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Let $B_F$ denote the bipartite graph with partite sets $V(F)$ and $F$ and with an edge between $v \\in V(F)$ and $c \\in F$ if $v \\in c$ or $\\bar{v} \\in c$. The matching number $\\nu(F)$ of $F$ is the size of a maximum matching in $B_F$. In our main result, we prove that the following parameterization of {\\sc MaxSat} (denoted by $(\\nu(F)+k)$-\\textsc{SAT}) is fixed-parameter tractable: Given a formula $F$, decide whether we can satisfy at least $\\nu(F)+k$ clauses in "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1212.0106","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-12-01T12:53:29Z","cross_cats_sorted":[],"title_canon_sha256":"2484b1c7b11d4f989dc35117c76fa92c410657015bc433f5d6ce3fa77d33e780","abstract_canon_sha256":"4156dff1004ba7afdf4f5d238293f08eb762bf409f57e73be53c11c32f7308be"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:39:27.267017Z","signature_b64":"zbXm14nfgzQAOYBMO51GhDnO8j7Rt1/P3j8RdRmhFvQ45UEqFsBS7KTGa8niYJQbEGiul5tOThz107/6TGnfBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f97a695b75ef60527785d5fde0f1986e458344145075a924b9f96cd2cf5e242e","last_reissued_at":"2026-05-18T03:39:27.266307Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:39:27.266307Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fixed-parameter tractability of satisfying beyond the number of variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"A. Yeo, G. Gutin, M. Jones, R. Crowston, S. Saurabh, V. Raman","submitted_at":"2012-12-01T12:53:29Z","abstract_excerpt":"We consider a CNF formula $F$ as a multiset of clauses: $F=\\{c_1,..., c_m\\}$. The set of variables of $F$ will be denoted by $V(F)$. Let $B_F$ denote the bipartite graph with partite sets $V(F)$ and $F$ and with an edge between $v \\in V(F)$ and $c \\in F$ if $v \\in c$ or $\\bar{v} \\in c$. The matching number $\\nu(F)$ of $F$ is the size of a maximum matching in $B_F$. In our main result, we prove that the following parameterization of {\\sc MaxSat} (denoted by $(\\nu(F)+k)$-\\textsc{SAT}) is fixed-parameter tractable: Given a formula $F$, decide whether we can satisfy at least $\\nu(F)+k$ clauses in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.0106","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1212.0106","created_at":"2026-05-18T03:39:27.266421+00:00"},{"alias_kind":"arxiv_version","alias_value":"1212.0106v1","created_at":"2026-05-18T03:39:27.266421+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.0106","created_at":"2026-05-18T03:39:27.266421+00:00"},{"alias_kind":"pith_short_12","alias_value":"7F5GSW3V55QF","created_at":"2026-05-18T12:26:56.085431+00:00"},{"alias_kind":"pith_short_16","alias_value":"7F5GSW3V55QFE54F","created_at":"2026-05-18T12:26:56.085431+00:00"},{"alias_kind":"pith_short_8","alias_value":"7F5GSW3V","created_at":"2026-05-18T12:26:56.085431+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7F5GSW3V55QFE54F2X66B4MYNZ","json":"https://pith.science/pith/7F5GSW3V55QFE54F2X66B4MYNZ.json","graph_json":"https://pith.science/api/pith-number/7F5GSW3V55QFE54F2X66B4MYNZ/graph.json","events_json":"https://pith.science/api/pith-number/7F5GSW3V55QFE54F2X66B4MYNZ/events.json","paper":"https://pith.science/paper/7F5GSW3V"},"agent_actions":{"view_html":"https://pith.science/pith/7F5GSW3V55QFE54F2X66B4MYNZ","download_json":"https://pith.science/pith/7F5GSW3V55QFE54F2X66B4MYNZ.json","view_paper":"https://pith.science/paper/7F5GSW3V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1212.0106&json=true","fetch_graph":"https://pith.science/api/pith-number/7F5GSW3V55QFE54F2X66B4MYNZ/graph.json","fetch_events":"https://pith.science/api/pith-number/7F5GSW3V55QFE54F2X66B4MYNZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7F5GSW3V55QFE54F2X66B4MYNZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7F5GSW3V55QFE54F2X66B4MYNZ/action/storage_attestation","attest_author":"https://pith.science/pith/7F5GSW3V55QFE54F2X66B4MYNZ/action/author_attestation","sign_citation":"https://pith.science/pith/7F5GSW3V55QFE54F2X66B4MYNZ/action/citation_signature","submit_replication":"https://pith.science/pith/7F5GSW3V55QFE54F2X66B4MYNZ/action/replication_record"}},"created_at":"2026-05-18T03:39:27.266421+00:00","updated_at":"2026-05-18T03:39:27.266421+00:00"}