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Computing the Grothendieck constant of some graph classes

Monique Laurent (M.Laurent***at***cwi.nl)
Antonios Varvitsiotis (A.Varvitsiotis***at***cwi.nl)

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.

Keywords: Grothendieck constant, elliptope, cut polytope, clique-web inequality

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Combinatorial Optimization


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Entry Submitted: 06/10/2011
Entry Accepted: 06/12/2011
Entry Last Modified: 06/13/2011

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