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A Sum of Squares Characterization of Perfect Graphs

Amir Ali Ahmadi(aaa***at***princeton.edu)
Cemil Dibek(cdibek***at***princeton.edu)

Abstract: We present an algebraic characterization of perfect graphs, i.e., graphs for which the clique number and the chromatic number coincide for every induced subgraph. We show that a graph is perfect if and only if certain nonnegative polynomials associated with the graph are sums of squares. As a byproduct, we obtain several infinite families of nonnegative polynomials that are not sums of squares through graph-theoretic constructions. We also characterize graphs for which the associated polynomials belong to certain structured subsets of sum of squares polynomials. Finally, we reformulate some well-known results from the theory of perfect graphs as statements about sum of squares proofs of nonnegativity of certain polynomials.

Keywords: Nonnegative and sum of squares polynomials, perfect graphs, matrix copositivity, semidefinite programming, convex relaxations for the clique number

Category 1: Combinatorial Optimization

Category 2: Linear, Cone and Semidefinite Programming

Category 3: Convex and Nonsmooth Optimization

Citation:

Download: [PDF]

Entry Submitted: 10/17/2021
Entry Accepted: 10/18/2021
Entry Last Modified: 10/17/2021

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