Stable Set Polytopes with Rank $|V(G)|/3$ for the Lov\'{a}sz--Schrijver SDP Operator

TL;DR

This paper characterizes the minimal size of graphs with a given lift-and-project rank for the stable set polytope under the Lovász--Schrijver SDP operator, settling longstanding conjectures.

Contribution

It proves the sharp bound that the smallest graph with LS_+-rank  contains 3 vertices, confirming a 2003 conjecture and addressing a 1994 problem.

Findings

Smallest graph with LS_+-rank  has 3 vertices.

For each , there exists a vertex-transitive graph with at most 4+12 vertices and LS_+-rank .

Abstract

We study the lift-and-project rank of the stable set polytope of graphs with respect to the Lov\'{a}sz--Schrijver SDP operator $\text{LS}_+$ applied to the fractional stable set polytope. In particular, we show that for every positive integer $\ell$, the smallest possible graph with $\text{LS}_+$-rank $\ell$ contains $3\ell$ vertices. This result is sharp and settles a conjecture posed by Lipt\'{a}k and the second author in 2003, as well as answers a generalization of a problem posed by Knuth in 1994. We also show that for every positive integer $\ell$ there exists a vertex-transitive graph on at most $4\ell+12$ vertices with $\text{LS}_+$-rank at least $\ell$.