- Computing the Grothendieck constant of some graph classes Monique Laurent (M.Laurentcwi.nl) Antonios Varvitsiotis (A.Varvitsiotiscwi.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 Citation: Download: [PDF]Entry Submitted: 06/10/2011Entry Accepted: 06/12/2011Entry Last Modified: 06/13/2011Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.