The maximum clique problem
TL;DR: A survey of results concerning algorithms, complexity, and applications of the maximum clique problem is presented and enumerative and exact algorithms, heuristics, and a variety of other proposed methods are discussed.
Abstract: The maximum clique problem is a classical problem in combinatorial optimization which finds important applications in different domains. In this paper we try to give a survey of results concerning algorithms, complexity, and applications of this problem, and also provide an updated bibliography. Of course, we build upon precursory works with similar goals [39, 232, 266].
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TL;DR: A comprehensive and structured analysis of various graph embedding techniques proposed in the literature, and the open-source Python library, named GEM (Graph Embedding Methods, available at https://github.com/palash1992/GEM ), which provides all presented algorithms within a unified interface to foster and facilitate research on the topic.
Abstract: Graphs, such as social networks, word co-occurrence networks, and communication networks, occur naturally in various real-world applications. Analyzing them yields insight into the structure of society, language, and different patterns of communication. Many approaches have been proposed to perform the analysis. Recently, methods which use the representation of graph nodes in vector space have gained traction from the research community. In this survey, we provide a comprehensive and structured analysis of various graph embedding techniques proposed in the literature. We first introduce the embedding task and its challenges such as scalability, choice of dimensionality, and features to be preserved, and their possible solutions. We then present three categories of approaches based on factorization methods, random walks, and deep learning, with examples of representative algorithms in each category and analysis of their performance on various tasks. We evaluate these state-of-the-art methods on a few common datasets and compare their performance against one another. Our analysis concludes by suggesting some potential applications and future directions. We finally present the open-source Python library we developed, named GEM (Graph Embedding Methods, available at https://github.com/palash1992/GEM ), which provides all presented algorithms within a unified interface to foster and facilitate research on the topic.
1,553 citations
Cites background from "The maximum clique problem"
...Over the years, many researchers have used aggregation based methods [38], [39], [40] to compress graphs....
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TL;DR: This survey overviews the definitions and methods for graph clustering, that is, finding sets of ''related'' vertices in graphs, and presents global algorithms for producing a clustering for the entire vertex set of an input graph.
1,329 citations
TL;DR: A depth-first search algorithm for generating all maximal cliques of an undirected graph, in which pruning methods are employed as in the Bron-Kerbosch algorithm, which proves that its worst-case time complexity is O(3n/3) for an n-vertex graph.
748 citations
Cites background or methods from "The maximum clique problem"
...[23] was reviewed by Pardalos and Xue [19] and Bomze et al....
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...A number of algorithms have been presented and evaluated experimentally or theoretically for this problem [5–7,9,26,13,19,12]....
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...In particular, Pardalas and Xue [19], and Bomze et al....
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TL;DR: This bibliography provides a classification of a comprehensive list of 1380 references on the theory and application of metaheuristics that have had widespread successes in attacking a variety of difficult combinatorial optimization problems that arise in many practical areas.
Abstract: Metaheuristics are the most exciting development in approximate optimization techniques of the last two decades. They have had widespread successes in attacking a variety of difficult combinatorial optimization problems that arise in many practical areas. This bibliography provides a classification of a comprehensive list of 1380 references on the theory and application of metaheuristics. Metaheuristics include but are not limited to constraint logic programming; greedy random adaptive search procedures; natural evolutionary computation; neural networks; non-monotonic search strategies; space-search methods; simulated annealing; tabu search; threshold algorithms and their hybrids. References are presented in alphabetical order under a number of subheadings.
646 citations
Additional excerpts
...Clique Partitioning: (Areibi and Vannelli, 1993); (Deamorim, Barthelemy and Ribeiro, 1992); (Gendreau, Soriano and Salvail, 1993), (Pardalos and Xue, 1994); (Soriano and Gendreau, 1996)....
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TL;DR: A branch-and-bound algorithm for the maximum clique problem--which is computationally equivalent to the maximum independent (stable) set problem--is presented with the vertex order taken from a coloring of the vertices and with a new pruning strategy.
645 citations
References
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Book•
01 Sep 1988
TL;DR: In this article, the authors present the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields, including computer programming and mathematics.
Abstract: From the Publisher:
This book brings together - in an informal and tutorial fashion - the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields
Major concepts are illustrated with running examples, and major algorithms are illustrated by Pascal computer programs No prior knowledge of GAs or genetics is assumed, and only a minimum of computer programming and mathematics background is required
52,797 citations
01 Jan 1979
42,654 citations
TL;DR: There is a deep and useful connection between statistical mechanics and multivariate or combinatorial optimization (finding the minimum of a given function depending on many parameters), and a detailed analogy with annealing in solids provides a framework for optimization of very large and complex systems.
Abstract: There is a deep and useful connection between statistical mechanics (the behavior of systems with many degrees of freedom in thermal equilibrium at a finite temperature) and multivariate or combinatorial optimization (finding the minimum of a given function depending on many parameters). A detailed analogy with annealing in solids provides a framework for optimization of the properties of very large and complex systems. This connection to statistical mechanics exposes new information and provides an unfamiliar perspective on traditional optimization problems and methods.
41,772 citations
Book•
01 Jan 1979
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
Abstract: This is the second edition of a quarterly column the purpose of which is to provide a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’’ W. H. Freeman & Co., San Francisco, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed. Readers having results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C355, Bell Laboratories, Murray Hill, NJ 07974, including details, or at least sketches, of any new proofs (full papers are preferred). In the case of unpublished results, please state explicitly that you would like the results mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this journal.
40,020 citations
01 Jan 1989
TL;DR: This book brings together the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields.
Abstract: From the Publisher:
This book brings together - in an informal and tutorial fashion - the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields.
Major concepts are illustrated with running examples, and major algorithms are illustrated by Pascal computer programs. No prior knowledge of GAs or genetics is assumed, and only a minimum of computer programming and mathematics background is required.
33,034 citations