{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:KFR43SP4SGLHGJL2VH5UJDIIE4","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":"cf4f5c40618594e24c02f41454ec6aaeeab4a77aab9779e61a5c56110d631341","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-07-25T07:24:13Z","title_canon_sha256":"b50e034d67adacbcb2c854417b161afc8f5b9839fdbe8fd81ac4840ff439a41c"},"schema_version":"1.0","source":{"id":"2407.17821","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.17821","created_at":"2026-07-05T08:48:22Z"},{"alias_kind":"arxiv_version","alias_value":"2407.17821v1","created_at":"2026-07-05T08:48:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.17821","created_at":"2026-07-05T08:48:22Z"},{"alias_kind":"pith_short_12","alias_value":"KFR43SP4SGLH","created_at":"2026-07-05T08:48:22Z"},{"alias_kind":"pith_short_16","alias_value":"KFR43SP4SGLHGJL2","created_at":"2026-07-05T08:48:22Z"},{"alias_kind":"pith_short_8","alias_value":"KFR43SP4","created_at":"2026-07-05T08:48:22Z"}],"graph_snapshots":[{"event_id":"sha256:8673f169fef58263b2959cc3985b7da699945e905906454fb824ae202bf540d3","target":"graph","created_at":"2026-07-05T08:48:22Z","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/2407.17821/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $G=(V,E)$ be a graph with four distinguished vertices, two sources $s_1, s_2$ and two sinks $t_1,t_2$, let $c:\\, E \\rightarrow \\mathbb Z_+$ be a capacity function, and let ${\\cal P}$ be the set of all simple paths in $G$ from $s_1$ to $t_1$ or from $s_2$ to $t_2$. A biflow (or $2$-commodity flow) in $G$ is an assignment $f:\\, {\\cal P}\\rightarrow \\mathbb R_+$ such that $\\sum_{e \\in Q \\in {\\cal P}}\\, f(Q) \\le c(e)$ for all $e \\in E$, whose value is defined to be $\\sum_{Q \\in {\\cal P}}\\, f(Q)$. A bicut in $G$ is a subset $K$ of $E$ that contains at least one edge from each member of ${\\cal P}","authors_text":"Guoli Ding, Mengxi Yang, Rongchuan Tao, Wenan Zang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-07-25T07:24:13Z","title":"Integral Biflow Maximization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.17821","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:aead1eae56d8b3789e9894745e06d51ff6277ad51198ec5c3cfe5f25d5bf6979","target":"record","created_at":"2026-07-05T08:48:22Z","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":"cf4f5c40618594e24c02f41454ec6aaeeab4a77aab9779e61a5c56110d631341","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-07-25T07:24:13Z","title_canon_sha256":"b50e034d67adacbcb2c854417b161afc8f5b9839fdbe8fd81ac4840ff439a41c"},"schema_version":"1.0","source":{"id":"2407.17821","kind":"arxiv","version":1}},"canonical_sha256":"5163cdc9fc919673257aa9fb448d082715a068e14c9307512a9ce6a744b469b6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5163cdc9fc919673257aa9fb448d082715a068e14c9307512a9ce6a744b469b6","first_computed_at":"2026-07-05T08:48:22.098030Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:48:22.098030Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Z/cjbbQoSaXzWePB57m/rN8N+laGPbyHChZ51t9rcLRRZDCAcYvBK1GWS2BWbZnH23veMUqTv8Z2t4JFLn2YCw==","signature_status":"signed_v1","signed_at":"2026-07-05T08:48:22.098499Z","signed_message":"canonical_sha256_bytes"},"source_id":"2407.17821","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aead1eae56d8b3789e9894745e06d51ff6277ad51198ec5c3cfe5f25d5bf6979","sha256:8673f169fef58263b2959cc3985b7da699945e905906454fb824ae202bf540d3"],"state_sha256":"926e55fb372c277e1ac4ca0dcb59e7abdb12193828c619605447737dc92a87c5"}