{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:CS7BDJWIR64UIO7A4UMZGYG3F4","short_pith_number":"pith:CS7BDJWI","schema_version":"1.0","canonical_sha256":"14be11a6c88fb9443be0e5199360db2f19a2729095da14f0d0fd8d0c3486d3c3","source":{"kind":"arxiv","id":"1504.07830","version":3},"attestation_state":"computed","paper":{"title":"On k-Submodular Relaxation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.DS"],"primary_cat":"math.OC","authors_text":"Hiroshi Hirai, Yuni Iwamasa","submitted_at":"2015-04-29T12:27:44Z","abstract_excerpt":"$k$-submodular functions, introduced by Huber and Kolmogorov, are functions defined on $\\{0, 1, 2, \\dots, k\\}^n$ satisfying certain submodular-type inequalities. $k$-submodular functions typically arise as relaxations of NP-hard problems, and the relaxations by $k$-submodular functions play key roles in design of efficient, approximation, or fixed-parameter tractable algorithms. Motivated by this, we consider the following problem: Given a function $f : \\{1, 2, \\dots, k\\}^n \\rightarrow \\mathbb{R} \\cup \\{\\infty\\}$, determine whether $f$ is extended to a $k$-submodular function $g : \\{0, 1, 2, \\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1504.07830","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-04-29T12:27:44Z","cross_cats_sorted":["cs.CC","cs.DS"],"title_canon_sha256":"9234e8bbc966f44d20c3133f21d0bf84fb3bc17a49ca62d9efddb55c68d0e765","abstract_canon_sha256":"a0577a80e6761563ad1eeed92e9f5d6d46b352174b1e2806d346da24c9b005eb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:54.046711Z","signature_b64":"X8PmCEFtmTyw5jUltJc/6ozlx3yWQ0m8uXqogFLT75MKlqglLdRNi3JJUdHWNWkQFftZK8K15qVLtzTTcBbwBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14be11a6c88fb9443be0e5199360db2f19a2729095da14f0d0fd8d0c3486d3c3","last_reissued_at":"2026-05-18T01:04:54.046062Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:54.046062Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On k-Submodular Relaxation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.DS"],"primary_cat":"math.OC","authors_text":"Hiroshi Hirai, Yuni Iwamasa","submitted_at":"2015-04-29T12:27:44Z","abstract_excerpt":"$k$-submodular functions, introduced by Huber and Kolmogorov, are functions defined on $\\{0, 1, 2, \\dots, k\\}^n$ satisfying certain submodular-type inequalities. $k$-submodular functions typically arise as relaxations of NP-hard problems, and the relaxations by $k$-submodular functions play key roles in design of efficient, approximation, or fixed-parameter tractable algorithms. Motivated by this, we consider the following problem: Given a function $f : \\{1, 2, \\dots, k\\}^n \\rightarrow \\mathbb{R} \\cup \\{\\infty\\}$, determine whether $f$ is extended to a $k$-submodular function $g : \\{0, 1, 2, \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.07830","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":""},"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":"1504.07830","created_at":"2026-05-18T01:04:54.046163+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.07830v3","created_at":"2026-05-18T01:04:54.046163+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.07830","created_at":"2026-05-18T01:04:54.046163+00:00"},{"alias_kind":"pith_short_12","alias_value":"CS7BDJWIR64U","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_16","alias_value":"CS7BDJWIR64UIO7A","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_8","alias_value":"CS7BDJWI","created_at":"2026-05-18T12:29:17.054201+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/CS7BDJWIR64UIO7A4UMZGYG3F4","json":"https://pith.science/pith/CS7BDJWIR64UIO7A4UMZGYG3F4.json","graph_json":"https://pith.science/api/pith-number/CS7BDJWIR64UIO7A4UMZGYG3F4/graph.json","events_json":"https://pith.science/api/pith-number/CS7BDJWIR64UIO7A4UMZGYG3F4/events.json","paper":"https://pith.science/paper/CS7BDJWI"},"agent_actions":{"view_html":"https://pith.science/pith/CS7BDJWIR64UIO7A4UMZGYG3F4","download_json":"https://pith.science/pith/CS7BDJWIR64UIO7A4UMZGYG3F4.json","view_paper":"https://pith.science/paper/CS7BDJWI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.07830&json=true","fetch_graph":"https://pith.science/api/pith-number/CS7BDJWIR64UIO7A4UMZGYG3F4/graph.json","fetch_events":"https://pith.science/api/pith-number/CS7BDJWIR64UIO7A4UMZGYG3F4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CS7BDJWIR64UIO7A4UMZGYG3F4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CS7BDJWIR64UIO7A4UMZGYG3F4/action/storage_attestation","attest_author":"https://pith.science/pith/CS7BDJWIR64UIO7A4UMZGYG3F4/action/author_attestation","sign_citation":"https://pith.science/pith/CS7BDJWIR64UIO7A4UMZGYG3F4/action/citation_signature","submit_replication":"https://pith.science/pith/CS7BDJWIR64UIO7A4UMZGYG3F4/action/replication_record"}},"created_at":"2026-05-18T01:04:54.046163+00:00","updated_at":"2026-05-18T01:04:54.046163+00:00"}