The maximum clique problem

In this paper we present a survey of results concerning algorithms, complexity, and applications of the maximum clique problem. We discuss enumerative and exact algorithms, heuristics, and a variety of other proposed methods. An up to date bibliography on the maximum clique and related problems is also provided.

[1]  C. Chevalley,et al.  The algebraic theory of spinors , 1954 .

[2]  Arun Jagota,et al.  Approximating maximum clique with a Hopfield network , 1995, IEEE Trans. Neural Networks.

[3]  L. Beineke A survey of packings and coverings of graphs , 1969 .

[4]  Eugene L. Lawler,et al.  Generating all Maximal Independent Sets: NP-Hardness and Polynomial-Time Algorithms , 1980, SIAM J. Comput..

[5]  T. Motzkin,et al.  Maxima for Graphs and a New Proof of a Theorem of Turán , 1965, Canadian Journal of Mathematics.

[6]  R. Osteen Clique Detection Algorithms Based on Line Addition and Line Removal , 1974 .

[7]  Henry Meyniel,et al.  A New Property of Critical Imperfect Graphs and some Consequences , 1987, Eur. J. Comb..

[8]  C. Bron,et al.  Algorithm 457: finding all cliques of an undirected graph , 1973 .

[9]  Avi Wigderson,et al.  Improving the performance guarantee for approximate graph coloring , 1983, JACM.

[10]  Frank Harary,et al.  A Procedure for Clique Detection Using the Group Matrix , 1957 .

[11]  James E. Burns The maximum independent set problem for cubic planar graphs , 1989, Networks.

[12]  Anders Krogh,et al.  Introduction to the theory of neural computation , 1994, The advanced book program.

[13]  Panos M. Pardalos,et al.  Active constraints, indefinite quadratic test problems, and complexity , 1991 .

[14]  E. Kay,et al.  Graph Theory. An Algorithmic Approach , 1975 .

[15]  Mouloud Boulala,et al.  Polytope des independants d'un graphe serie-parallele , 1979, Discret. Math..

[16]  Egon Balas,et al.  On graphs with polynomially solvable maximum-weight clique problem , 1989, Networks.

[17]  A. Jagota,et al.  Feasible and infeasible maxima in a quadratic program for maximum clique , 1996 .

[18]  M K Brown,et al.  The Extraction of Curved Surface Features with Generic Range Sensors , 1986 .

[19]  D. Rose Triangulated graphs and the elimination process , 1970 .

[20]  Dana H. Ballard,et al.  Graph Problems and Connectionist Architectures , 1987 .

[21]  Laura A. Sanchis,et al.  Some Experimental and Theoretical Results on Test Case Generators for the Maximum Clique Problem , 1996, INFORMS J. Comput..

[22]  P. Erdös On an extremal problem in graph theory , 1970 .

[23]  E. Loukakis,et al.  Determining the number of internal stability of a graph , 1982 .

[24]  C. De Simone,et al.  On the vertex packing problem , 1993, Graphs Comb..

[25]  Peter L. Hammer,et al.  Discrete Applied Mathematics , 1993 .

[26]  Stefan Felsner,et al.  Trapezoid Graphs and Generalizations, Geometry and Algorithms , 1994, Discret. Appl. Math..

[27]  Herbert S. Wilf,et al.  The number of maximal independent sets in a tree , 1986 .

[28]  G. Tintner,et al.  Economic Applications of the Theory of Graphs. , 1963 .

[29]  S. L. Hakimi,et al.  Upper Bounds on the Order of a Clique of a Graph , 1972 .

[30]  Michael Doob,et al.  Spectra of graphs , 1980 .

[31]  A. Kaufmann,et al.  Methods and models of operations research , 1963 .

[32]  Edmund R. Peay,et al.  Hierarchical Clique Structures , 1974 .

[33]  Leslie E. Trotter,et al.  Vertex packings: Structural properties and algorithms , 1975, Math. Program..

[34]  Giorgio Ausiello,et al.  Theoretical Computer Science Approximate Solution of Np Optimization Problems * , 2022 .

[35]  M. Padberg,et al.  Degree-two Inequalities, Clique Facets, and Biperfect Graphs , 1982 .

[36]  Egon Balas,et al.  Addendum: Minimum Weighted Coloring of Triangulated Graphs, with Application to Maximum Weight Vertex Packing and Clique Finding in Arbitrary Graphs , 1992, SIAM J. Comput..

[37]  O. Perron,et al.  Über lückenlose Ausfüllung desn-dimensionalen Raumes durch kongruente Würfel. II , 1940 .

[38]  Kazuo Nakajima,et al.  An Optimal Algorithm for Finding a Maximum Independent Set of a Circular-Arc Graph , 1988, SIAM J. Comput..

[39]  Chee-Kit Looi,et al.  Neural network methods in combinatorial optimization , 1992, Comput. Oper. Res..

[40]  László Lovász,et al.  Approximating clique is almost NP-complete , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[41]  B. Pittel On the probable behaviour of some algorithms for finding the stability number of a graph , 1982, Mathematical Proceedings of the Cambridge Philosophical Society.

[42]  László Lovász,et al.  Algorithmic theory of numbers, graphs and convexity , 1986, CBMS-NSF regional conference series in applied mathematics.

[43]  Gottfried Tinhofer,et al.  A branch and bound algorithm for the maximum clique problem , 1990, ZOR Methods Model. Oper. Res..

[44]  R. Möhring Algorithmic graph theory and perfect graphs , 1986 .

[45]  Paul Erdös,et al.  Clique numbers of graphs , 1986, Discret. Math..

[46]  Mark K. Goldberg,et al.  Constructing cliques using restricted backtracking , 1993, Cliques, Coloring, and Satisfiability.

[47]  Fanica Gavril,et al.  Algorithms on circular-arc graphs , 1974, Networks.

[48]  Stephen H. Unger,et al.  Minimizing the Number of States in Incompletely Specified Sequential Switching Functions , 1959, IRE Trans. Electron. Comput..

[49]  Frank Harary,et al.  Graph Theory , 2016 .

[50]  John Michael Robson,et al.  Algorithms for Maximum Independent Sets , 1986, J. Algorithms.

[51]  B. Bollobás,et al.  Cliques in random graphs , 1976, Mathematical Proceedings of the Cambridge Philosophical Society.

[52]  Daniel Brélaz,et al.  New methods to color the vertices of a graph , 1979, CACM.

[53]  Emile H. L. Aarts,et al.  Simulated Annealing: Theory and Applications , 1987, Mathematics and Its Applications.

[54]  Pierluigi Crescenzi,et al.  Introduction to the theory of complexity , 1994, Prentice Hall international series in computer science.

[55]  Manfred W. Padberg (1,k)-configurations and facets for packing problems , 1980, Math. Program..

[56]  Michael Luby A Simple Parallel Algorithm for the Maximal Independent Set Problem , 1986, SIAM J. Comput..

[57]  László Lovász,et al.  Interactive proofs and the hardness of approximating cliques , 1996, JACM.

[58]  Norishige Chiba,et al.  An Approximation Algorithm for the Maximum Independent Set Problem on Planar Graphs , 1982, SIAM J. Comput..

[59]  Parthasarathy Guturu,et al.  Clique finding-a genetic approach , 1994, Proceedings of the First IEEE Conference on Evolutionary Computation. IEEE World Congress on Computational Intelligence.

[60]  Reuven Bar-Yehuda,et al.  A Local-Ratio Theorem for Approximating the Weighted Vertex Cover Problem , 1983, WG.

[61]  Laura A. Sanchis,et al.  On the Complexity of Test Case Generation for NP-Hard Problems , 1990, Inf. Process. Lett..

[62]  Egon Balas,et al.  Finding large cliques in arbitrary graphs by bipartite matching , 1993, Cliques, Coloring, and Satisfiability.

[63]  Stephan Olariu,et al.  Weak bipolarizable graphs , 1989, Discret. Math..

[64]  Vera T. Sós,et al.  Extremals of functions on graphs with applications to graphs and hypergraphs , 1982, J. Comb. Theory, Ser. B.

[65]  Herbert S. Wilf,et al.  Spectral bounds for the clique and independence numbers of graphs , 1986, J. Comb. Theory, Ser. B.

[66]  Michael Luby,et al.  A simple parallel algorithm for the maximal independent set problem , 1985, STOC '85.

[67]  Shuji Tsukiyama,et al.  A New Algorithm for Generating All the Maximal Independent Sets , 1977, SIAM J. Comput..

[68]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[69]  George L. Nemhauser,et al.  A polynomial algorithm for maximum weighted vertex packings on graphs without long odd cycles , 1981, Math. Program..

[70]  Bruce E. Sagan,et al.  A Note on Independent Sets in Trees , 1988, SIAM J. Discret. Math..

[71]  O. J. Murphy,et al.  Graph theoretic algorithms for the PLA folding problem , 1989, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[72]  Salah Dowaji,et al.  Concurrent data structures and load balancing strategies for parallel branch-and-bound/A* algorithms , 1994, Parallel Algorithms.

[73]  Alain Billionnet An upper bound on the size of the largest cliques in a graph , 1981, J. Graph Theory.

[74]  GERNOT METZE,et al.  On the Connection Assignment Problem of Diagnosable Systems , 1967, IEEE Trans. Electron. Comput..

[75]  L. Lovász,et al.  Polynomial Algorithms for Perfect Graphs , 1984 .

[76]  Mihalis Yannakakis,et al.  On the approximation of maximum satisfiability , 1992, SODA '92.

[77]  V. Degot,et al.  De l'utilisation de la notion de clique (sous-graphe complet symétrique) en matière de typologie de populations , 1975 .

[78]  Béla Bollobás,et al.  Random Graphs , 1985 .

[79]  Kamil John Counterexample to a conjecture of Grothendieck , 1983 .

[80]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[81]  Yoshiyasu Takefuji,et al.  Neural network parallel computing , 1992, The Kluwer international series in engineering and computer science.

[82]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[83]  Norishige Chiba,et al.  An algorithm for finding a large independent set in planar graphs , 1983, Networks.

[84]  Douglas M. Blough,et al.  Fault detection and diagnosis in multiprocessor systems , 1988 .

[85]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[86]  Mark K. Goldberg,et al.  A new parallel algorithm for the maximal independent set problem , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[87]  R. Tarjan,et al.  A Separator Theorem for Planar Graphs , 1977 .

[88]  David S. Johnson,et al.  Approximation algorithms for combinatorial problems , 1973, STOC.

[89]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[90]  Peter L. Hammer,et al.  Stability in Circular Arc Graphs , 1988, J. Algorithms.

[91]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[92]  Leslie E. Trotter,et al.  On stable set polyhedra for K1, 3-free graphs , 1981, J. Comb. Theory, Ser. B.

[93]  Etsuji Tomita,et al.  A Simple Algorithm for Finding a Maximum Clique and Its Worst-Case Time Complexity , 1990, Systems and Computers in Japan.

[94]  Marek Kubale,et al.  A generalized implicit enumeration algorithm for graph coloring , 1985, CACM.

[95]  M. Padberg On the Complexity of Set Packing Polyhedra , 1977 .

[96]  Egon Balas,et al.  Minimum Weighted Coloring of Triangulated Graphs, with Application to Maximum Weight Vertex Packing and Clique Finding in Arbitrary Graphs , 1991, SIAM J. Comput..

[97]  Joseph C. Culberson,et al.  Camouflaging independent sets in quasi-random graphs , 1993, Cliques, Coloring, and Satisfiability.

[98]  Panos M. Pardalos,et al.  Constrained Global Optimization: Algorithms and Applications , 1987, Lecture Notes in Computer Science.

[99]  M. Padberg Covering, Packing and Knapsack Problems , 1979 .

[100]  D. Kozen A clique problem equivalent to graph isomorphism , 1978, SIGA.

[101]  J. Trotter Solution characteristics and algorithms for the vertex packing problem. , 1973 .

[102]  Bert Gerards,et al.  Matrices with the edmonds—Johnson property , 1986, Comb..

[103]  Kihong Park,et al.  How good are genetic algorithms at finding large cliques: an experimental , 1993 .

[104]  E. A. Akkoyunlu,et al.  The Enumeration of Maximal Cliques of Large Graphs , 1973, SIAM J. Comput..

[105]  Robert C. Bolles,et al.  3DPO: A Three- Dimensional Part Orientation System , 1986, IJCAI.

[106]  Egon Balas,et al.  A node covering algorithm , 1977 .

[107]  Philip N. Klein Efficient Parallel Algorithms for Chordal Graphs , 1996, SIAM J. Comput..

[108]  V. Chvátal,et al.  Topics on perfect graphs , 1984 .

[109]  Michel Gendreau,et al.  Tabu search algorithms for the maximum clique problem , 1993, Cliques, Coloring, and Satisfiability.

[110]  Caterina De Simone,et al.  Stability Number of Bull- and Chair-Free Graphs , 1993, Discret. Appl. Math..

[111]  Derek G. Corneil,et al.  Corrections to Bierstone's Algorithm for Generating Cliques , 1972, J. ACM.

[112]  N. Biggs THE TRAVELING SALESMAN PROBLEM A Guided Tour of Combinatorial Optimization , 1986 .

[113]  Johan Håstad Testing of the long code and hardness for clique , 1996, STOC '96.

[114]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[115]  Charles Fleurent,et al.  Object-oriented implementation of heuristic search methods for Graph Coloring, Maximum Clique, and Satisfiability , 1993, Cliques, Coloring, and Satisfiability.

[116]  Moni Naor,et al.  Fast parallel algorithms for chordal graphs , 1987, STOC '87.

[117]  Mihalis Yannakakis,et al.  On Generating All Maximal Independent Sets , 1988, Inf. Process. Lett..

[118]  Francisco Barahona,et al.  Habitat Dispersion in Forest Planning and the Stable Set Problem , 1992, Oper. Res..

[119]  Jue Xue,et al.  Edge-maximal triangulated subgraphs and heuristics for the maximum clique problem , 1994, Networks.

[120]  David S. Johnson,et al.  The Complexity of Near-Optimal Graph Coloring , 1976, J. ACM.

[121]  Thang Nguyen Bui,et al.  A Hybrid Genetic Algorithm for the Maximum Clique Problem , 1995, ICGA.

[122]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[123]  Claude E. Shannon,et al.  Certain Results in Coding Theory for Noisy Channels , 1957, Inf. Control..

[124]  Robert J. Vanderbei,et al.  An Interior-Point Method for Semidefinite Programming , 1996, SIAM J. Optim..

[125]  T. Grossman Applying The INN Model to the MaxClique Problem , 1993 .

[126]  Michel Gendreau,et al.  Diversification strategies in tabu search algorithms for the maximum clique problem , 1996, Ann. Oper. Res..

[127]  G. Nemhauser,et al.  A Strong Cutting Plane/Branch-and-Bound Algorithm for Node Packing , 1992 .

[128]  Jorge Urrutia,et al.  Finding maximum cliques in circle graphs , 1981, Networks.

[129]  Alan M. Frieze,et al.  On the independence number of random graphs , 1990, Discret. Math..

[130]  George J. Minty,et al.  On maximal independent sets of vertices in claw-free graphs , 1980, J. Comb. Theory B.

[131]  N. Linial Hard enumeration problems in geometry and combinatorics , 1986 .

[132]  Gerard J. Chang,et al.  Algorithmic Aspects of the Generalized Clique-transversal Problem on Chordal Graphs , 1996, Discret. Appl. Math..

[133]  L M Adleman,et al.  Molecular computation of solutions to combinatorial problems. , 1994, Science.

[134]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[135]  Mauricio G. C. Resende,et al.  A Greedy Randomized Adaptive Search Procedure for Maximum Independent Set , 1994, Oper. Res..

[136]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988, Wiley interscience series in discrete mathematics and optimization.

[137]  Ioan Tomescu Problems in combinatorics and graph theory , 1985 .

[138]  Harry G. Barrow,et al.  A Versatile Computer-Controlled Assembly System , 1973, IJCAI.

[139]  P. Hammer,et al.  Vertices Belonging to All or to No Maximum Stable Sets of a Graph , 1982 .

[140]  Ron Y. Pinter,et al.  Optimal Chaining of CMOS Transistors in a Functional Cell , 1987, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[141]  Sudhakar M. Reddy,et al.  Guaranteed convergence in a class of Hopfield networks , 1992, IEEE Trans. Neural Networks.

[142]  Panos M. Pardalos,et al.  Test case generators and computational results for the maximum clique problem , 1993, J. Glob. Optim..

[143]  Pierre Hansen,et al.  Two Algorithms for Maximum Cliques in Dense Graphs , 1992 .

[144]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[145]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[146]  Giuseppe Avondo Bodino,et al.  Economic applications of the theory of graphs , 1962 .

[147]  Martin Vingron,et al.  Multiple Sequence Comparison and Consistency on Multipartite Graphs , 1995 .

[148]  Bernd Radig,et al.  Image sequence analysis using relational structures , 1984, Pattern Recognit..

[149]  Joseph Y.-T. Leung,et al.  Efficient algorithms for interval graphs and circular-arc graphs , 1982, Networks.

[150]  Uriel Feige,et al.  Two prover protocols: low error at affordable rates , 1994, STOC '94.

[151]  Louis J. Billera,et al.  The Combinatorics of Permutation Polytopes , 1994, Formal Power Series and Algebraic Combinatorics.

[152]  Desh Ranjan,et al.  Quantifiers and approximation , 1990, STOC '90.

[153]  Charles H. Hubbell An Input-Output Approach to Clique Identification , 1965 .

[154]  H. S. Lee,et al.  Finding a Maximum Set of Independent Chords in a Circle , 1992, Inf. Process. Lett..

[155]  Michael O. Albertson,et al.  On the independence ratio of a graph , 1978, J. Graph Theory.

[156]  Michel Gendreau,et al.  Solving the maximum clique problem using a tabu search approach , 1993, Ann. Oper. Res..

[157]  Ryan B. Hayward,et al.  Weakly triangulated graphs , 1985, J. Comb. Theory B.

[158]  M. Bellare,et al.  Efficient probabilistic checkable proofs and applications to approximation , 1994, STOC '94.

[159]  Fuyau Lin,et al.  A parallel computation network for the maximum clique problem , 1993, 1993 IEEE International Symposium on Circuits and Systems.

[160]  Manfred W. Padberg,et al.  On the facial structure of set packing polyhedra , 1973, Math. Program..

[161]  William I. Gasarch,et al.  On bounded queries and approximation , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[162]  P. Pardalos,et al.  A global optimization approach for solving the maximum clique problem , 1990 .

[163]  Norishige Chiba,et al.  Arboricity and Subgraph Listing Algorithms , 1985, SIAM J. Comput..

[164]  Immanuel M. Bomze,et al.  A finite algorithm for solving general quadratic problems , 1994, J. Glob. Optim..

[165]  Lin-Yu Tseng,et al.  Sequential and Parallel Algorithms for the Maximum-Weight Independent Set Problem on Permutation Graphs , 1993, Inf. Process. Lett..

[166]  Tal Grossman,et al.  Applying the INN model to the Maximum Clique problem , 1993, Cliques, Coloring, and Satisfiability.

[167]  Rajeev Motwani,et al.  Clique partitions, graph compression and speeding-up algorithms , 1991, STOC '91.

[168]  Kihong Park,et al.  On the effectiveness of genetic search in combinatorial optimization , 1995, SAC '95.

[169]  Robert E. Tarjan,et al.  Applications of a planar separator theorem , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[170]  M. P. Marcus Derivation of Maximal Compatibles Using Boolean Algebra , 1964, IBM J. Res. Dev..

[171]  Richard J. Lipton,et al.  On Proving that a Graph has no Large Clique: A Connection with Ramsey Theory , 1996, Information Processing Letters.

[172]  Ravi B. Boppana,et al.  Approximating maximum independent sets by excluding subgraphs , 1992, BIT Comput. Sci. Sect..

[173]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[174]  Mark K. Goldberg,et al.  Constructing a Maximal Independent Set in Parallel , 1989, SIAM J. Discret. Math..

[175]  Jack Minker,et al.  An Analysis of Some Graph Theoretical Cluster Techniques , 1970, JACM.

[176]  Gilbert Laporte,et al.  An exact quadratic 0-1 algorithm for the stable set problem , 1993, Cliques, Coloring, and Satisfiability.

[177]  Fanica Gavril,et al.  Algorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph , 1972, SIAM J. Comput..

[178]  William R. Pulleyblank Minimum node covers and 2-bicritical graphs , 1979, Math. Program..

[179]  E. Loukakis,et al.  An algorithm for the maximum internally stable set in a weighted graph , 1983 .

[180]  O. Perron,et al.  Über lückenlose Ausfüllung desn-dimensionalen Raumes durch kongruente Würfel , 1940 .

[181]  Roberto Tadei,et al.  A multi-KP modeling for the maximum-clique problem , 1994 .

[182]  Donald E. Knuth The Sandwich Theorem , 1994, Electron. J. Comb..

[183]  Laura A. Sanchis,et al.  Approximately solving Maximum Clique using neural network and related heuristics , 1993, Cliques, Coloring, and Satisfiability.

[184]  Zoltán Füredi,et al.  The number of maximal independent sets in connected graphs , 1987, J. Graph Theory.

[185]  Joel Spencer On cliques in graphs , 1971 .

[186]  Henry Meyniel,et al.  On the perfect graph conjecture , 1976, Discret. Math..

[187]  Robert E. Tarjan,et al.  Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs , 1984, SIAM J. Comput..

[188]  S. F. Kapoor,et al.  The Many Facets of Graph Theory , 1969 .

[189]  Wen-Lian Hsu,et al.  Maximum Weight Clique Algorithms for Circular-Arc Graphs and Circle Graphs , 1985, SIAM J. Comput..

[190]  Viggo Kann,et al.  On the Approximability of the Maximum Common Subgraph Problem , 1992, STACS.

[191]  Paulis Kilkusts Another Algorithm Determining the Independence Number of a Graph , 1986, J. Inf. Process. Cybern..

[192]  Radu Horaud,et al.  Stereo Correspondence Through Feature Grouping and Maximal Cliques , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[193]  J. C. Picard,et al.  On the integer-valued variables in the linear vertex packing problem , 1977, Math. Program..

[194]  Farid Alizadeh,et al.  A sublinear-time randomized parallel algorithm for the maximum clique problem in perfect graphs , 1991, SODA '91.

[195]  H Frank,et al.  Maximum internally stable sets of a graph , 1969 .

[196]  Sheng-Roan Kai,et al.  Design and Implementation of an Interactive Optimization System for Telephone Network Planning , 1992, Oper. Res..

[197]  S. Hyakin,et al.  Neural Networks: A Comprehensive Foundation , 1994 .

[198]  Jue Xue Fast algorithms for vertex packing and related problems , 1991 .

[199]  R. Luce,et al.  Connectivity and generalized cliques in sociometric group structure , 1950, Psychometrika.

[200]  Jerrold R. Griggs,et al.  The number of maximal independent sets in a connected graph , 1988, Discret. Math..

[201]  Leslie E. Trotter,et al.  Properties of vertex packing and independence system polyhedra , 1974, Math. Program..

[202]  Carlo Mannino,et al.  Edge projection and the maximum cardinality stable set problem , 1993, Cliques, Coloring, and Satisfiability.

[203]  Mihalis Yannakakis,et al.  The Clique Problem for Planar Graphs , 1981, Inf. Process. Lett..

[204]  W.A.A. Nuij Solution to Problem E2422 [1973, 691] - Tiling with incomparable rectangles , 1974 .

[205]  V. Chvátal On certain polytopes associated with graphs , 1975 .

[206]  P. Pardalos,et al.  An exact algorithm for the maximum clique problem , 1990 .

[207]  Narendra Karmarkar,et al.  An Interior-Point Approach to NP-Complete Problems , 1990, Conference on Integer Programming and Combinatorial Optimization.

[208]  B. Mohar,et al.  Eigenvalues in Combinatorial Optimization , 1993 .

[209]  Michel Gendreau,et al.  An Efficient Implicit Enumeration Algorithm for the Maximum Clique Problem , 1988 .

[210]  Wen-Lian Hsu,et al.  The coloring and maximum independent set problems on planar perfect graphs , 1988, JACM.

[211]  L. Wolsey,et al.  On the greedy heuristic for continuous covering and packing problems , 1982 .

[212]  Jeffrey C. Lagarias,et al.  Keller’s cube-tiling conjecture is false in high dimensions , 1992 .

[213]  Min-Hong Han,et al.  The use of maximum curvature points for the recognition of partially occluded objects , 1990, Pattern Recognit..

[214]  Oleg Verbitsky,et al.  On the Hardness of Approximating Some Optimization Problems That Are Supposedly Easier Than MAX CLIQUE , 1995, Combinatorics, Probability and Computing.

[215]  Giorgio Ausiello,et al.  Structure Preserving Reductions among Convex Optimization Problems , 1980, J. Comput. Syst. Sci..

[216]  James A. McHugh,et al.  Algorithmic Graph Theory , 1986 .

[217]  Saharon Shelah,et al.  Expected Computation Time for Hamiltonian Path Problem , 1987, SIAM J. Comput..

[218]  Bruce Hedman The maximum number of cliques in dense graphs , 1985, Discret. Math..

[219]  M. Pelillo Relaxation labeling networks for the maximum clique problem , 1996 .

[220]  Fanica Gavril,et al.  Algorithms for a maximum clique and a maximum independent set of a circle graph , 1973, Networks.

[221]  I. Heller On linear systems with integral valued solutions. , 1957 .

[222]  Panos M. Pardalos,et al.  Continuous Approaches to Discrete Optimization Problems , 1996 .

[223]  Thomas Bäck,et al.  An evolutionary heuristic for the maximum independent set problem , 1994, Proceedings of the First IEEE Conference on Evolutionary Computation. IEEE World Congress on Computational Intelligence.

[224]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[225]  Najiba Sbihi,et al.  Algorithme de recherche d'un stable de cardinalite maximum dans un graphe sans etoile , 1980, Discret. Math..

[226]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[227]  Owen J. Murphy,et al.  Computing independent sets in graphs with large girth , 1992, Discret. Appl. Math..

[228]  Chính T. Hoàng,et al.  Optimizing weakly triangulated graphs , 1990, Graphs Comb..

[229]  Jerrold R. Griggs Lower bounds on the independence number in terms of the degrees , 1983, J. Comb. Theory, Ser. B.

[230]  Vasek Chvátal,et al.  A Greedy Heuristic for the Set-Covering Problem , 1979, Math. Oper. Res..

[231]  Robert C. Bolles,et al.  Locating Partially Visible Objects: The Local Feature Focus Method , 1980, AAAI.

[232]  W. Meeusen,et al.  Clique detection in directed graphs: a new algorithm , 1975 .

[233]  Kihong Park,et al.  How good are genetic algorithms at finding large cliques: an experimental , 1993 .

[234]  Martin Vingron,et al.  Multiple Sequence Comparison and n-Dimensional Image Reconstruction , 1993, CPM.

[235]  Hideo Ogawa Labeled point pattern matching by Delaunay triangulation and maximal cliques , 1986, Pattern Recognit..

[236]  P. Pardalos,et al.  Parallel branch and bound algorithms for quadratic zero–one programs on the hypercube architecture , 1990 .

[237]  Harry G. Barrow,et al.  Subgraph Isomorphism, Matching Relational Structures and Maximal Cliques , 1976, Inf. Process. Lett..

[238]  J. Scott Provan,et al.  The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected , 1983, SIAM J. Comput..

[239]  Cornelis Hoede Hard graphs for the maximum clique problem , 1988, Discret. Math..

[240]  D. E. Goldberg,et al.  Genetic Algorithms in Search , 1989 .

[241]  Yoshiyasu Takefuji,et al.  A Neural Network Model for Finding a Near-Maximum Clique , 1992, J. Parallel Distributed Comput..

[242]  Pierluigi Crescenzi,et al.  A Note on the Approximation of the MAX CLIQUE Problem , 1991, Inf. Process. Lett..

[243]  William R. Pulleyblank,et al.  Random near-regular graphs and the node packing problem , 1985 .

[244]  Robert E. Tarjan,et al.  Finding a Maximum Independent Set , 1976, SIAM J. Comput..

[245]  N. Sloane Unsolved Problems in Graph Theory Arising from the Study of Codes , 1989 .

[246]  Richard M. Karp,et al.  A fast parallel algorithm for the maximal independent set problem , 1984, STOC '84.

[247]  Dieter Jungnickel,et al.  Graphs, Networks, and Algorithms , 1980 .

[248]  Richard Chang,et al.  On the query complexity of clique size and maximum satisfiability , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[249]  Azriel Rosenfeld,et al.  Computer Vision , 1988, Adv. Comput..

[250]  F. Escalante Über iterierte Clique-Graphen , 1973 .

[251]  E. Reingold,et al.  Combinatorial Algorithms: Theory and Practice , 1977 .

[252]  Egon Balas,et al.  Polyhedral methods for the maximum clique problem , 1994, Cliques, Coloring, and Satisfiability.

[253]  Mark Jerrum,et al.  Large Cliques Elude the Metropolis Process , 1992, Random Struct. Algorithms.

[254]  Claude E. Shannon,et al.  The zero error capacity of a noisy channel , 1956, IRE Trans. Inf. Theory.

[255]  Soura Dasgupta,et al.  Neural network solutions to a graph theoretic problem , 1990, IEEE International Symposium on Circuits and Systems.

[256]  Yoshiyasu Takefuji,et al.  Parallel algorithms for finding a near-maximum independent set of a circle graph , 1990, IEEE Trans. Neural Networks.

[257]  David J. Houck,et al.  An Algorithm for the Vertex Packing Problem , 1977, Oper. Res..

[258]  Egon Balas,et al.  On the Maximum Weight Clique Problem , 1987, Math. Oper. Res..

[259]  J. Edmonds Covers and packings in a family of sets , 1962 .

[260]  Gang Yu,et al.  Weighted vertex packing problem for specially structured geometric graphs , 1995 .

[261]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[262]  A. Jagota Efficiently approximating max-clique in a Hopfield-style network , 1992, [Proceedings 1992] IJCNN International Joint Conference on Neural Networks.

[263]  T. Fukao,et al.  New method to the solution of maximum clique problem: mean-field approximation algorithm and its experimentation , 1994, Proceedings of IEEE International Conference on Systems, Man and Cybernetics.

[264]  Pavol Hell,et al.  A Linear Algorithm for Maximum Weight Cliques in Proper Circular Arc Graphs , 1996, SIAM J. Discret. Math..

[265]  Alain Hertz,et al.  Tabaris: An exact algorithm based on tabu search for finding a maximum independent set in a graph , 1990, Comput. Oper. Res..

[266]  P. Turán On the theory of graphs , 1954 .

[267]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[268]  Paul Erdös,et al.  Optima of dual integer linear programs , 1988, Comb..

[269]  N.R. Malik,et al.  Graph theory with applications to engineering and computer science , 1975, Proceedings of the IEEE.

[270]  Marek Karpinski,et al.  A Fast Parallel Algorithm for Computing all Maximal Cliques in a Graph and the Related Problems (Extended Abstract) , 1988, SWAT.

[271]  Laura A. Sanchis,et al.  Test Case Construction for the Vertex Cover Problem , 1992, Computational Support for Discrete Mathematics.

[272]  C. S. Edwards,et al.  Lower bounds for the clique and the chromatic numbers of a graph , 1983, Discret. Appl. Math..

[273]  W. Staton Some Ramsey-type numbers and the independence ratio , 1979 .

[274]  Webb Miller Building multiple alignments from pairwise alignments , 1993, Comput. Appl. Biosci..

[275]  John Franco,et al.  An approximation algorithm for the maximum independent set problem in cubic planar graphs , 1986, Networks.

[276]  Johan Håstad,et al.  A Simple Lower Bound for Monotone Clique Using a Communication Game , 1992, Inf. Process. Lett..

[277]  Toshinobu Kashiwabara,et al.  Efficient algorithms for finding maximum cliques of an overlap graph , 1990, Networks.

[278]  Kenji Yoshino,et al.  Covering , 1912, The Indian medical gazette.

[279]  Leslie E. Trotter,et al.  A class of facet producing graphs for vertex packing polyhedra , 1975, Discret. Math..

[280]  R. Tarjan Finding a Maximum Clique. , 1972 .

[281]  Mihalis Yannakakis,et al.  Expressing combinatorial optimization problems by linear programs , 1991, STOC '88.

[282]  Egon Balas,et al.  Finding a Maximum Clique in an Arbitrary Graph , 1986, SIAM J. Comput..

[283]  Sumio Masuda,et al.  Generation of Maximum Independent Sets of a Bipartite Graph and Maximum Cliques of a Circular-Arc Graph , 1992, J. Algorithms.

[284]  Fred Glover,et al.  Tabu Search - Part II , 1989, INFORMS J. Comput..

[285]  P. Hammer,et al.  Pseudo-Boolean functions and stability of graphs , 1984 .

[286]  Panos M. Pardalos,et al.  An algorithm for finding a maximum weighted independent set in an arbitrary graph , 1991, Int. J. Comput. Math..

[287]  S. Szabó A reduction of Keller's conjecture , 1986 .

[288]  Carsten Lund,et al.  Efficient probabilistically checkable proofs and applications to approximations , 1993, STOC.

[289]  G. Dirac A Property of 4-Chromatic Graphs and some Remarks on Critical Graphs , 1952 .

[290]  R. Bolles,et al.  Recognizing and Locating Partially Visible Objects: The Local-Feature-Focus Method , 1982 .

[291]  Panos M. Pardalos,et al.  A continuous based heuristic for the maximum clique problem , 1993, Cliques, Coloring, and Satisfiability.

[292]  Alexander Schrijver,et al.  Relaxations of vertex packing , 1986, J. Comb. Theory B.

[293]  N. Shor Dual quadratic estimates in polynomial and Boolean programming , 1991 .

[294]  Marcus Peinado,et al.  Experiments with polynomial-time CLIQUE approximation algorithms on very large graphs , 1993, Cliques, Coloring, and Satisfiability.

[295]  Egon Balas A fast algorithm for finding an edge-maximal subgraph with a TR-formative coloring , 1986, Discret. Appl. Math..

[296]  G. Dirac On rigid circuit graphs , 1961 .

[297]  M. Burlet,et al.  Polynomial algorithm to recognize a Meyniel graph , 1984 .

[298]  H. Wilf The Eigenvalues of a Graph and Its Chromatic Number , 1967 .

[299]  Yves Crama,et al.  Local Search in Combinatorial Optimization , 2018, Artificial Neural Networks.

[300]  Alexander Schrijver,et al.  Cones of Matrices and Set-Functions and 0-1 Optimization , 1991, SIAM J. Optim..

[301]  Richard C. T. Lee,et al.  Efficient Parallel Algorithms for Finding Maximal Cliques, Clique Trees, and Minimum Coloring on Chordal Graphs , 1988, Inf. Process. Lett..

[302]  Emile H. L. Aarts,et al.  Neural Networks for Combinatorial Optimization , 2009, Encyclopedia of Optimization.

[303]  Richard M. Karp,et al.  A fast parallel algorithm for the maximal independent set problem , 1985, JACM.

[304]  E. Loukakis A new backtracking algorithm for generating the family of maximal independent sets of a graph , 1983 .

[305]  Mihir Bellare,et al.  Improved non-approximability results , 1994, STOC '94.

[306]  Piotr Berman,et al.  On the Complexity of Approximating the Independent Set Problem , 1989, Inf. Comput..

[307]  K. Corrádi,et al.  A combinatorial approach for Keller's conjecture , 1990 .

[308]  Vasek Chvátal,et al.  Determining the Stability Number of a Graph , 1976, SIAM J. Comput..

[309]  J. Moon,et al.  On cliques in graphs , 1965 .

[310]  P. Argos,et al.  Motif recognition and alignment for many sequences by comparison of dot-matrices. , 1991, Journal of molecular biology.

[311]  Raymond E. Bonner,et al.  On Some Clustering Techniques , 1964, IBM J. Res. Dev..

[312]  N. J. A. Sloane,et al.  A new table of constant weight codes , 1990, IEEE Trans. Inf. Theory.

[313]  Patrick Doreman,et al.  A Note on the Detection of Cliques in Valued Graphs , 1969 .