Orthonormal representations, vector chromatic number, and extension complexity

TL;DR

This paper introduces a bipartite generalization of nearly orthogonal vectors to establish new bounds on various graph parameters and polytope extension complexities, advancing understanding in combinatorial optimization.

Contribution

It presents a novel bipartite construction that yields strong bounds on the Lovász theta function, vector chromatic number, and extension complexity, with new lower bounds for the latter.

Findings

Derived lower bounds for the vector chromatic number

Established bounds for Lovász theta function and extension complexity

Introduced bipartite generalization of nearly orthogonal vectors

Abstract

We construct a bipartite generalization of Alon and Szegedy's nearly orthogonal vectors, thereby obtaining strong bounds for several extremal problems involving the Lov\'asz theta function, vector chromatic number, minimum semidefinite rank, nonnegative rank, and extension complexity of polytopes. In particular, we derive a couple of general lower bounds for the vector chromatic number which may be of independent interest.