot be ap... |

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Citation Context ...ield theory. Computer vision and pattern recognition problems for which the maximum clique formulation has proven to be effective include, for example, stereo correspondence [176], object recognition =-=[51]-=-, [52], [156], point pattern matching [248], and motion analysis [283], [280]. The framework has recently been extended to handle the problem of matching hierarchical relational structures, such as tr... |

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Citation Context .... [112]). The work of Arora et al. stimulated much research, and many investigators have progressively refined the exact approximation ratio for which approximating maximum clique becomes intractable =-=[40, 42, 114, 41, 161]-=-. The best polynomial-time approximation algorithm for the maximum clique problem was developed by Boppana and Halld'orsson [66], and achieves an approximation ratio of n 1\Gammao(1) . In [162], Hasta... |

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Citation Context ...164], [165], perfectly orderable graphs [95], and unimodular graphs [166]. A class of graphs that is closely related to the perfect graphs is the t-perfect graphs. This class of graphs was defined in =-=[92]-=-. Polynomial algorithms for the maximum weight independent set problem on t-perfect graphs exist [152]. The class of t-perfect graphs contains bipartite graphs, seriesparallel graphs [107], [92], [67]... |

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Citation Context ...ass of planar counting problems which includes the minimum vertex cover problem see [183]. Interest therefore has soon shifted towards characterizing the approximation properties of this problem (see =-=[256, 70, 16]-=- for an introduction to approximation complexity in optimization problems). Early works in this area go back to mid-1970's when Garey and Johnson [126] proved that if the maximum clique problem admits... |

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Citation Context ...que problems on certain graphs. In Coding Theory, one wishes to find a binary code as large as possible that can correct a certain number of errors for a given size of the binary words (vectors), see =-=[74, 297]-=-. In order to correct errors, the code must consist of binary words among which any two differ in a certain number of positions so that a misspelled word can be detected and corrected. A misspelled wo... |

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Citation Context ...ng formulation of the maximum clique problem. In their work, the merit of two different branching rules, greedy and nongreedy, are tested. In the 1990's, more algorithms were proposed, for example in =-=[262, 122, 80, 21, 19, 322, 99]-=-. Pardalos and Phillips [262] formulate the maximum clique problem as an indefinite quadratic global optimization problem with linear constrains. The algorithm of Friden et al. [122] is a branch and b... |

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Citation Context ...onal TNP. The solution of the MC problem provides the largest subset ¯z j1 � ¯z j2 �����¯z jk of ¯z’s which define a TNP contained in V; [2]. SPINORS, WITT AND FOCK BASIS Cartan was the first to show =-=[3]-=- that the geometry of null vectors can be treated elegantly with spinors; we will follow this road. A spinor Φ is a vector belonging to the spaces S of endomorphism of Cl 2n �EndS and is defined by th... |

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Citation Context ... smallest maximal clique and it shows that its size is almost surely M(n; ffi)=2 in the limit of large n. Another interesting result for these graphs is known as the Jerrum conjecture [188] (see also =-=[10]-=- for recent, related results). It states that in large random graphs of density ffi = 1=2 there is no polynomial time algorithm that, with probability greater than 1=2, can find a clique larger than t... |

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Citation Context ...to be very effective. More recently, Soriano and Gendreau [298] tested their algorithms over the DIMACS benchmark graphs and the results confirmed the early conclusions. Recently, Battiti and Protasi =-=[37]-=- extended the tabu search framework by introducing a reactive local search method. They modified a previously introduced reactive scheme by exploiting the particular neighborhood structure of the maxi... |

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Citation Context ... scheme for MAX 3-SAT, where n is the number of vertices in the graph (see also Feige et al. [112], where a connection between approximation complexity and interactive proof systems is discussed). In =-=[7], Alo-=-n et al. derandomized Berman and Schnitger's reduction by using the so-called expander graphs (also used by Arora and Safra in [14]), and obtained the following result: there exists an " ? 0 such... |

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Citation Context ...all constant factors. The question of whether such approximation schemes can be found for the complete problems in this class was left unresolved. By using randomized reductions, Berman and Schnitger =-=[45, 46] have show-=-n that even an O(n " )-approximation algorithm for maximum clique (for some small enough ") would yield a randomized polynomial-time approximation scheme for MAX 3-SAT, where n is the number... |

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Citation Context ...s the maximum common subgraph problem, which is known to be NP -complete [127]. Obviously, for this reason, the relational structure matching problem too is NP -complete. Ambler et al. [11] (see also =-=[36]-=- and [207]) introduced the notion of association graph as an auxiliary structure for matching relational structures. The association graph of two relational structures S 0 and S 00 is the undirected g... |

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Citation Context ...attributes that represent special characteristics of the application at hand. Understanding the structure of the underlying digraph is essential for storage organization and information retrieval. In =-=[4]-=- experiments with data from telecommunications traffic, the corresponding multigraph has 53,767,087 vertices and over 170 million of edges. A giant connected component with 44,989,297 vertices was com... |

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Citation Context ...nal tests on graphs with up to 100 vertices were reported in [111]. One of the most important contributions in the 1980's on practical algorithms for the maximum clique problem is due to Balas and Yu =-=[31]-=-. In their algorithm, the implicit enumeration was implemented in a new way. The idea of their approach is as follows. First, find a maximal induced triangulated subgraph D of G. Once D is found, find... |

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48 |
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Citation Context ...ain sharper bounds. A much studied case is that of randomly generated graphs. For these graphs there exists a well established theory [53] and research proceeds in several directions; for example see =-=[55]-=-, [123], [194], [230] and [237] for eigenvalues of random graphs. In the specific field of maximum cliques a well known result, due to Matula, accurately predicts the size of the maximum clique when t... |

44 |
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Citation Context ... problem becomes the maximum common subgraph problem, which is known to be NP -complete [127]. Obviously, for this reason, the relational structure matching problem too is NP -complete. Ambler et al. =-=[11]-=- (see also [36] and [207]) introduced the notion of association graph as an auxiliary structure for matching relational structures. The association graph of two relational structures S 0 and S 00 is t... |

41 |
Simulated Annealing and Boltzmann
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Citation Context ...ct solution, faster cooling schedules, producing acceptably good results, are in widespread use. Introductory textbooks describing both theoretical and practical issues of the algorithm are [209] and =-=[1]-=-. Aarts and Korst [1], without presenting any experimental result, suggested the use of simulated annealing for solving the independent set problem, using a penalty function approach. Here, the soluti... |

38 |
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Citation Context ...20], [288], [289], [307]. Examples of more recently found perfect graphs are Meyniel graphs [234], [78], quasi parity graphs [235], weakly triangulated graphs [164], [165], perfectly orderable graphs =-=[95]-=-, and unimodular graphs [166]. A class of graphs that is closely related to the perfect graphs is the t-perfect graphs. This class of graphs was defined in [92]. Polynomial algorithms for the maximum ... |

36 |
Evolution towards the maximum clique
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Citation Context ...a problem since, while providing information about the size of the maximum clique, they do not allow us to easily extract its vertices. The spurious solution problem has recently been solved by Bomze =-=[56]-=-. Motivated by a different characterization of maximal cliques due to Comtet [96], he considers the following regularized version of function g:sg(x) = x T AGx + 1 2 x T x (11) which is obtained from ... |

34 | Camou independent sets in quasi-random graphs
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Citation Context ...lity [190]. These include graphs arising from coding theory [160], artificially generated graphs with known clique size, graphs in which the expected clique number is much smaller than the actual one =-=[72]-=-, etc. These data are available at the following WWW address: http://dimacs.rutgers.edu/Challenges/index.html along with additional useful material. 6.1 Sequential Greedy Heuristics Many approximation... |

30 |
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Citation Context ...the literature [146], among them bipartite graphs, interval graphs, and triangulated graphs [128], [120], [288], [289], [307]. Examples of more recently found perfect graphs are Meyniel graphs [234], =-=[78]-=-, quasi parity graphs [235], weakly triangulated graphs [164], [165], perfectly orderable graphs [95], and unimodular graphs [166]. A class of graphs that is closely related to the perfect graphs is t... |

29 |
and unweighted maximum clique algorithms with upper bounds from fractional coloring. Algorithmica 15
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Citation Context ... efficiency of the branch and bound algorithm are observed because of the use of this new bounding procedure. Details about the method and how to extended it to the weighted case can also be found in =-=[30]-=-. Della Croce and Tadei reformulate in [99] the maximum clique problem into a multivariate binary knapsack problem (cf. also [253]) and combine a greedy algorithm with branch and bound methods. In [69... |

27 |
Approximate Solution of NP Optimization Problems
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Citation Context ...ass of planar counting problems which includes the minimum vertex cover problem see [183]. Interest therefore has soon shifted towards characterizing the approximation properties of this problem (see =-=[256, 70, 16]-=- for an introduction to approximation complexity in optimization problems). Early works in this area go back to mid-1970's when Garey and Johnson [126] proved that if the maximum clique problem admits... |

27 |
On standard quadratic optimization problems
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(Show Context)
Citation Context ...] have recently characterized these spurious solutions and have introduced and studied a regularized version which avoids this kind of problems, exactly as in the unweighted case (for proofs see also =-=[58]-=-): instead of the Motzkin-Straus class M(G;w) here a different class C(G; w) of matrices is considered to be used as input data for problem (14): let C(G) = f(c ij ) i;j2V : c ijsc ii + c jj if (i; j)... |

25 |
A hybrid genetic algorithm for the maximum clique problem
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Citation Context ...er," and a modified mutation operator. However, they presented results over very small (i.e., up to 50 vertices) graphs, thereby making it difficult to properly evaluate the algorithm. Bui and Ep=-=pley [77]-=- obtained encouraging results by using a hybrid strat34 egy which incorporates a local optimization step at each generation of the genetic algorithm, and a vertex-ordering preprocessing phase. They te... |

22 |
A fast algorithm for the maximum weight clique problem
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Citation Context ...oblems. The largest graphs they tried to solve had up to 120 vertices. Similar approaches, partly only for the unweighted case, can be found in [245, 18, 24]. In an algorithm published in 1994, Babel =-=[20]-=- uses a branch and bound approach as follows: upper and lower bounds for the maximum weight of cliques are found by coloring the weighted graph, where the number of colors represents the total sum of ... |

21 |
On some clustering techniques
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Citation Context ...cial case. Their work was stimulated by the matrix manipulation problem of sociometric data to find a complete identification of cliques. Early works following that of Harary and Ross can be found in =-=[223, 269, 65, 227, 38, 109]-=-. What Paull and Unger [269], and Marcus [227] proposed were algorithms to minimize the number of rows in a flow table for a sequential switching function. Bonner addresses in [65] the clustering prob... |

20 |
Polyhedral methods for the maximum clique problem.DIMACS
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- 1996
(Show Context)
Citation Context ... too many iterations in solving the linear relaxation problems. The largest graphs they tried to solve had up to 120 vertices. Similar approaches, partly only for the unweighted case, can be found in =-=[245, 18, 24]-=-. In an algorithm published in 1994, Babel [20] uses a branch and bound approach as follows: upper and lower bounds for the maximum weight of cliques are found by coloring the weighted graph, where th... |

20 | Annealed replication: A new heuristic for the maximum clique problem
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(Show Context)
Citation Context ...so [64]. One may wonder whether a subtle choice of initial conditions and/or a variant of the dynamics may significantly improve the results, but experiments in [63] indicate this is not the case. In =-=[59]-=- the properties of the following function are studiedsg ff (x) = x T AGx + ffx T x and a heuristic is proposed which is based on the modification of the parametersff in the course of the optimization ... |

20 |
Evolutionary approach to the maximum clique problem: Empirical evidence on a larger scale
- Bomze, Pelillo, et al.
- 1997
(Show Context)
Citation Context ... set. A careful choice of k is therefore needed. The resulting algorithm was tested over various DIMACS benchmark graphs [190] and the results obtained confirmed the effectiveness of the approach. In =-=[61]-=-, replicator equations are used in conjunction to the spurious-free formulation given in Theorem 2.6 to find maximal cliques of G. Note that here the nodes comprising the clique are directly given by ... |

19 | An evolutionary heuristic for the maximum independent set problem
- Bäck, Khuri
- 1994
(Show Context)
Citation Context ...roaches, and that they are computationally very expensive. In a later study [268], genetic algorithms were proven to be less effective than simulated annealing. At almost the same time Back and Khuri =-=[22]-=-, working on the maximum independent set problem, arrived at the opposite conclusion. By using a straightforward, general-purpose genetic algorithm called GENEsYs and a suitable fitness function which... |

19 |
Minimum Weighted Coloring of Triangulated Graphs, with Application to Maximum Weight Vertex Packing and Clique Finding in Arbitrary Graphs
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(Show Context)
Citation Context ...e literature until late 1980's and early 1990's. Recently published algorithms we are aware of for the maximum weight clique/independent set problem are due to Pardalos and Desai [261], Balas and Xue =-=[29]-=-, and Nemhauser and Sigismondi [244]. The algorithm proposed by Pardalos and Desai [261] was based on an unconstrained quadratic 0-1 formulation of the maximum weight independent set problem. Their al... |