Continuous characterizations of the maximum clique problem

Continuous characterizations of the maximum clique problem

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Article ID: iaor2004669
Country: United States
Volume: 22
Issue: 3
Start Page Number: 754
End Page Number: 768
Publication Date: Aug 1997
Journal: Mathematics of Operations Research
Authors: , , ,
Keywords: programming: quadratic
Abstract:

Given a graph G whose adjacency matrix is A, the Motzkin–Strauss formulation of the Maximum-Clique Problem is the quadratic program (QP) max (T)(x)(A)(x) | (x) (T) (e) = 1, x ≥ 0. It is well known that the global optimum value of this QP is (1 − 1/ω(G)), where ω(G) is the clique number of G. Here, we characterize the following properties pertaining to the above QP: (1) first order optimality, (2) second order optimality, (3) local optimality, (4) strict local optimality. These characterizations reveal interesting underlying discrete structures, and are polynomial time verifiable. 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.

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