{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:MZUB7YRG5L2EUS572FVV4LHFOE","short_pith_number":"pith:MZUB7YRG","canonical_record":{"source":{"id":"1511.08647","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-11-27T12:40:24Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0f239e733d6314030891509850c8723f933c25ca942a9876c931b3b3384f25a6","abstract_canon_sha256":"133b1b85250ca751a5aabb209b1f5c0f563f0ba7c005bd1fa055b352c39df00e"},"schema_version":"1.0"},"canonical_sha256":"66681fe226eaf44a4bbfd16b5e2ce5710038e4c2dcf1bd18bade7ec8366f97cf","source":{"kind":"arxiv","id":"1511.08647","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.08647","created_at":"2026-05-18T00:28:57Z"},{"alias_kind":"arxiv_version","alias_value":"1511.08647v3","created_at":"2026-05-18T00:28:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.08647","created_at":"2026-05-18T00:28:57Z"},{"alias_kind":"pith_short_12","alias_value":"MZUB7YRG5L2E","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"MZUB7YRG5L2EUS57","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"MZUB7YRG","created_at":"2026-05-18T12:29:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:MZUB7YRG5L2EUS572FVV4LHFOE","target":"record","payload":{"canonical_record":{"source":{"id":"1511.08647","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-11-27T12:40:24Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0f239e733d6314030891509850c8723f933c25ca942a9876c931b3b3384f25a6","abstract_canon_sha256":"133b1b85250ca751a5aabb209b1f5c0f563f0ba7c005bd1fa055b352c39df00e"},"schema_version":"1.0"},"canonical_sha256":"66681fe226eaf44a4bbfd16b5e2ce5710038e4c2dcf1bd18bade7ec8366f97cf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:28:57.178456Z","signature_b64":"yx5bJqKw2/EZr9S8TPmbiBXEdo9JhB9Z43/l59XlfgNPsyG5x+TfbHXh6Ha6iRiB6Wri+I+5HdmrGxVrXxt3Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"66681fe226eaf44a4bbfd16b5e2ce5710038e4c2dcf1bd18bade7ec8366f97cf","last_reissued_at":"2026-05-18T00:28:57.177996Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:28:57.177996Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.08647","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:28:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oHrpf31N85/LAW15LFmhbfpAS8n4Y3BGS9x+pGiAZymFCd2K5OcX/1qcC4UO8Li4eS2clNSrGuBYz1+ETbdABw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T07:53:19.847660Z"},"content_sha256":"cbda3f2365bdb07f88e364fca754fd91504423171aca783c80ea1a4ffe7832bb","schema_version":"1.0","event_id":"sha256:cbda3f2365bdb07f88e364fca754fd91504423171aca783c80ea1a4ffe7832bb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:MZUB7YRG5L2EUS572FVV4LHFOE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tight Bounds for Gomory-Hu-like Cut Counting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DS","authors_text":"Lior Kamma, Rajesh Chitnis, Robert Krauthgamer","submitted_at":"2015-11-27T12:40:24Z","abstract_excerpt":"By a classical result of Gomory and Hu (1961), in every edge-weighted graph $G=(V,E,w)$, the minimum $st$-cut values, when ranging over all $s,t\\in V$, take at most $|V|-1$ distinct values. That is, these $\\binom{|V|}{2}$ instances exhibit redundancy factor $\\Omega(|V|)$. They further showed how to construct from $G$ a tree $(V,E',w')$ that stores all minimum $st$-cut values. Motivated by this result, we obtain tight bounds for the redundancy factor of several generalizations of the minimum $st$-cut problem.\n  1. Group-Cut: Consider the minimum $(A,B)$-cut, ranging over all subsets $A,B\\subset"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08647","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:28:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2UWefj1niunoV8yTz7rtZ/uGfy/oNFfXJXE670diclmaqd8shdvqWwc+z4iInrFiR7CViS1Mqb6gKiRKNK2QAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T07:53:19.848409Z"},"content_sha256":"c98ef7b028d6a1fa8942b05588e1f58525b8ad659b16c3ffee66a2980db03208","schema_version":"1.0","event_id":"sha256:c98ef7b028d6a1fa8942b05588e1f58525b8ad659b16c3ffee66a2980db03208"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MZUB7YRG5L2EUS572FVV4LHFOE/bundle.json","state_url":"https://pith.science/pith/MZUB7YRG5L2EUS572FVV4LHFOE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MZUB7YRG5L2EUS572FVV4LHFOE/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":"2026-06-11T07:53:19Z","links":{"resolver":"https://pith.science/pith/MZUB7YRG5L2EUS572FVV4LHFOE","bundle":"https://pith.science/pith/MZUB7YRG5L2EUS572FVV4LHFOE/bundle.json","state":"https://pith.science/pith/MZUB7YRG5L2EUS572FVV4LHFOE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MZUB7YRG5L2EUS572FVV4LHFOE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:MZUB7YRG5L2EUS572FVV4LHFOE","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":"133b1b85250ca751a5aabb209b1f5c0f563f0ba7c005bd1fa055b352c39df00e","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-11-27T12:40:24Z","title_canon_sha256":"0f239e733d6314030891509850c8723f933c25ca942a9876c931b3b3384f25a6"},"schema_version":"1.0","source":{"id":"1511.08647","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.08647","created_at":"2026-05-18T00:28:57Z"},{"alias_kind":"arxiv_version","alias_value":"1511.08647v3","created_at":"2026-05-18T00:28:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.08647","created_at":"2026-05-18T00:28:57Z"},{"alias_kind":"pith_short_12","alias_value":"MZUB7YRG5L2E","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"MZUB7YRG5L2EUS57","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"MZUB7YRG","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:c98ef7b028d6a1fa8942b05588e1f58525b8ad659b16c3ffee66a2980db03208","target":"graph","created_at":"2026-05-18T00:28:57Z","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":"By a classical result of Gomory and Hu (1961), in every edge-weighted graph $G=(V,E,w)$, the minimum $st$-cut values, when ranging over all $s,t\\in V$, take at most $|V|-1$ distinct values. That is, these $\\binom{|V|}{2}$ instances exhibit redundancy factor $\\Omega(|V|)$. They further showed how to construct from $G$ a tree $(V,E',w')$ that stores all minimum $st$-cut values. Motivated by this result, we obtain tight bounds for the redundancy factor of several generalizations of the minimum $st$-cut problem.\n  1. Group-Cut: Consider the minimum $(A,B)$-cut, ranging over all subsets $A,B\\subset","authors_text":"Lior Kamma, Rajesh Chitnis, Robert Krauthgamer","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-11-27T12:40:24Z","title":"Tight Bounds for Gomory-Hu-like Cut Counting"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08647","kind":"arxiv","version":3},"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:cbda3f2365bdb07f88e364fca754fd91504423171aca783c80ea1a4ffe7832bb","target":"record","created_at":"2026-05-18T00:28:57Z","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":"133b1b85250ca751a5aabb209b1f5c0f563f0ba7c005bd1fa055b352c39df00e","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-11-27T12:40:24Z","title_canon_sha256":"0f239e733d6314030891509850c8723f933c25ca942a9876c931b3b3384f25a6"},"schema_version":"1.0","source":{"id":"1511.08647","kind":"arxiv","version":3}},"canonical_sha256":"66681fe226eaf44a4bbfd16b5e2ce5710038e4c2dcf1bd18bade7ec8366f97cf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"66681fe226eaf44a4bbfd16b5e2ce5710038e4c2dcf1bd18bade7ec8366f97cf","first_computed_at":"2026-05-18T00:28:57.177996Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:28:57.177996Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yx5bJqKw2/EZr9S8TPmbiBXEdo9JhB9Z43/l59XlfgNPsyG5x+TfbHXh6Ha6iRiB6Wri+I+5HdmrGxVrXxt3Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:28:57.178456Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.08647","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cbda3f2365bdb07f88e364fca754fd91504423171aca783c80ea1a4ffe7832bb","sha256:c98ef7b028d6a1fa8942b05588e1f58525b8ad659b16c3ffee66a2980db03208"],"state_sha256":"3bdcdc7e39238674e8febd40048e34ffd6c0ca03e6e21552580105ef53d70609"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7oTuyaX0EsN+zuVuBGVgDUTNqPitKOQK0MPobloVVB14y6Vh1nhfJ1wTG6UBUeZtw4UDA/j+0t9ss+X81pgCBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T07:53:19.852586Z","bundle_sha256":"bb4cb9f3978a5494e1eb72fe93aa0b2e3ee56816ffc7bbb5635ebe59d9022831"}}