Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. In: D.Z. Du, P.M. Pardalos (eds.) Handbook of Combinatorial Optimization, pp. 1--74. Kluwer Academic Publishers, Dordrecht, The Netherlands (1999)  Home/Search   Document Details and Download   Summary   Related Articles   Check

....other techniques have been proposed, including approximation algorithms, heuristics, or branch and bound structured methods. A survey of di#erent formulations, complete methods and heuristics for the maximum clique problem is given by Pardalos and Xue [23] and, more recently, by Bomze et al. [4]. Although semidefinite programs can be solved in polynomial time theoretically, it lasted until a few years ago until fast solvers for this purpose were implemented. Until then, application inside a branch and bound framework was unrealistic. Still, solving a semidefinite program takes ....

I.M. Bomze, M. Budinich, P.M. Pardalos, and M. Pelillo. The Maximum Clique Problem. In D.-Z. Du and P.M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer Academic Publishers, Boston, MA, 1999.

....strictly included in another clique; otherwise it is maximal. The goal of the maximum clique problem is to nd a clique of maximum cardinality. This problem is one of the rst problems shown to be NP complete, and moreover, it does not admit a polynomial time approximation algorithm (unless P=NP) [2]. Hence, complete approaches usually based on a branch and bound tree search become intractable when the number of vertices increases, and much e ort has recently been directed on heuristic incomplete approaches. These approaches leave out exhaustivity and use heuristics to guide the search ....

I. Bomze, M. Budinich, P. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer Academic Publishers, Boston, MA, 1999.

....improves the scalability of a powerful backtracking algorithm on a hard optimisation problem. 4 Application to maximum cliques The Maximum Clique Problem (MCP) was one of the rst problems shown to be NP complete. A recent survey of its applications, algorithms and complexity results is given in [6]. The applications include computer vision, coding theory, tiling, fault diagnosis and the analysis of biological and archaeological data, and it provides a lower bound for the chromatic number of a graph. It was one of the three problems proposed in a DIMACS workshop [22] as a way of comparing ....

I. M. Bomze, M. Budinich, P. M. Pardalos, M. Pelillo. The Maximum Clique Problem. In D.-Z. Du, P. M. Pardalos (eds.), Handbook of Combinatorial Optimization volume 4, Kluwer Academic Publishers, Boston, MA, 1999. 10

....4.3 follows that the maximum of implies the maximum of (C) Thus C is a maximum M clique with (C ) This proves that is surjective. A neural maximum M clique solver Many different neural network approaches and techniques have been proposed to solve the maximum clique problem [2]. We customize the Hopfield Clique (HC) network of [25] to solve the more general maximum M clique problem. The HC network is extremely fast and outperforms greedy heuristics with respect to both speed and solution quality. The efficacy of the HC algorithm is based on a fast annealing schedule and ....

I.M. Bomze, M. Budinich, P.M. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P.M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4, pages 1--74. Kluwer Academic Publishers, Boston, MA, 1999.

....and graph colouring. Many algorithms have been applied to the MCP on a common benchmark set, and its history, applicability and rich set of available results make the MCP ideal for evaluating new approaches. A recent survey of its applications, algorithms and complexity results is given in [7]. The MCP is defined as follows. A graph G (V , E) consists of a set V of vertices and a set E of edges between vertices. Two vertices connected by an edge are said to be adjacent.Aclique is a subset of V whose vertices are pairwise adjacent. A maximum clique is a clique of maximum ....

I.M. Bomze, M. Budinich, P.M. Pardalos and M. Pelillo, The Maximum Clique Problem, in: Handbook of Combinatorial Optimization, Vol. 4, eds. D.-Z. Du and P.M. Pardalos (Kluwer Academic, Boston, MA, 1999).

....of structured objects [14] 15] graph coloring, disposal systems, VLSI circuit design, and biological systems [5] Due to its applicability in various elds many heuristics have been devised to approximately solve the maximum clique problem. For details we refer to a survey by Bomze et al. [2]. Among other heuristics arti cial neural networks have been successfully employed to nd good approximations of a maximum clique in a given graph [2] 16] Following the seminal paper of Hop eld and Tank [7] the general approach to solve combinatorial optimization problems maps the objective ....

....various elds many heuristics have been devised to approximately solve the maximum clique problem. For details we refer to a survey by Bomze et al. 2] Among other heuristics arti cial neural networks have been successfully employed to nd good approximations of a maximum clique in a given graph [2], 16] Following the seminal paper of Hop eld and Tank [7] the general approach to solve combinatorial optimization problems maps the objective function of the optimization problem onto an energy function of a neural network. The constraints of the problem are included in the energy function as ....

[Article contains additional citation context not shown here]

I.M. Bomze, M. Budinich, P.M. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P.M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4, pages 1-74. Kluwer Academic Publishers, Boston, MA, 1999.

....all maximal cliques 5 of a graph is well known to be NP hard, greedy or approximation algorithms can be used to provide a subset of them, assuming that the heuristic will be e cient for the particular structure of the graph. We use a very simple greedy algorithm, similar to the ones featured in [4], to build a set of cliques, trying to nd the largest ones. From each node of the graph, a clique is incrementally built from candidates belonging to its adjacency list, starting with the node which has the largest intersection between its neighbors and the other candidates. This algorithm has ....

I. Bomze, M. Budinich, P. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer Academic Publishers, Boston, MA, 1999.

....a maximum clique of the underlying unweighted graph, but it is certainly maximal. Also this problem has several computationally equivalent variants, one being the weighted node packing problem (WNP) Various issues of the maximum clique problem and the maximum weight clique problem are surveyed in [7, 20]. A new exact algorithm for the maximum weight clique problem is presented in this paper. Previously, good test graphs for maximum weight clique algorithms have been lacking, and in many studies the only test graphs have been various types of random graphs. The DIMACS test graphs are not weighted ....

Bomze, I. M., Budinich, M., Pardalos, P. M., and Pelillo, M. 1999. The maximum clique problem. In Handbook of Combinatorial Optimization, Sup- MAXIMUM-WEIGHT CLIQUE PROBLEM 13 plement Volume A, Du, D.-Z. and Pardalos, P. M., Editors. Kluwer, Dordrecht, 1-74.

....favorably to other heuristics, i.e. ones that are not based on the SDP relaxation. It is known that, besides Max Cut, a number of other combinatorial optimization problems can also be formulated as unconstrained binary quadratic programs in the form of (2. 2) such as the Max Clique problem (see [4], for example) These are potential candidates for which the rank two relaxation approach may also produce high performance heuristic algorithms. Further investigation in this direction will be worthwhile. Acknowledgment. We are grateful to an anonymous referee for valuable comments and ....

I. Bomze and M. Budinich and P. Pardalos and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer Academic Publishers, Boston, MA, 1999.

....to be sucient for generating good approximate solutions to the discrete problem. It is known that, besides Max Cut, a number of other combinatorial optimization problems can also be formulated as unconstrained binary quadratic programs in the form of (2) such as the Max Clique problem (see [2], for example) These are potential candidates for which the rank two relaxation approach may also produce high performance heuristic algorithms. Further investigations in this direction are certainly worthwhile. ....

I. Bomze and M. Budinich and P. Pardalos and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer Academic Publishers, Boston, MA, 1999.

....to be su#cient for generating good approximate solutions to the discrete problem. It is known that, besides Max Cut, a number of other combinatorial optimization problems can also be formulated as unconstrained binary quadratic programs in the form of (2) such as the Max Clique problem (see [2], for example) These are potential candidates for which the rank two relaxation approach may also produce high performance heuristic algorithms. Further investigations in this direction are certainly worthwhile. ....

I. Bomze and M. Budinich and P. Pardalos and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer Academic Publishers, Boston, MA, 1999.

....terminating conditions of Unscramble are met. 4.2.4. Complexity issues and merits of Unscramble The enumeration of maximal cliques in an undirected graph is known to be an NP complete problem; an extended overview of algorithms for maximum and maximal clique finding algorithms is presented in (Bomze et al., 1999). However, for small graphs this need not be a serious problem. Most musical categorisation tasks would involve tens or maybe a few hundreds of musical segments rather than thousands. It is also possible to consider using a semi incremental version of the algorithm whereby objects are clustered in ....

Bomze, I.M., Budinich, M., Pardalos, P.M. and Pelillo, M. (1999) The Maximum Clique Problem. In Handbook of Combinatorial Optimisation, D.-Z. Du and P.M. Pardalos (Eds.), Kluwer Academic Publishers, Dordrecht, The Netherlands.

....to be sufficient for generating good approximate solutions to the discrete problem. It is known that, besides Max Cut, a number of other combinatorial optimization problems can also be formulated as unconstrained binary quadratic programs in the form of (2) such as the Max Clique problem (see [2], for example) These are potential candidates for which the rank two relaxation approach may also produce high performance heuristic algorithms. Further investigations in this direction are certainly worthwhile. ....

I. Bomze and M. Budinich and P. Pardalos and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer Academic Publishers, Boston, MA, 1999.

....performed on a class of benchmark instances from the literature assess the effectiveness of the proposed algorithm. 1 Introduction The Maximum Clique Problem (MCP in the following) is a well known problem in combinatorial optimization which finds important applications in many different domains [2]. Let G = V; E) be an undirected graph, where V = f1; ng is the set of vertices, and E V V is the set of edges. A clique C of G is a subset of V in which every pair of vertices is connected by an edge. A clique is called maximal if no strict superset of C is a clique, A maximum ....

I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo, "The maximum clique problem", in D.-Z. Du and P. M. Pardalos (Eds.), Handbook of Combinatorial Optimization (Supplement Volume A), Kluwer Academic Publishers, Boston, MA, 1999.

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I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, pages 1--74. Kluwer Acad. Publishers, 1999.

No context found.

Bomze IM, Budinich M, Pardalos PM, Pelillo M. The maximum clique problem. In: Du D-Z, Pardalos PM, editors. Handbook of combinatorial optimization. Dordrecht: Kluwer Academic Publishers; 1999. p. 1--74.

....directed towards devising efficient clique finding heuristics, for which no formal guarantee of performance may be provided, but are anyway of interest in practical applications. In the neural network community, there has also been much recent interest around this important problem. We refer to [2] for a recent review concerning algorithms, applications, and complexity issues related to the MCP. In the mid 1960s, Motzkin and Straus [11] established a remarkable connection between the MCP and a quadratic programming problem on the standard simplex. The MotzkinStraus formulation, and ....

I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo, "The maximum clique problem," In Handbook of Combinatorial Optimization (Suppl. Vol. A), D.-Z. Du and P. M. Pardalos, Eds. Boston, MA: Kluwer, 1999, pp. 1--74.

....theory and powerful algorithms have been developed. Note that, although the maximum clique problem is known to be NP hard, powerful heuristics exist which efficiently find good approximate solutions and there exist several classes of graphs for which the problem can be solved in polynomial time [5]. In many computer vision problems, the graphs at hand have a peculiar structure: they are connected and acyclic, i.e. they are free trees (see, e.g. 3] 16] 18] 26] Other application domains where free trees arise quite frequently are pattern recognition [9] and biochemistry [1] Note ....

....[2] A subset of vertices of a graph G is said to be a clique if all its nodes are mutually adjacent. A maximal clique is one which is not contained in any larger clique, while a maximum clique is a clique having the largest cardinality. The maximum clique problem is to find a maximum clique of G [5]. The main result of this section establishes a one to one correspondence between maximal maximum subtree isomorphisms and maximal maximum cliques in the FTAG. To prove it, we first need the following lemma. Lemma 1. Let u 1 ;v 1 ;w 1 ;z 1 V 1 and u 2 ;v 2 ;w 2 ;z 2 V 2 be distinct nodes of ....

I.M. Bomze, M. Budinich, P.M. Pardalos, and M. Pelillo, "The Maximum Clique Problem," Handbook of Combinatorial Optimization (supplement vol. A), D.-Z. Du and P.M. Pardalos, eds., pp. 1-74, Boston: Kluwer, 1999.

....graph G is equal to the minimum cardinality of a vertex cover. Practical applications of these optimization problems are abundant. They appear in information retrieval, signal transmission analysis, classification theory, economics, scheduling, experimental design, and computer vision. See [1, 2, 5, 3, 4, 9, 22, 26] for details. The remainder of this paper is organized as follows. In Section 2 we review some integer programming and continuous formulations of the maximum independent set problem. Two new polynomial formulations are proposed in Section 3. In Section 4, we show how the Motzkin Straus theorem ....

I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, pages 1--74. Kluwer Acad. Publishers, 1999.

....attractive because it casts graph matching as a pure graph theoretic problem, for which a solid theory and powerful algorithms have been developed. Note that, although the maximum clique problem is known to be NP hard, powerful heuristics exist which efficiently find good approximate solutions [4]. Motivated by our recent work on rooted tree matching [15] in this paper we propose a solution to the free tree matching problem by providing a straightforward way of deriving an association graph from two free trees. We prove that in the new formulation there is a one to one correspondence ....

I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization (Suppl. Vol. A), pages 1--74. Kluwer, Boston, MA, 1999.

....DIMACS challenge on cliques, coloring and satisfiability [54] In this article we provide an informal survey of recent heuristics for maximum clique and related problems, and up to date bibliographic pointers to the relevant literature. A more comprehensive review and bibliography can be found in [13]. Sequential greedy heuristics. Many approximation algorithms in the literature for the maximum clique problem are called sequential greedy heuristics. These heuristics generate a maximal clique through the repeated addition of a vertex into a partial clique, or the repeated deletion of a vertex ....

Bomze, I. M., Budinich, M., Pardalos, P. M., and Pelillo, M.: `The maximum clique problem', Handbook of Combinatorial Optimization (Supplement Volume A), in D.-Z. Du and P. M. Pardalos (eds.). Kluwer Academic Publishers, Boston, MA, 1999.

....graph, it consists of finding a subset of pairwise adjacent vertices (i.e. a clique) having largest cardinality. The MCP finds applications in a variety of practical problems in such diverse domains as computer vision, experimental design, information retrieval, fault tolerance, etc. see [19] and references therein) In addition, many important intractable problems turn out to be easily reducible to the MCP, and these include, for example, the Boolean satisfiability problem, the independent set problem, the subgraph isomorphism problem, and the vertex covering problem. Due to the ....

.... along these lines can be found in [15, 16, 34] In the light of these negative results, much effort has recently been directed towards devising efficient heuristics for the MCP, for which no formal guarantee of performance may be provided, but are anyway of interest in practical applications [19]. An important generalization of the MCP which is receiving increasing attention arises when positive weights are associated to the vertices of the graph. In this case, the problem is known as the maximum weight clique problem (MWCP) and consists of finding a clique in the graph which has ....

[Article contains additional citation context not shown here]

I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo, "The maximum clique problem," in Handbook of Combinatorial Optimization (Vol. 4), D.-Z. Du and P. M. Pardalos (Eds.), Kluwer Academic Publishers, Boston, MA, 1999 (to appear).

....onto a flat optimization network. In this paper, we consider isomorphism problems on graphs and trees. We present a new continuous framework for these problems which is based on the idea of reducing it to the maximum clique problem, another well known combinatorial optimization problem [6]. Central to our approach is a powerful result originally proved by Motzkin and Straus [34] and recently extended in various ways [5, 14, 15, 42] which allows us to formulate the maximum clique problem in terms of an indefinite quadratic program. In the proposed formulation an elegant one to one ....

.... of distinct nodes u 0 u 1 : u n such that for all i = 1 : n, u i Gamma1 u i ; in this case, the length of the path 1 It should be pointed out, however, that these are worst case results, and there are certain classes of graphs for which the problem is solvable in polynomial time [17, 6]. is n. If u 0 = u n the path is called a cycle. A graph is said to be connected if any pair of nodes is joined by a path. The distance between two nodes u and v, denoted by d(u; v) is the length of the shortest path joining them (by convention d(u; v) 1, if there is no such path) Given a ....

I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer Academic Publishers, Boston, MA, 1999.

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Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. In: D.Z. Du, P.M. Pardalos (eds.) Handbook of Combinatorial Optimization, pp. 1--74. Kluwer Academic Publishers, Dordrecht, The Netherlands (1999)

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I.M. Bomze, M. Budinich, P.M. Pardalos, and M. Pelillo. The Maximum Clique Problem. In D.-Z. Du and P.M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer, 1999.

No context found.

I.M. Bomze, M. Budinich, P.M. Pardalos, and M. Pelillo. The Maximum Clique Problem. In D.-Z. Du and P.M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer Academic Publishers, Boston, MA, 1999.

No context found.

I. Bomze, M. Budinich, P. Pardalos, and M. Pelillo. The maximum clique problem. In Du and Pardalos [28], pages 1--74.

No context found.

I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, pages 1--74. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.

No context found.

Bomze, I. M., Budinich, M., Pardalos, P. M., Pelillo, M., 1999. The maximum clique problem. In: Du, D.-Z., Pardalos, P. M. (Eds.), Handbook of Combinatorial Optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 1--74.

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Bomze, I. M., Budinich, M., Pardalos, P. M. and Pelillo, M., 1999, The maximum clique problem. In: D.-Z. Du and P. M. Pardalos (Eds) Handbook of Combinatorial Optimization (Dordrecht, The Netherlands: Kluwer Academic), pp. 1--74.

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I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, pages 1--74. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.

No context found.

I.M. Bomze, M. Budinich, P.M. Pardalos, and M. Pelillo. The Maximum Clique Problem. In D.-Z. Du and P.M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer Academic Publishers, Boston, MA, 1999.

No context found.

I.M. Bomze, M. Budinich, P.M. Pardalos, and M. Pelillo. The Maximum Clique Problem. In D.-Z. Du and P.M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer, 1999.

No context found.

I. Bomze, M. Budinich, P. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer Academic Publishers, 1999.

No context found.

I.M. Bomze, M. Budinich, P.M. Pardalos, and M. Pelillo. The maximum clique problem. In D.Z. Du and P.M. Pardalos, editors, Handbook of Combinatorial Optimization. Kluwer Academic Publishers, 1999.

No context found.

Bomze, I. M., Budinich, M., Pardalos, P. M., and Pelillo, M. 1999. The maximum clique problem. Handbook of Combinatorial Optimization (Supplement Volume A). D.-Z. Du and P. M. Pardalos (Eds.). Kluwer Academic Publishers. Boston, MA. pp. 1-74

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I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, volume Suppl. A, pages 1--74. Kluwer Academic Publishers, Boston, MA, USA, 1999.

No context found.

I. Bomze, M. Budinich, P. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer Academic Publishers, Boston, MA, 1999.

No context found.

I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo, "The maximum clique problem", in D.-Z. Du and P. M. Pardalos (Eds.), Handbook of Combinatorial Optimization (Supplement Volume A), Kluwer Academic Publishers, Boston, MA, 1999.

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