{"paper":{"title":"On the Lov\\'asz Theta function for Independent Sets in Sparse Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Anupam Gupta, Guru Guruganesh, Nikhil Bansal","submitted_at":"2015-04-18T21:55:07Z","abstract_excerpt":"We consider the maximum independent set problem on graphs with maximum degree~$d$. We show that the integrality gap of the Lov\\'asz $\\vartheta$-function based SDP is $\\widetilde{O}(d/\\log^{3/2} d)$. This improves on the previous best result of $\\widetilde{O}(d/\\log d)$, and almost matches the integrality gap of $\\widetilde{O}(d/\\log^2 d)$ recently shown for stronger SDPs, namely those obtained using poly-$(\\log(d))$ levels of the $SA^+$ semidefinite hierarchy. The improvement comes from an improved Ramsey-theoretic bound on the independence number of $K_r$-free graphs for large values of $r$.\n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04767","kind":"arxiv","version":2},"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"}