{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2004:72DNILSMDOFY5GBG5E35IQO2YI","short_pith_number":"pith:72DNILSM","canonical_record":{"source":{"id":"math/0408331","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2004-08-24T13:42:31Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"741f081f5b3425863cf0857de8237d4ea009655d110954d512a9e9cb32a79897","abstract_canon_sha256":"f53e41027561bb33d5c9f9a30defb64c45cbdc771245d9e23a09f8678c497ff3"},"schema_version":"1.0"},"canonical_sha256":"fe86d42e4c1b8b8e9826e937d441dac2336f3152618600eff9a23ac33aa0c616","source":{"kind":"arxiv","id":"math/0408331","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0408331","created_at":"2026-07-04T14:39:01Z"},{"alias_kind":"arxiv_version","alias_value":"math/0408331v1","created_at":"2026-07-04T14:39:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0408331","created_at":"2026-07-04T14:39:01Z"},{"alias_kind":"pith_short_12","alias_value":"72DNILSMDOFY","created_at":"2026-07-04T14:39:01Z"},{"alias_kind":"pith_short_16","alias_value":"72DNILSMDOFY5GBG","created_at":"2026-07-04T14:39:01Z"},{"alias_kind":"pith_short_8","alias_value":"72DNILSM","created_at":"2026-07-04T14:39:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2004:72DNILSMDOFY5GBG5E35IQO2YI","target":"record","payload":{"canonical_record":{"source":{"id":"math/0408331","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2004-08-24T13:42:31Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"741f081f5b3425863cf0857de8237d4ea009655d110954d512a9e9cb32a79897","abstract_canon_sha256":"f53e41027561bb33d5c9f9a30defb64c45cbdc771245d9e23a09f8678c497ff3"},"schema_version":"1.0"},"canonical_sha256":"fe86d42e4c1b8b8e9826e937d441dac2336f3152618600eff9a23ac33aa0c616","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:39:01.113048Z","signature_b64":"jLy6dV+btSJEM4VsKOllXpByrGkO5BFPnWix1mVMowM5JT4QJEkO2jjN0hJoY/L45+znic3vHzFo8YcDMMOnAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe86d42e4c1b8b8e9826e937d441dac2336f3152618600eff9a23ac33aa0c616","last_reissued_at":"2026-07-04T14:39:01.112716Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:39:01.112716Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0408331","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":"2026-07-04T14:39:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nUzAzr+a2jrditMqUlcenNWP1F+uupxpSRMecQRqOu65vTOo+Cvt4338vAmxT4/c9ANFgcAiru6EIPKqiW4qAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T08:14:47.206118Z"},"content_sha256":"dc8224d571f563f648fce0c368195cf1049e3121513e701ff827b6c90fd7bb20","schema_version":"1.0","event_id":"sha256:dc8224d571f563f648fce0c368195cf1049e3121513e701ff827b6c90fd7bb20"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2004:72DNILSMDOFY5GBG5E35IQO2YI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Computing Optimal Morse Matchings","license":"","headline":"","cross_cats":["math.OC"],"primary_cat":"math.CO","authors_text":"Marc E. Pfetsch, Michael Joswig","submitted_at":"2004-08-24T13:42:31Z","abstract_excerpt":"Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NP-hard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0408331","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0408331/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T14:39:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vwhlkHuEAdOnr0hA9Suwy0E7kG1qgay8MCOEO4wQGGPZ9eQ4JD0Ux9YJACcsEhPubB916huPdR5JODZWjUikBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T08:14:47.206725Z"},"content_sha256":"7972e73df597ad53623ed4c03d79fe6026ac0c1a8c6820bd2f1ab474dee4856a","schema_version":"1.0","event_id":"sha256:7972e73df597ad53623ed4c03d79fe6026ac0c1a8c6820bd2f1ab474dee4856a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/72DNILSMDOFY5GBG5E35IQO2YI/bundle.json","state_url":"https://pith.science/pith/72DNILSMDOFY5GBG5E35IQO2YI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/72DNILSMDOFY5GBG5E35IQO2YI/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":"2026-07-07T08:14:47Z","links":{"resolver":"https://pith.science/pith/72DNILSMDOFY5GBG5E35IQO2YI","bundle":"https://pith.science/pith/72DNILSMDOFY5GBG5E35IQO2YI/bundle.json","state":"https://pith.science/pith/72DNILSMDOFY5GBG5E35IQO2YI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/72DNILSMDOFY5GBG5E35IQO2YI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:72DNILSMDOFY5GBG5E35IQO2YI","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":"f53e41027561bb33d5c9f9a30defb64c45cbdc771245d9e23a09f8678c497ff3","cross_cats_sorted":["math.OC"],"license":"","primary_cat":"math.CO","submitted_at":"2004-08-24T13:42:31Z","title_canon_sha256":"741f081f5b3425863cf0857de8237d4ea009655d110954d512a9e9cb32a79897"},"schema_version":"1.0","source":{"id":"math/0408331","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0408331","created_at":"2026-07-04T14:39:01Z"},{"alias_kind":"arxiv_version","alias_value":"math/0408331v1","created_at":"2026-07-04T14:39:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0408331","created_at":"2026-07-04T14:39:01Z"},{"alias_kind":"pith_short_12","alias_value":"72DNILSMDOFY","created_at":"2026-07-04T14:39:01Z"},{"alias_kind":"pith_short_16","alias_value":"72DNILSMDOFY5GBG","created_at":"2026-07-04T14:39:01Z"},{"alias_kind":"pith_short_8","alias_value":"72DNILSM","created_at":"2026-07-04T14:39:01Z"}],"graph_snapshots":[{"event_id":"sha256:7972e73df597ad53623ed4c03d79fe6026ac0c1a8c6820bd2f1ab474dee4856a","target":"graph","created_at":"2026-07-04T14:39:01Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0408331/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NP-hard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results.","authors_text":"Marc E. Pfetsch, Michael Joswig","cross_cats":["math.OC"],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"2004-08-24T13:42:31Z","title":"Computing Optimal Morse Matchings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0408331","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:dc8224d571f563f648fce0c368195cf1049e3121513e701ff827b6c90fd7bb20","target":"record","created_at":"2026-07-04T14:39:01Z","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":"f53e41027561bb33d5c9f9a30defb64c45cbdc771245d9e23a09f8678c497ff3","cross_cats_sorted":["math.OC"],"license":"","primary_cat":"math.CO","submitted_at":"2004-08-24T13:42:31Z","title_canon_sha256":"741f081f5b3425863cf0857de8237d4ea009655d110954d512a9e9cb32a79897"},"schema_version":"1.0","source":{"id":"math/0408331","kind":"arxiv","version":1}},"canonical_sha256":"fe86d42e4c1b8b8e9826e937d441dac2336f3152618600eff9a23ac33aa0c616","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fe86d42e4c1b8b8e9826e937d441dac2336f3152618600eff9a23ac33aa0c616","first_computed_at":"2026-07-04T14:39:01.112716Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:39:01.112716Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jLy6dV+btSJEM4VsKOllXpByrGkO5BFPnWix1mVMowM5JT4QJEkO2jjN0hJoY/L45+znic3vHzFo8YcDMMOnAg==","signature_status":"signed_v1","signed_at":"2026-07-04T14:39:01.113048Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0408331","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc8224d571f563f648fce0c368195cf1049e3121513e701ff827b6c90fd7bb20","sha256:7972e73df597ad53623ed4c03d79fe6026ac0c1a8c6820bd2f1ab474dee4856a"],"state_sha256":"4186444854e6709b1fe8dfefecf943226b0a71a0f48d797e350e67a9dc7dd876"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"acCLfoKvyazuhr2sjZuyQu1Zs37jMOB1K/RvoArDhcNkBZIhHPgzXgl1gMoO1RbT8hclqBvjVBH2H4zpyqTTBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T08:14:47.209343Z","bundle_sha256":"15b4c750bcae2d70fba9a5d99574e29fc5de704c6be571796efcb399458fdca8"}}