Combinatorial Bounds on Nonnegative Rank and Extended Formulations

Abstract

An extended formulation of a polytope P is a polytope Q which can be projected onto P. Extended formulations of small size (i.e., number of facets) are of interest, as they allow to model corresponding optimization problems as linear programs of small sizes. The main known lower bounds on the minimum sizes of extended formulations for fixed polytope P (Yannakakis 1991) are closely related to the concept of nondeterministic communication complexity. We study the relative power and limitations of the bounds on several examples.

Revision, 25pp

References in corpus (2)

• Some 0/1 polytopes need exponential size extended formulations
• Finding Descriptions of Polytopes via Extended Formulations and Liftings

Cited by in corpus (7)

• Lower bounds on nonnegative rank via nonnegative nuclear norms
• The matching polytope does not admit fully-polynomial size relaxation schemes
• Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix
• Exponential Lower Bounds for Polytopes in Combinatorial Optimization
• Fooling sets and rank
• Fooling-sets and rank in nonzero characteristic (extended abstract)
• An algebraic approach to symmetric extended formulations