{"paper":{"title":"Random MAX SAT, Random MAX CUT, and Their Phase Transitions","license":"","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"David Gamarnik, Don Coppersmith, Gregory B. Sorkin, Mohammad Hajiaghayi","submitted_at":"2003-06-02T21:03:14Z","abstract_excerpt":"Given a 2-SAT formula $F$ consisting of $n$ variables and $\\cn$ random clauses, what is the largest number of clauses $\\max F$ satisfiable by a single assignment of the variables? We bound the answer away from the trivial bounds of $(3/4)cn$ and $cn$. We prove that for $c<1$, the expected number of clauses satisfiable is $\\cn-\\Theta(1/n)$; for large $c$, it is $((3/4)c + \\Theta(\\sqrt{c}))n$; for $c = 1+\\eps$, it is at least $(1+\\eps-O(\\eps^3))n$ and at most $(1+\\eps-\\Omega(\\eps^3/\\ln \\eps))n$; and in the ``scaling window'' $c= 1+\\Theta(n^{-1/3})$, it is $cn-\\Theta(1)$. In particular, just as t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0306047","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"}