{"paper":{"title":"Tight Lower Bounds for Planted Clique in the Degree-4 SOS Program","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"cs.DS","authors_text":"Prasad Raghavendra, Tselil Schramm","submitted_at":"2015-07-18T00:32:28Z","abstract_excerpt":"We give a lower bound of $\\tilde{\\Omega}(\\sqrt{n})$ for the degree-4 Sum-of-Squares SDP relaxation for the planted clique problem. Specifically, we show that on an Erd\\\"os-R\\'enyi graph $G(n,\\tfrac{1}{2})$, with high probability there is a feasible point for the degree-4 SOS relaxation of the clique problem with an objective value of $\\tilde{\\Omega}(\\sqrt{n})$, so that the program cannot distinguish between a random graph and a random graph with a planted clique of size $\\tilde{O}(\\sqrt{n})$. This bound is tight.\n  We build on the works of Deshpande and Montanari and Meka et al., who give lowe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.05136","kind":"arxiv","version":3},"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"}