{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:IDJDHB5IMPZKKZX54BF7YCHADS","short_pith_number":"pith:IDJDHB5I","canonical_record":{"source":{"id":"1309.5469","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2013-09-21T12:37:35Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0e44446f59b6c522fb9c9a44dc3622e03febb9ccc989f730bf035263590458e9","abstract_canon_sha256":"799b5d502ace6e44247222e898af2504d2b75f6f68ca6ddfc2124e0d6531f270"},"schema_version":"1.0"},"canonical_sha256":"40d23387a863f2a566fde04bfc08e01ca2ea1882ec4d1e5a2570a2d8fe1ba521","source":{"kind":"arxiv","id":"1309.5469","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.5469","created_at":"2026-05-18T03:12:34Z"},{"alias_kind":"arxiv_version","alias_value":"1309.5469v1","created_at":"2026-05-18T03:12:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5469","created_at":"2026-05-18T03:12:34Z"},{"alias_kind":"pith_short_12","alias_value":"IDJDHB5IMPZK","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"IDJDHB5IMPZKKZX5","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"IDJDHB5I","created_at":"2026-05-18T12:27:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:IDJDHB5IMPZKKZX54BF7YCHADS","target":"record","payload":{"canonical_record":{"source":{"id":"1309.5469","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2013-09-21T12:37:35Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0e44446f59b6c522fb9c9a44dc3622e03febb9ccc989f730bf035263590458e9","abstract_canon_sha256":"799b5d502ace6e44247222e898af2504d2b75f6f68ca6ddfc2124e0d6531f270"},"schema_version":"1.0"},"canonical_sha256":"40d23387a863f2a566fde04bfc08e01ca2ea1882ec4d1e5a2570a2d8fe1ba521","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:34.536888Z","signature_b64":"bDBa3QosNV9/J4jVrAQhsHfLg+TLqdgGKHUAV40iLfjDP5P1zXXnx4QiIpQBNRumjN/3xvBl2tpvhQjfwRA1Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"40d23387a863f2a566fde04bfc08e01ca2ea1882ec4d1e5a2570a2d8fe1ba521","last_reissued_at":"2026-05-18T03:12:34.536083Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:34.536083Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.5469","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:12:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CTzy8FhnlArV4rg6eWuYIqzq8hdHwO0MZA+G2hEXBNNJ/4NgRIDULRrr5f83Wxl2Fm89qqj2fPRKCUe/tKNUAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T06:04:46.119876Z"},"content_sha256":"ecd80b11db7604f87f5d0acd7bf963bad0be9dbcd0e8775e22ebcbc5234e51b9","schema_version":"1.0","event_id":"sha256:ecd80b11db7604f87f5d0acd7bf963bad0be9dbcd0e8775e22ebcbc5234e51b9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:IDJDHB5IMPZKKZX54BF7YCHADS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Towards Minimizing k-Submodular Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Anna Huber, Vladimir Kolmogorov","submitted_at":"2013-09-21T12:37:35Z","abstract_excerpt":"In this paper we investigate k-submodular functions. This natural family of discrete functions includes submodular and bisubmodular functions as the special cases k = 1 and k = 2 respectively.\n  In particular we generalize the known Min-Max-Theorem for submodular and bisubmodular functions. This theorem asserts that the minimum of the (bi)submodular function can be found by solving a maximization problem over a (bi)submodular polyhedron. We define and investigate a k-submodular polyhedron and prove a Min-Max-Theorem for k-submodular functions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5469","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:12:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XC/+Kr9Zfk6D8k3mhUXmODSnUSYQiysd2oXMIT3dU1kgtJyFCU6pWwTYp7Uq0/rjnsE1GS3Jm3hUQgN16wkXCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T06:04:46.120644Z"},"content_sha256":"aab9cf94e776ab7a687731c292b7a2b804d35ffd4d33bea47b31d7b6a724aaa7","schema_version":"1.0","event_id":"sha256:aab9cf94e776ab7a687731c292b7a2b804d35ffd4d33bea47b31d7b6a724aaa7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IDJDHB5IMPZKKZX54BF7YCHADS/bundle.json","state_url":"https://pith.science/pith/IDJDHB5IMPZKKZX54BF7YCHADS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IDJDHB5IMPZKKZX54BF7YCHADS/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-04T06:04:46Z","links":{"resolver":"https://pith.science/pith/IDJDHB5IMPZKKZX54BF7YCHADS","bundle":"https://pith.science/pith/IDJDHB5IMPZKKZX54BF7YCHADS/bundle.json","state":"https://pith.science/pith/IDJDHB5IMPZKKZX54BF7YCHADS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IDJDHB5IMPZKKZX54BF7YCHADS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:IDJDHB5IMPZKKZX54BF7YCHADS","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"799b5d502ace6e44247222e898af2504d2b75f6f68ca6ddfc2124e0d6531f270","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2013-09-21T12:37:35Z","title_canon_sha256":"0e44446f59b6c522fb9c9a44dc3622e03febb9ccc989f730bf035263590458e9"},"schema_version":"1.0","source":{"id":"1309.5469","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.5469","created_at":"2026-05-18T03:12:34Z"},{"alias_kind":"arxiv_version","alias_value":"1309.5469v1","created_at":"2026-05-18T03:12:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5469","created_at":"2026-05-18T03:12:34Z"},{"alias_kind":"pith_short_12","alias_value":"IDJDHB5IMPZK","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"IDJDHB5IMPZKKZX5","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"IDJDHB5I","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:aab9cf94e776ab7a687731c292b7a2b804d35ffd4d33bea47b31d7b6a724aaa7","target":"graph","created_at":"2026-05-18T03:12:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper we investigate k-submodular functions. This natural family of discrete functions includes submodular and bisubmodular functions as the special cases k = 1 and k = 2 respectively.\n  In particular we generalize the known Min-Max-Theorem for submodular and bisubmodular functions. This theorem asserts that the minimum of the (bi)submodular function can be found by solving a maximization problem over a (bi)submodular polyhedron. We define and investigate a k-submodular polyhedron and prove a Min-Max-Theorem for k-submodular functions.","authors_text":"Anna Huber, Vladimir Kolmogorov","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2013-09-21T12:37:35Z","title":"Towards Minimizing k-Submodular Functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5469","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ecd80b11db7604f87f5d0acd7bf963bad0be9dbcd0e8775e22ebcbc5234e51b9","target":"record","created_at":"2026-05-18T03:12:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"799b5d502ace6e44247222e898af2504d2b75f6f68ca6ddfc2124e0d6531f270","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2013-09-21T12:37:35Z","title_canon_sha256":"0e44446f59b6c522fb9c9a44dc3622e03febb9ccc989f730bf035263590458e9"},"schema_version":"1.0","source":{"id":"1309.5469","kind":"arxiv","version":1}},"canonical_sha256":"40d23387a863f2a566fde04bfc08e01ca2ea1882ec4d1e5a2570a2d8fe1ba521","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"40d23387a863f2a566fde04bfc08e01ca2ea1882ec4d1e5a2570a2d8fe1ba521","first_computed_at":"2026-05-18T03:12:34.536083Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:12:34.536083Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bDBa3QosNV9/J4jVrAQhsHfLg+TLqdgGKHUAV40iLfjDP5P1zXXnx4QiIpQBNRumjN/3xvBl2tpvhQjfwRA1Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:12:34.536888Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.5469","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ecd80b11db7604f87f5d0acd7bf963bad0be9dbcd0e8775e22ebcbc5234e51b9","sha256:aab9cf94e776ab7a687731c292b7a2b804d35ffd4d33bea47b31d7b6a724aaa7"],"state_sha256":"8d54476c06dbc8b8316ed987998ed5649facd42b3f869c6f654d2ba04b03f642"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"czjlCgUQB84LmqpMHB51yXA0TtBsn9oIlSEk4orY5mV5Zc4jEHDGcaWYKKvxet77N3jNqJwv3zOJOMPYCy+HCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T06:04:46.124151Z","bundle_sha256":"861ed64e4e1a80c6bf9e974c70adfb35a3149596563bc3a42ff5ae82be6cbf1f"}}