{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:O2QLVQRLWVUBWI4OWZZSZCK35Q","short_pith_number":"pith:O2QLVQRL","canonical_record":{"source":{"id":"2607.25540","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2026-07-28T10:20:56Z","cross_cats_sorted":[],"title_canon_sha256":"475bcb8e2db6a32a3e1312b0aa756b08ce0c31c0b9cdbb1f5e062e37080d65cb","abstract_canon_sha256":"08207894bbb514253fdf15d175770d514ef6c53341c7c46ca638015c66a2a262"},"schema_version":"1.0"},"canonical_sha256":"76a0bac22bb5681b238eb6732c895bec046de45408aedc6d039e9367a576d24e","source":{"kind":"arxiv","id":"2607.25540","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.25540","created_at":"2026-07-29T01:25:45Z"},{"alias_kind":"arxiv_version","alias_value":"2607.25540v1","created_at":"2026-07-29T01:25:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.25540","created_at":"2026-07-29T01:25:45Z"},{"alias_kind":"pith_short_12","alias_value":"O2QLVQRLWVUB","created_at":"2026-07-29T01:25:45Z"},{"alias_kind":"pith_short_16","alias_value":"O2QLVQRLWVUBWI4O","created_at":"2026-07-29T01:25:45Z"},{"alias_kind":"pith_short_8","alias_value":"O2QLVQRL","created_at":"2026-07-29T01:25:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:O2QLVQRLWVUBWI4OWZZSZCK35Q","target":"record","payload":{"canonical_record":{"source":{"id":"2607.25540","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2026-07-28T10:20:56Z","cross_cats_sorted":[],"title_canon_sha256":"475bcb8e2db6a32a3e1312b0aa756b08ce0c31c0b9cdbb1f5e062e37080d65cb","abstract_canon_sha256":"08207894bbb514253fdf15d175770d514ef6c53341c7c46ca638015c66a2a262"},"schema_version":"1.0"},"canonical_sha256":"76a0bac22bb5681b238eb6732c895bec046de45408aedc6d039e9367a576d24e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-29T01:25:45.722839Z","signature_b64":"E0TAUVbxQAs+M6VbEhegOPzsa2toMotediwytAXjXB97RPDG2lp5Rosnrphmy4W1UbI1zEVeZUPVWFQAnttNCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76a0bac22bb5681b238eb6732c895bec046de45408aedc6d039e9367a576d24e","last_reissued_at":"2026-07-29T01:25:45.722011Z","signature_status":"signed_v1","first_computed_at":"2026-07-29T01:25:45.722011Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.25540","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":"2026-07-29T01:25:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UehSWP3FsRz5efEaAN7Pt6GPGkyPxLLk9Pb8qhvEoT1acLay2vXmKLsZ4fwitiZJQudcareX0pyDqTjkcGj9Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T03:17:16.549600Z"},"content_sha256":"b8a2c41ff8bda0a52fbee33e1f1551f65e6ddc79a941c7b344e7c26478556f5a","schema_version":"1.0","event_id":"sha256:b8a2c41ff8bda0a52fbee33e1f1551f65e6ddc79a941c7b344e7c26478556f5a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:O2QLVQRLWVUBWI4OWZZSZCK35Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Kemeny Rank Aggregation is NP-Hard for Three Voters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.GT","authors_text":"Dominik Peters","submitted_at":"2026-07-28T10:20:56Z","abstract_excerpt":"Rank aggregation is the task of combining $n$ input rankings (linear orders) of alternatives into a single output ranking. The Kemeny rank aggregation rule selects the output ranking that minimizes the total Kendall-tau distance to the input rankings, i.e., the total number of adjacent swaps that need to be performed across input rankings so that they are all equal to the output ranking. Dwork et al. (2001) proved that the problem of computing such a ranking is NP-complete for every even $n \\ge 4$ and asked whether hardness holds even for $n = 3$. We give a hardness reduction from MAX CUT that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.25540","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/2607.25540/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":"2026-07-29T01:25:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r9QQ+Oxx3iqm/CvfHDW9IA+zqiri4UfRwvrq4c+hfO7CADPGsSUm3iEQke6wBet1joCGzlEfxzDFXTztDFW9DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T03:17:16.550097Z"},"content_sha256":"a3b5faec97d45a5977b9e919bff0475cab0bac2b578036ecd12677515bc6ff19","schema_version":"1.0","event_id":"sha256:a3b5faec97d45a5977b9e919bff0475cab0bac2b578036ecd12677515bc6ff19"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O2QLVQRLWVUBWI4OWZZSZCK35Q/bundle.json","state_url":"https://pith.science/pith/O2QLVQRLWVUBWI4OWZZSZCK35Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O2QLVQRLWVUBWI4OWZZSZCK35Q/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":"2026-08-06T03:17:16Z","links":{"resolver":"https://pith.science/pith/O2QLVQRLWVUBWI4OWZZSZCK35Q","bundle":"https://pith.science/pith/O2QLVQRLWVUBWI4OWZZSZCK35Q/bundle.json","state":"https://pith.science/pith/O2QLVQRLWVUBWI4OWZZSZCK35Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O2QLVQRLWVUBWI4OWZZSZCK35Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:O2QLVQRLWVUBWI4OWZZSZCK35Q","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":"08207894bbb514253fdf15d175770d514ef6c53341c7c46ca638015c66a2a262","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2026-07-28T10:20:56Z","title_canon_sha256":"475bcb8e2db6a32a3e1312b0aa756b08ce0c31c0b9cdbb1f5e062e37080d65cb"},"schema_version":"1.0","source":{"id":"2607.25540","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.25540","created_at":"2026-07-29T01:25:45Z"},{"alias_kind":"arxiv_version","alias_value":"2607.25540v1","created_at":"2026-07-29T01:25:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.25540","created_at":"2026-07-29T01:25:45Z"},{"alias_kind":"pith_short_12","alias_value":"O2QLVQRLWVUB","created_at":"2026-07-29T01:25:45Z"},{"alias_kind":"pith_short_16","alias_value":"O2QLVQRLWVUBWI4O","created_at":"2026-07-29T01:25:45Z"},{"alias_kind":"pith_short_8","alias_value":"O2QLVQRL","created_at":"2026-07-29T01:25:45Z"}],"graph_snapshots":[{"event_id":"sha256:a3b5faec97d45a5977b9e919bff0475cab0bac2b578036ecd12677515bc6ff19","target":"graph","created_at":"2026-07-29T01:25:45Z","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/2607.25540/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Rank aggregation is the task of combining $n$ input rankings (linear orders) of alternatives into a single output ranking. The Kemeny rank aggregation rule selects the output ranking that minimizes the total Kendall-tau distance to the input rankings, i.e., the total number of adjacent swaps that need to be performed across input rankings so that they are all equal to the output ranking. Dwork et al. (2001) proved that the problem of computing such a ranking is NP-complete for every even $n \\ge 4$ and asked whether hardness holds even for $n = 3$. We give a hardness reduction from MAX CUT that","authors_text":"Dominik Peters","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2026-07-28T10:20:56Z","title":"Kemeny Rank Aggregation is NP-Hard for Three Voters"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.25540","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:b8a2c41ff8bda0a52fbee33e1f1551f65e6ddc79a941c7b344e7c26478556f5a","target":"record","created_at":"2026-07-29T01:25:45Z","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":"08207894bbb514253fdf15d175770d514ef6c53341c7c46ca638015c66a2a262","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2026-07-28T10:20:56Z","title_canon_sha256":"475bcb8e2db6a32a3e1312b0aa756b08ce0c31c0b9cdbb1f5e062e37080d65cb"},"schema_version":"1.0","source":{"id":"2607.25540","kind":"arxiv","version":1}},"canonical_sha256":"76a0bac22bb5681b238eb6732c895bec046de45408aedc6d039e9367a576d24e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"76a0bac22bb5681b238eb6732c895bec046de45408aedc6d039e9367a576d24e","first_computed_at":"2026-07-29T01:25:45.722011Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-29T01:25:45.722011Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"E0TAUVbxQAs+M6VbEhegOPzsa2toMotediwytAXjXB97RPDG2lp5Rosnrphmy4W1UbI1zEVeZUPVWFQAnttNCQ==","signature_status":"signed_v1","signed_at":"2026-07-29T01:25:45.722839Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.25540","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b8a2c41ff8bda0a52fbee33e1f1551f65e6ddc79a941c7b344e7c26478556f5a","sha256:a3b5faec97d45a5977b9e919bff0475cab0bac2b578036ecd12677515bc6ff19"],"state_sha256":"cfc5bc2051a6f50bd52f1f1e49ad1f6809f1369667b4f5d898042ea5cb027004"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lWhd/AWXQCgjHlaIZFNpTH3tei0ugLCxgq92yux3oFbw7fHZKO3ycfRo22F5Oameu3aVIHIFIumZpNhJgjZqDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T03:17:16.555301Z","bundle_sha256":"9422d0e4e9cd0f0f919144d0e05908646799c166af21a4574cdde73cb073dd89"}}