Continuous Characterizations of the Maximum Clique
نویسندگان
چکیده
Given a graph G whose adjacency matrix is A, the Motzkin-Strauss formulation of the Maximum-Clique Problem is the quadratic program maxfxTAxjxT e = 1; x 0g. It is well known that the global optimum value of this QP is (1 1=!(G)), where !(G) is the clique number of G. Here, we characterize the following: 1) rst order optimality 2) second order optimality 3) local optimality 4) strict local. These characterizations reveal interesting underlying discrete structures, and are polynomial time veri able. A parametrization of the Motzkin-Strauss QP is then introduced and its properties are investigated. Finally, an extension of the Motzkin-Strauss formulation is provided for the weighted clique number of a graph and this is used to derive a maximin characterization of perfect graphs.
منابع مشابه
Continuous Characterizations of the Maximum Clique Problem 1
Given a graph G whose adjacency matrix is A, the Motzkin-Strauss formulation of the Maximum-Clique Problem is the quadratic program maxfx T Axjx T e = 1; x 0g. It is well known that the global optimum value of this QP is (1 ? 1=!(G)), where !(G) is the clique number of G. Here, we characterize the following: 1) rst order optimality 2) second order optimality 3) local optimality 4) strict local ...
متن کاملApproximating the maximum weight clique using replicator dynamics
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 the 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 ve...
متن کاملPartial characterizations of clique-perfect and coordinated graphs: superclasses of triangle-free graphs
A graph is clique-perfect if the cardinality of a maximum clique-independent set equals the cardinality of a minimum clique-transversal, for all its induced subgraphs. A graph G is coordinated if the chromatic number of the clique graph of H equals the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. Coordinated graphs are a subclass of perfect graphs. The...
متن کاملThe Weighted Exponentiated Family of Distributions: Properties, Applications and Characterizations
In this paper a new method of introducing an additional parameter to a continuous distribution is proposed, which leads to a new class of distributions, called the weighted exponentiated family. A special sub-model is discussed. General expressions for some of the mathematical properties of this class such as the moments, quantile function, generating function and order statistics are derived;...
متن کاملPartial characterizations of clique-perfect graphs II: Diamond-free and Helly circular-arc graphs
A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. A graph G is clique-perfect if the sizes of a minimum clique-transversal and a maximum clique-independent set are equal for every induced subgraph ofG. The list of minimal forbidden induced subgraphs for the class of clique-perf...
متن کامل