Combinatorial optimization: algorithms and complexity
Combinatorial optimization: algorithms and complexity
Simulated annealing and Boltzmann machines: a stochastic approach to combinatorial optimization and neural computing
A neural network model for finding a near-maximum clique
Journal of Parallel and Distributed Computing - Special issue on neural computing on massively parallel processing
Polyhedral combinatorics and neural networks
Neural Computation
Relaxation labeling networks for the maximum clique problem
Journal of Artificial Neural Networks - Special issue: neural networks for optimization
Feasible and infeasible maxima in a quadratic program for maximum clique
Journal of Artificial Neural Networks - Special issue: neural networks for optimization
Continuous characterizations of the maximum clique problem
Mathematics of Operations Research
Matching Hierarchical Structures Using Association Graphs
IEEE Transactions on Pattern Analysis and Machine Intelligence
Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, Workshop, October 11-13, 1993
Evolution towards the Maximum Clique
Journal of Global Optimization
Performance of Neural Net Heuristics for Maximum Clique on DiverseHighly Compressible Graphs
Journal of Global Optimization
On Standard Quadratic Optimization Problems
Journal of Global Optimization
Matching Free Trees, Maximal Cliques, and Monotone Game Dynamics
IEEE Transactions on Pattern Analysis and Machine Intelligence
Annealed replication: a new heuristic for the maximum clique problem
Discrete Applied Mathematics
Clique is hard to approximate within n1-
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Replicator Equations, Maximal Cliques, and Graph Isomorphism
Neural Computation
Approximating the maximum weight clique using replicator dynamics
IEEE Transactions on Neural Networks
Approximating maximum clique with a Hopfield network
IEEE Transactions on Neural Networks
Discovering Shape Classes using Tree Edit-Distance and Pairwise Clustering
International Journal of Computer Vision
A game-theoretic approach to partial clique enumeration
Image and Vision Computing
A continuous-based approach for partial clique enumeration
GbRPR'07 Proceedings of the 6th IAPR-TC-15 international conference on Graph-based representations in pattern recognition
Graph-based quadratic optimization: A fast evolutionary approach
Computer Vision and Image Understanding
A Scale Independent Selection Process for 3D Object Recognition in Cluttered Scenes
International Journal of Computer Vision
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Evolutionary game-theoretic models and, in particular, the so-called replicator equations have recently proven to be remarkably effective at approximately solving the maximum clique and related problems. The approach is centered around a classic result from graph theory that formulates the maximum clique problem as a standard (continuous) quadratic program and exploits the dynamical properties of these models, which, under a certain symmetry assumption, possess a Lyapunov function. In this letter, we generalize previous work along these lines in several respects. We introduce a wide family of game-dynamic equations known as payoff-monotonic dynamics, of which replicator dynamics are a special instance, and show that they enjoy precisely the same dynamical properties as standard replicator equations. These properties make any member of this family a potential heuristic for solving standard quadratic programs and, in particular, the maximum clique problem. Extensive simulations, performed on random as well as DIMACS benchmark graphs, show that this class contains dynamics that are considerably faster than and at least as accurate as replicator equations. One problem associated with these models, however, relates to their inability to escape from poor local solutions. To overcome this drawback, we focus on a particular subclass of payoff-monotonic dynamics used to model the evolution of behavior via imitation processes and study the stability of their equilibria when a regularization parameter is allowed to take on negative values. A detailed analysis of these properties suggests a whole class of annealed imitation heuristics for the maximum clique problem, which are based on the idea of varying the parameter during the imitation optimization process in a principled way, so as to avoid unwanted inefficient solutions. Experiments show that the proposed annealing procedure does help to avoid poor local optima by initially driving the dynamics toward promising regions in state space. Furthermore, the models outperform state-of-the-art neural network algorithms for maximum clique, such as mean field annealing, and compare well with powerful continuous-based heuristics.