On Minrank and the Lov\'asz Theta Function

TL;DR

This paper explores the relationship between the Lovász theta function and the minrank parameter of graphs, providing explicit constructions that highlight limitations of theta-based algorithms in approximating minrank.

Contribution

It introduces explicit graph constructions with constant theta and high minrank, revealing fundamental limitations of theta-based approximation methods.

Findings

Constructed graphs with constant theta and high minrank

Demonstrated limitations of theta-based algorithms for minrank approximation

Used polynomial spaces and incidence matrices in proofs

Abstract

Two classical upper bounds on the Shannon capacity of graphs are the $\vartheta$-function due to Lov\'asz and the minrank parameter due to Haemers. We provide several explicit constructions of $n$-vertex graphs with a constant $\vartheta$-function and minrank at least $n^\delta$ for a constant $\delta>0$ (over various prime order fields). This implies a limitation on the $\vartheta$-function-based algorithmic approach to approximating the minrank parameter of graphs. The proofs involve linear spaces of multivariate polynomials and the method of higher incidence matrices.