{"paper":{"title":"A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Aaron Potechin, Ankur Moitra, Boaz Barak, Jonathan Kelner, Pravesh K. Kothari, Samuel B. Hopkins","submitted_at":"2016-04-11T19:49:03Z","abstract_excerpt":"We prove that with high probability over the choice of a random graph $G$ from the Erd\\H{o}s-R\\'enyi distribution $G(n,1/2)$, the $n^{O(d)}$-time degree $d$ Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least $n^{1/2-c(d/\\log n)^{1/2}}$ for some constant $c>0$. This yields a nearly tight $n^{1/2 - o(1)}$ bound on the value of this program for any degree $d = o(\\log n)$. Moreover we introduce a new framework that we call \\emph{pseudo-calibration} to construct Sum of Squares lower bounds. This framework is inspired by taking a computational ana"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.03084","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"}