A note on the approximation on the MAX CLIQUE problem
Information Processing Letters
A cluster-based approach for routing in dynamic networks
ACM SIGCOMM Computer Communication Review
Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, Workshop, October 11-13, 1993
Approximation of the Stability Number of a Graph via Copositive Programming
SIAM Journal on Optimization
IEEE Intelligent Systems
Simple and Fast: Improving a Branch-And-Bound Algorithm for Maximum Clique
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
A fast algorithm for the maximum clique problem
Discrete Applied Mathematics - Sixth Twente Workshop on Graphs and Combinatorial Optimization
Free bits, PCPs and non-approximability-towards tight results
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Variable neighborhood search for the maximum clique
Discrete Applied Mathematics - The fourth international colloquium on graphs and optimisation (GO-IV)
Optimal Protein Structure Alignment Using Maximum Cliques
Operations Research
A Convex Quadratic Characterization of the Lovász Theta Number
SIAM Journal on Discrete Mathematics
An Efficient Branch-and-bound Algorithm for Finding a Maximum Clique with Computational Experiments
Journal of Global Optimization
Computing the Stability Number of a Graph Via Linear and Semidefinite Programming
SIAM Journal on Optimization
Simple ingredients leading to very efficient heuristics for the maximum clique problem
Journal of Heuristics
A sequential elimination algorithm for computing bounds on the clique number of a graph
Discrete Optimization
Cliques with maximum/minimum edge neighborhood and neighborhood density
Computers and Operations Research
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The maximum clique (MC) problem has long been concentrating the attention of many researchers within the combinatorial optimization community. Most formulations proposed in the literature for the MC problem were adapted from the maximum independent set (MIS) problem, which is known to be equivalent to the MC. In fact, independent sets can be easily modelled within the natural variables space. In the present paper we propose new formulations for the MC problem, using additional and extra indexed variables, leading to extended and discretized formulations. The number of variables and constraints of the new models depend on the range of variation of an interval containing the clique number (@w) of the graph. Therefore, tight lower and upper bounds for @w may strongly benefit the dimension of the new models. Computational results have been conducted on some DIMACS benchmark and biological instances. These results indicate that, among sparse graphs, the linear programming relaxation of the discretized formulations may produce stronger upper bounds than the known models, being faster to find an optimum clique when embedded into the branch-and-bound. Furthermore, the new models can be used to address other related problems, namely to find a maximum size k-plex, or a maximum size component involving two overlapping cliques.