{"paper":{"title":"NP-hard sets are not sparse unless P=NP: An exposition of a simple proof of Mahaney's Theorem, with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.RT"],"primary_cat":"cs.CC","authors_text":"Joshua A. Grochow","submitted_at":"2016-10-18T23:01:55Z","abstract_excerpt":"Mahaney's Theorem states that, assuming $\\mathsf{P} \\neq \\mathsf{NP}$, no NP-hard set can have a polynomially bounded number of yes-instances at each input length. We give an exposition of a very simple unpublished proof of Manindra Agrawal whose ideas appear in Agrawal-Arvind (\"Geometric sets of low information content,\" Theoret. Comp. Sci., 1996). This proof is so simple that it can easily be taught to undergraduates or a general graduate CS audience - not just theorists! - in about 10 minutes, which the author has done successfully several times. We also include applications of Mahaney's Th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.05825","kind":"arxiv","version":1},"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"}