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A New Trust Region Technique for the Maximum Weight Clique Problem
 Discrete Applied Mathematics
, 2002
"... A new simple generalization of the MotzkinStraus theorem for the maximum weight clique problem is formulated and directly proved. Within this framework a new trust region heuristic is developed. In contrast to usual trust region methods, it regards not only the global optimum of a quadratic objecti ..."
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Cited by 31 (2 self)
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A new simple generalization of the MotzkinStraus theorem for the maximum weight clique problem is formulated and directly proved. Within this framework a new trust region heuristic is developed. In contrast to usual trust region methods, it regards not only the global optimum of a quadratic objective over a sphere, but also a set of other stationary points of the program. We formulate and prove a condition when a MotzkinStraus optimum coincides with such a point. The developed method has complexity O(n ), where n is the number of graph vertices. It was implemented in a publicly available software package QUALEXMS.
Maximum stable set formulations and heuristics based on continuous optimization
 MATH. PROGRAM., SER. A 94: 137–166 (2002)
, 2002
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Finding Maximum Clique in Stochastic Graphs Using Distributed Learning Automata
 International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems
, 2015
"... Because of unpredictable, uncertain and timevarying nature of real networks it seems that stochastic graphs, in which weights associated to the edges are random variables, may be a better candidate as a graph model for real world networks. Once the graph model is chosen to be a stochastic graph, ev ..."
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Cited by 4 (2 self)
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Because of unpredictable, uncertain and timevarying nature of real networks it seems that stochastic graphs, in which weights associated to the edges are random variables, may be a better candidate as a graph model for real world networks. Once the graph model is chosen to be a stochastic graph, every feature of the graph such as path, clique, spanning tree and dominating set, to mention a few, should be treated as a stochastic feature. For example, choosing stochastic graph as the graph model of an online social network and defining community structure in terms of clique, and the associations among the individuals within the community as random variables, the concept of stochastic clique may be used to study community structure properties. In this paper maximum clique in stochastic graph is first defined and then several learning automatabased algorithms are proposed for solving maximum clique problem in stochastic graph where the probability distribution functions of the weights associated with the edges of the graph are unknown. It is shown that by a proper choice of the parameters of the proposed algorithms, one can make the probability of finding maximum clique in stochastic graph as close to unity as possible. Experimental results show that the proposed algorithms significantly reduce the number of samples needed to be taken from the edges of the stochastic graph as compared to the number of samples needed by standard sampling method at a given confidence level.
Graph polynomials from principal pivoting
, 2006
"... www.elsevier.com/locate/disc The recursive computation of the interlace polynomial introduced by Arratia, Bollobás and Sorkin is defined in terms of a new pivoting operation on undirected simple graphs. In this paper, we interpret the new pivoting operation on graphs in terms of standard pivoting (o ..."
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www.elsevier.com/locate/disc The recursive computation of the interlace polynomial introduced by Arratia, Bollobás and Sorkin is defined in terms of a new pivoting operation on undirected simple graphs. In this paper, we interpret the new pivoting operation on graphs in terms of standard pivoting (on matrices). Specifically, we show that, up to swapping vertex labels, Arratia et al.’s pivoting operation on a graph is equivalent to a principal pivot transform on the graph’s adjacency matrix, provided that all computations are performed in the Galois field F2. Principal pivoting on adjacency matrices over F2 has a natural counterpart on isotropic systems. Thus, our view of the interlace polynomial is closely related to the one by Aigner and van der Holst. The observations that adjacency matrices of undirected simple graphs are skewsymmetric in F2 and that principal pivoting preserves skewsymmetry in all fields suggest to extend Arratia et al.’s pivoting operation to fields other than F2. Thus, the interlace polynomial extends to polynomials on gain graphs, namely bidirected edgeweighted graphs whereby reversed edges carry nonzero weights that differ only by their sign. Extending a proof by Aigner and van der Holst, we show that the extended interlace polynomial can be represented in a nonrecursive form analogous to the nonrecursive form of the original interlace polynomial, i.e., the Martin polynomial. For infinite fields it is shown that the extended interlace polynomial does not depend on the (nonzero) gains, as long as they obey a nonsingularity condition. These gain graphs are all supported by a single undirected simple graph. Thus, a new graph polynomial
c ○ World Scientific Publishing Company MATCHING SEGMENTATION HIERARCHIES ∗
"... When matching regions from “similar ” images, one typically has the problem of missing counterparts due to local or even global variations of segmentation fineness. Matching segmentation hierarchies, however, not only increases the chances of finding counterparts, but also allows us to exploit the m ..."
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When matching regions from “similar ” images, one typically has the problem of missing counterparts due to local or even global variations of segmentation fineness. Matching segmentation hierarchies, however, not only increases the chances of finding counterparts, but also allows us to exploit the manifold constraints coming from the topological relations between any two regions in a hierarchy. To define the topological relations we represent a plane image I by a plane attributed graph G and derive a finite topology O from G. In particular, segmenting I corresponds to taking a topological minor of G which, in turn, is equivalent to coarsening O. Moreover, each finite topology involved is a coarsening of the standard topology on IR 2. Then, we construct a weighted association graph GA, the nodes of which represent potential matches and the edges of which indicate topological consistency with respect to O. Specifically, a maximal weight clique of GA corresponds to a topologically consistent mapping with maximal total similarity. To find “heavy ” cliques, we extend a greedy pivotingbased heuristic to the weighted case. Experiments on pairs of stereo images, on a video sequence of a cluttered outdoor scene, and on a sequence of panoramic images demonstrate the effectiveness of our method. Keywords: Graphbased segmentation; structural matching; topological relations between regions.
A New Trust Region Technique for the Maximum Weight Clique Problem
, 2002
"... Abstract A new simple generalization of the MotzkinStraus theorem for the maximum weight clique problem is formulated and directly proved. Within this framework a new trust region heuristic is developed. In contrast to usual trust region methods, it regards not only the global optimum of a quadrati ..."
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Abstract A new simple generalization of the MotzkinStraus theorem for the maximum weight clique problem is formulated and directly proved. Within this framework a new trust region heuristic is developed. In contrast to usual trust region methods, it regards not only the global optimum of a quadratic objective over a sphere, but also a set of other stationary points of the program. We formulate and prove a condition when a MotzkinStraus optimum coincides with such a point. The developed method has complexity O(n 3), where n is the number of graph vertices. It was implemented in a publicly available software package QUALEXMS. Computational experiments evidence that the algorithm is exact on small graphs and exceptionally efficient on DIMACS benchmark graphs and various random maximum weight clique problem instances.
A Hybrid Evolutionary Approach to Maximum Weight Clique Problem
"... Abstract: In this paper we propose a hybrid evolutionary approach combining steadystate genetic algorithm and a greedy heuristic for the maximum weight clique problem. The genetic algorithm generates cliques that are then extended into maximum weight clique by the heuristic. Tests on a variety of b ..."
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Abstract: In this paper we propose a hybrid evolutionary approach combining steadystate genetic algorithm and a greedy heuristic for the maximum weight clique problem. The genetic algorithm generates cliques that are then extended into maximum weight clique by the heuristic. Tests on a variety of benchmark problem instances demonstrate the effectiveness of our approach.
OPTIMIZATION WITH LINEAR COMPLEMENTARITY CONSTRAINTS
, 2014
"... A Mathematical Program with Linear Complementarity Constraints (MPLCC) is an optimization problem where a continuously differentiable function is minimized on a set defined by linear constraints and complementarity conditions on pairs of complementary variables. This problem finds many applications ..."
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A Mathematical Program with Linear Complementarity Constraints (MPLCC) is an optimization problem where a continuously differentiable function is minimized on a set defined by linear constraints and complementarity conditions on pairs of complementary variables. This problem finds many applications in several areas of science, engineering and economics and is also an important tool for the solution of some NPhard structured and nonconvex optimization problems, such as bilevel, bilinear and nonconvex quadratic programs and the eigenvalue complementarity problem. In this paper some of the most relevant applications of the MPLCC and formulations of nonconvex optimization problems as MPLCCs are first presented. Algorithms for computing a feasible solution, a stationary point and a global minimum for the MPLCC are next discussed.The most important nonlinear programming methods, complementarity algorithms, enumerative techniques and 0−1 integer programming approaches for the MPLCC are reviewed. Some comments about the computational performance of these algorithms and a few topics for future research are also included in this survey.