## The Maximum Clique Problem (1999)

Citations: | 144 - 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 . . . . . . . . . . . . . . . . . . . . . . . .

### Citations

1789 | Random Graphs
- BOLLOBAS
- 2001
(Show Context)
Citation Context ...phs one can exploit the characteristics of the given subset to obtain 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... |

1136 |
Extremal graph theory
- Bollobas
- 1978
(Show Context)
Citation Context ...e maximum clique problem and a certain standard quadratic programming problem providing an alternative proof of a slightly weaker version of the fundamental Tur'an theorem [312] (for a discussion see =-=[54]-=-). Let G = (V; E) be an undirected (unweighted) graph, and let \Delta denote the standard simplex in the n-dimensional Euclidean space IR n , as well as \Delta S the face of \Delta corresponding to a ... |

828 |
Computer Vision
- Ballard, Brown
- 1982
(Show Context)
Citation Context ...orm on such graphs. 7.4 Computer Vision and Pattern Recognition Many fundamental problems in computer vision and pattern recognition can be formulated as the problem of matching relational structures =-=[33]-=-. In the computer vision terminology, a relational structure is a triple S = (U; P; R), where U is a set of units, P = fP 1 ; : : : ; P l g is a set of properties, and R = fR 1 ; : : : ; R k g is a se... |

717 | Proof verification and the hardness of approximation problems
- Arora, Lund, et al.
- 1998
(Show Context)
Citation Context ...thin a factor of 2 log 1\Gamma" n , then any NP problem can be solved in "quasi-polynomial" time (i.e., in 2 log O(1) n time). A breakthrough in approximation complexity is the result b=-=y Arora et al. [13]-=-, [14]. It is shown that the maximum number of satisfiable clauses in a 3-SAT formula (MAX 3-SAT) cannot be approximated to arbitrary small constants (unless P = NP ), thus resolving the open question... |

614 | Molecular computation of solutions to combinatorial problems
- Adleman
- 1994
(Show Context)
Citation Context ...The results obtained over a few DIMACS graphs are comparable with those obtained using continuous-based heuristics but are inferior to those obtained by reactive local search. Recently, DNA computing =-=[5]-=- has also emerged as a potential technique for the maximum clique problem [252], [326]. The major advantage of DNA computing is its high parallelism, but at present the size of graphs this algorithm c... |

375 |
A greedy heuristic for the set-covering problem
- Chvatal
(Show Context)
Citation Context ...is algorithm can handle is limited to a few tens. Additional heuristics for the maximum clique/independent set and related problems on arbitrary or special class of graphs can be found in [89], [91], =-=[94]-=-, [117]. Among others, further (partly randomized) parallel algorithms for the (weighted) maximum clique problem are proposed in [196], [221], [102], [144], [145], [81], [264], [6], and [100]. 7 Selec... |

363 | Probabilistic checking of proofs: a new characterization of NP
- Arora, Safra
- 1998
(Show Context)
Citation Context ...ation complexity and interactive proof systems is discussed). In [7], Alon et al. derandomized Berman and Schnitger's reduction by using the so-called expander graphs (also used by Arora and Safra in =-=[14]), and obt-=-ained the following result: there exists an " ? 0 such that if the maximum clique problem is approximable within O(n " ) then MAX 3-SAT is in PTAS. Panconesi and Ranjan [254, 255] expanded o... |

351 |
New methods to color the vertices of a graph
- Brelaz
(Show Context)
Citation Context ...ranch and bound algorithm for the maximum clique problem. The main ingredient of their algorithm is the use of a fast and relatively good heuristic for the minimum coloring problem proposed by Brelaz =-=[71]-=-. The coloring heuristic is called the degree of saturation largest first (DSATUR). Applying DSATUR to a graph, one can find an upper bound on the size of the maximum clique as well as a maximal cliqu... |

217 |
A fast and simple randomized parallel algorithm for the maximal independent set problem
- Alon, Babai, et al.
- 1986
(Show Context)
Citation Context ... is a bound derived from the probabilistic analysis of this class of graphs [53, 55, 123]. A different implementation of a randomized algorithm for the maximum independent set problem can be found in =-=[9]-=-. 6.3 Advanced Search Heuristics Local search algorithms are only capable of finding local solutions of an optimization problem. In the past few years, many powerful variations of the basic local sear... |

207 |
Algorithm 457: Finding All Cliques of an Undirected Graph
- Bron, Kerbosch
(Show Context)
Citation Context ... [15] had two errors that were corrected by Mulligan and Corneil [240] in 1972. Then in 1973, two new algorithms using the backtracking method were proposed by Akkoyunlu [8], and by Bron and Kerbosch =-=[73]-=-. The advantage of the backtracking method is the elimination of the redundancy in generating the same clique. What was more important for these two algorithms was their polynomial storage requirement... |

206 | Free bits, PCPs and non-approximability — towards tight results
- Bellare, Goldreich, et al.
- 1998
(Show Context)
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... |

182 |
Spectra of Graphs
- Cvetković, Doob, et al.
- 1980
(Show Context)
Citation Context ...e and dominating (see e.g. [177] pp. 507 ff.). Moreover there may be at most one negative eigenvalue \Gamma P with the same absolute value, and this happens if and only if the graph is bipartite (see =-=[101]-=-, Theorems 3.11 and 3.4). Finally, all the components of the only Perron eigenvector x P (such that AGx P =sP x P ) are strictly positive, i.e. x P ? 0. Straightforward calculation of Perron root and ... |

174 | Complexity and Approximation
- Ausiello, Crescenzi, et al.
- 1999
(Show Context)
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 ... |

171 | Efficient probabilistically checkable proofs and applications to approximations
- Bellare, Goldwasser, et al.
- 1993
(Show Context)
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... |

135 | Approximating maximum independent sets by excluding subgraphs
- Boppana, Halldorsson
- 1992
(Show Context)
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... |

133 |
3DPO: A three-dimensional part orientation system
- Bolles, Horaud
- 1987
(Show Context)
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... |

117 | Improved non-approximability results
- Bellare, Sudan
- 1994
(Show Context)
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... |

111 |
Recognizing and locating partially visible objects: The local-feature-focus method
- Bolles, Cain
- 1982
(Show Context)
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... |

103 |
On certain polytopes associated with graphs
- Chvátal
- 1975
(Show Context)
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]... |

94 | Introduction to the Theory of Complexity
- Bovet, Crescenzi
- 1993
(Show Context)
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... |

84 | A new table of constant weight codes
- Brouwer, Shearer, et al.
- 1990
(Show Context)
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... |

82 |
The theory of spinors
- Cartan
- 1966
(Show Context)
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... |

80 | Finding a large hidden clique in a random graph
- Alon, Krivelevich, et al.
- 1998
(Show Context)
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... |

80 |
An exact algorithm for the maximum clique problem
- Carraghan, Pardalos
- 1990
(Show Context)
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... |

75 | Reactive local search for the maximum clique problem, Algorithmica 29 (4
- Battiti, Protasi
- 2001
(Show Context)
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... |

73 | Derandomized Graph Products
- Alon, Feige, et al.
- 1995
(Show Context)
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... |

73 |
On the Complexity of Approximating the Independent Set Problem
- Berman, Schnitger
- 1992
(Show Context)
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... |

71 |
Arboricity and Subgraph Listing Algorithms
- Chiba, Nishizeki
- 1985
(Show Context)
Citation Context ...omputational results suggested their proposed algorithm was as efficient as that in [73] for general graphs, but more efficient on sparse graphs. In 1980's, other proposed algorithms include those in =-=[216, 215, 87, 310, 191]-=-. Loukakis and Tsouros [216] proposed a depth-first enumerative algorithm that generated all maximal independent sets lexicographically. They compared their algorithm with the algorithms of [73] and [... |

59 |
Subgraph isomorphism, matching relational structures, and maximal cliques
- Barrow, Burstall
- 1976
(Show Context)
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... |

57 |
Finding a maximum clique in an arbitrary graph
- Balas, Yu
- 1986
(Show Context)
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... |

51 | On maximum clique problems in very large graphs
- Abello, Pardalos, et al.
- 1999
(Show Context)
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... |

47 |
An analysis of some graph theoretical cluster techniques
- Augustson, Minker
- 1970
(Show Context)
Citation Context ... they are solving the same problem of enumerating all cliques of a graph. With the technology at that time, these early algorithms could only be tested on special graphs. In 1970, Auguston and Minker =-=[15]-=- investigated several graph theoretic clustering techniques used in information systems. In their work, the algorithm of Bierstone and that of Bonner were tested. The method used in both algorithms wa... |

47 |
Cliques in random graphs
- Bollobás, Erdős
- 1976
(Show Context)
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... |

42 |
A versatile computer-controlled assembly
- Ambler, Barrow, et al.
- 1982
(Show Context)
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... |

40 |
Simulated Annealing and Boltzmann
- Aarts, Korst
- 1989
(Show Context)
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 |
Perfectly ordered graphs
- Chvátal
- 1984
(Show Context)
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 ... |

33 | Camouflaging independent sets in quasi-random graphs
- Brockington, Culberson
(Show Context)
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... |

32 |
Evolution towards the maximum clique
- Bomze
- 1997
(Show Context)
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 ... |

28 |
M.: Approximate solutions of NP optimization problems
- Ausiello, Crescenzi, et al.
- 1995
(Show Context)
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 |
Weighted and unweighted maximum clique algorithms with upper bounds from fractional coloring
- Balas, Xue
- 1996
(Show Context)
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 |
Polynomial algorithm to recognize a Meyniel graph
- Burlet, Fonlupt
- 1984
(Show Context)
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... |

24 |
A hybrid genetic algorithm for the maximum clique problem
- Bui, Eppley
- 1995
(Show Context)
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 |
On standard quadratic optimization problems
- Bomze
- 1998
(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)... |

21 |
A fast algorithm for the maximum weight clique problem
- Babel
- 1994
(Show Context)
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 ... |

20 | Annealed replication: a new heuristic for the maximum clique problem
- Bomze, Budinich, et al.
(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, Dev. Global Optimiz
- 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 |
Polyhedral methods for the maximum clique problem
- Balas, Ceria, et al.
(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... |

18 | 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... |

18 |
Minimum weighted coloring of triangulated graphs, with application to maximum weight vertex packing and clique finding in arbitrary graphs
- Balas, Xue
- 1991
(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... |

18 |
On some clustering techniques
- Bonner
- 1964
(Show Context)
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... |