{"paper":{"title":"Computing the Grothendieck constant of some graph classes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.CO","authors_text":"Antonios Varvitsiotis, Monique Laurent","submitted_at":"2011-06-10T21:34:45Z","abstract_excerpt":"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 inequaliti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.2735","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}