Computing the Grothendieck constant of some graph classes

TL;DR

This paper derives a closed-form formula for the Grothendieck constant of certain graph classes, providing bounds and insights into the integrality gap of semidefinite relaxations for combinatorial optimization problems.

Contribution

It presents a closed-form formula for the Grothendieck constant of $K_5$-minor free graphs and establishes bounds based on the structure of the cut polytope.

Findings

Grothendieck constant of $K_5$-minor free graphs is at most 3/2.

The constant is bounded by $ ext{ka}(K_k)$ if the cut polytope is defined by inequalities supported by at most $k$ points.

The integrality ratio for clique-web inequalities is bounded by 3.

Abstract

Given a graph $G=([n],E)$ and $w\in\R^E$, consider the integer program ${\max}_{x\in \{\pm 1\}^n} \sum_{ij \in E} w_{ij}x_ix_j$ and its canonical semidefinite programming relaxation ${\max} \sum_{ij \in E} w_{ij}v_i^Tv_j$, where the maximum is taken over all unit vectors $v_i\in\R^n$. The integrality gap of this relaxation is known as the Grothendieck constant $\ka(G)$ of $G$. We present a closed-form formula for the Grothendieck constant of $K_5$-minor free graphs and derive that it is at most 3/2. Moreover, we show that $\ka(G)\le \ka(K_k)$ if the cut polytope of $G$ is defined by inequalities supported by at most $k$ points. Lastly, since the Grothendieck constant of $K_n$ grows as $\Theta(\log n)$, it is interesting to identify instances with large gap. However this is not the case for the clique-web inequalities, a wide class of valid inequalities for the cut polytope, whose integrality ratio is shown to be bounded by 3.