{"paper":{"title":"The Algorithmic Phase Transition of Random $k$-SAT for Low Degree Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math-ph","math.MP","math.PR","stat.ML"],"primary_cat":"cs.CC","authors_text":"Brice Huang, Guy Bresler","submitted_at":"2021-06-03T21:01:02Z","abstract_excerpt":"Let $\\Phi$ be a uniformly random $k$-SAT formula with $n$ variables and $m$ clauses. We study the algorithmic task of finding a satisfying assignment of $\\Phi$. It is known that satisfying assignments exist with high probability up to clause density $m/n = 2^k \\log 2 - \\frac12 (\\log 2 + 1) + o_k(1)$, while the best polynomial-time algorithm known, the Fix algorithm of Coja-Oghlan, finds a satisfying assignment at the much lower clause density $(1 - o_k(1)) 2^k \\log k / k$. This prompts the question: is it possible to efficiently find a satisfying assignment at higher clause densities?\n  We pro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.02129","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2106.02129/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}