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17
Matching Hierarchical Structures Using Association Graphs
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1998
"... this article, please send email to: tpami@computer.org, and reference IEEECS Log Number 108453 ..."
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Cited by 169 (26 self)
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this article, please send email to: tpami@computer.org, and reference IEEECS Log Number 108453
Replicator Equations, Maximal Cliques, and Graph Isomorphism
, 1999
"... We present a new energyminimization framework for the graph isomorphism problem that is based on an equivalent maximum clique formulation. The approach is centered around a fundamental result proved by Motzkin and Straus in the mid1960s, and recently expanded in various ways, which allows us to fo ..."
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Cited by 52 (11 self)
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We present a new energyminimization framework for the graph isomorphism problem that is based on an equivalent maximum clique formulation. The approach is centered around a fundamental result proved by Motzkin and Straus in the mid1960s, and recently expanded in various ways, which allows us to formulate the maximum clique problem in terms of a standard quadratic program. The attractive feature of this formulation is that a clear onetoone correspondence exists between the solutions of the quadratic program and those in the original, combinatorial problem. To solve the program we use the socalled replicator equations—a class of straightforward continuous and discretetime dynamical systems developed in various branches of theoretical biology. We show how, despite their inherent inability to escape from local solutions, they nevertheless provide experimental results that are competitive with those obtained using more elaborate meanfield annealing heuristics.
A New GraphTheoretic Approach to Clustering, with Applications to Computer Vision
, 2004
"... This work applies cluster analysis as a unified approach for a wide range of vision applications, thereby combining the research domain of computer vision and that of machine learning. Cluster analysis is the formal study of algorithms and methods for recovering the inherent structure within a given ..."
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Cited by 44 (4 self)
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This work applies cluster analysis as a unified approach for a wide range of vision applications, thereby combining the research domain of computer vision and that of machine learning. Cluster analysis is the formal study of algorithms and methods for recovering the inherent structure within a given dataset. Many problems of computer vision have precisely this goal, namely to find which visual entities belong to an inherent structure, e.g. in an image or in a database of images. For example, a meaningful structure in the context of image segmentation is a set of pixels which correspond to the same object in a scene. Clustering algorithms can be used to partition the pixels of an image into meaningful parts, which may correspond to different objects. In this work we focus on the problems of image segmentation and image database organization. The visual entities to consider are pixels and images, respectively. Our first contribution in this work is a novel partitional (flat) clustering algorithm. The algorithm uses pairwise representation, where the visual objects (pixels,
Dominant sets and pairwise clustering
 IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI
"... Abstract—We develop a new graphtheoretic approach for pairwise data clustering which is motivated by the analogies between the intuitive concept of a cluster and that of a dominant set of vertices, a notion introduced here which generalizes that of a maximal complete subgraph to edgeweighted graph ..."
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Cited by 30 (5 self)
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Abstract—We develop a new graphtheoretic approach for pairwise data clustering which is motivated by the analogies between the intuitive concept of a cluster and that of a dominant set of vertices, a notion introduced here which generalizes that of a maximal complete subgraph to edgeweighted graphs. We establish a correspondence between dominant sets and the extrema of a quadratic form over the standard simplex, thereby allowing the use of straightforward and easily implementable continuous optimization techniques from evolutionary game theory. Numerical examples on various pointset and image segmentation problems confirm the potential of the proposed approach. Index Terms—Clustering, quadratic optimization, evolutionary game dynamics, image segmentation, perceptual organization. 1
Relaxation Labeling Networks for the Maximum Clique Problem
 J. Artif. Neural Networks
, 1995
"... this paper, it is shown how to take advantage of the properties of these models to approximately solve the maximum clique problem, a wellknown intractable optimization problem which has practical applications in various fields. The approach is based on a result by Motzkin and Straus which naturally ..."
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Cited by 27 (17 self)
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this paper, it is shown how to take advantage of the properties of these models to approximately solve the maximum clique problem, a wellknown intractable optimization problem which has practical applications in various fields. The approach is based on a result by Motzkin and Straus which naturally leads to formulate the problem in a manner that is readily mapped onto a relaxation labeling network. Extensive simulations have demonstrated the validity of the proposed model, both in terms of quality of solutions and speed. Maximum clique problem, relaxation labeling processes, neural networks, optimization. 1 INTRODUCTION
Approximating the Maximum Weight Clique Using Replicator Dynamics
, 2000
"... Given an undirected graph with weights on the vertices, the maximum weight clique problem (MWCP) is to find a subset of mutually adjacent vertices (i.e., a clique) having largest total weight. This is a generalization of the classical problem of finding the maximum cardinality clique of an unweig ..."
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Cited by 25 (9 self)
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Given an undirected graph with weights on the vertices, the maximum weight clique problem (MWCP) is to find a subset of mutually adjacent vertices (i.e., a clique) having largest total weight. This is a generalization of the classical problem of finding the maximum cardinality clique of an unweighted graph, which arises as a special case of the MWCP when all the weights associated to the vertices are equal. The problem is known to be NP hard for arbitrary graphs and, according to recent theoretical results, so is the problem of approximating it within a constant factor. Although there has recently been much interest around neural network algorithms for the unweighted maximum clique problem, no effort has been directed so far towards its weighted counterpart. In this paper, we present a parallel, distributed heuristic for approximating the MWCP based on dynamics principles developed and studied in various branches of mathematical biology. The proposed framework centers aroun...
Annealed Replication: A New Heuristic for the Maximum Clique Problem
 Discr. Appl. Math
, 2000
"... In this paper, a new heuristic for approximating the maximum clique problem is proposed, based on a detailed analysis of a class of continuous optimization models which yield a complete solution to this NPhard combinatorial problem. The idea is to alter a regularization parameter iteratively in suc ..."
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Cited by 20 (11 self)
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In this paper, a new heuristic for approximating the maximum clique problem is proposed, based on a detailed analysis of a class of continuous optimization models which yield a complete solution to this NPhard combinatorial problem. The idea is to alter a regularization parameter iteratively in such a way that an iterative procedure with the updated parameter value would avoid unwanted, inefficient local solutions, i.e., maximal cliques which contain less than the maximum possible number of vertices. The local search procedure is performed with the help of the replicator dynamics, and the regularization parameter is chosen deliberately as to render dynamical instability of the (formerly) stable solutions which we want to discard in order to get an improvement. In this respect, the proposed procedure differs from usual simulated annealing approaches which mostly use a "blackbox" cooling schedule. To demonstrate the validity of this approach, we report on the performance applied to sel...
Maximum stable set formulations and heuristics based on continuous optimization
 MATH. PROGRAM., SER. A 94: 137–166 (2002)
, 2002
"... ..."
A Complementary Pivoting Approach to the Maximum Weight Clique Problem
 SIAM J. Optim
, 2002
"... Abstract. Given an undirected graph with positive weights on the vertices, the maximum weight clique problem (MWCP) is to find a subset of mutually adjacent vertices (i.e., a clique) having largest total weight. The problem is known to be NPhard, even toapproximate. Motivated by a recent quadratic ..."
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Cited by 8 (3 self)
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Abstract. Given an undirected graph with positive weights on the vertices, the maximum weight clique problem (MWCP) is to find a subset of mutually adjacent vertices (i.e., a clique) having largest total weight. The problem is known to be NPhard, even toapproximate. Motivated by a recent quadratic programming formulation, which generalizes an earlier remarkable result of Motzkin and Straus, in this paper we propose a new framework for the MWCP based on the corresponding linear complementarity problem (LCP). We show that, generically, all stationary points of the MWCP quadratic program exhibit strict complementarity. Despite this regularity result, however, the LCP turns out to be inherently degenerate, and we find that Lemke’s wellknown pivoting method, equipped with standard degeneracy resolution strategies, yields unsatisfactory experimental results. We exploit the degeneracy inherent in the problem to develop a variant of Lemke’s algorithm which incorporates a new and effective “lookahead ” pivot rule. The resulting algorithm is tested extensively on various instances of random as well as DIMACS benchmark graphs, and the results obtained show the effectiveness of our method.
Continuousbased Heuristics for Graph and Tree Isomorphisms, with Application to Computer Vision
 in NIPS 99 Workshop on Complexity and Neural Computation
, 1999
"... We present a new (continuous) quadratic programming approach for graph and treeisomorphism problems which is based on an equivalent maximum clique formulation. The approach is centered around a fundamental result proved by Motzkin and Straus in the mid1960s, and recently expanded in various ways, ..."
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Cited by 8 (0 self)
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We present a new (continuous) quadratic programming approach for graph and treeisomorphism problems which is based on an equivalent maximum clique formulation. The approach is centered around a fundamental result proved by Motzkin and Straus in the mid1960s, and recently expanded in various ways, which allows us to formulate the maximum clique problem in terms of a standard quadratic program. The attractive feature of this formulation is that a clear onetoone correspondence exists between the solutions of the quadratic programs and those in the original, combinatorial problems. To approximately solve the program we use the socalled "replicator" equations, a class of straightforward continuous and discretetime dynamical systems developed in various branches of theoretical biology. We show how, despite their inherent inability to escape from local solutions, they nevertheless provide experimental results which are competitive with those obtained using more sophisticated meanfiel...