{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:L4DI6PTFWFLYIX2EZUDWADSFBI","short_pith_number":"pith:L4DI6PTF","canonical_record":{"source":{"id":"2507.12304","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.DS","submitted_at":"2025-07-16T15:04:26Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"99c548e0eb6b480180dad1b2041dff7e8c76c5070eefa3c8695deef5c80cedc6","abstract_canon_sha256":"6c8db20a4a3c9e9addc6625785afc9c176e68077b3608c5d6f805f97364ea59a"},"schema_version":"1.0"},"canonical_sha256":"5f068f3e65b157845f44cd07600e450a2ff4bf1945f8d616989aec504281b1b1","source":{"kind":"arxiv","id":"2507.12304","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.12304","created_at":"2026-07-05T11:38:18Z"},{"alias_kind":"arxiv_version","alias_value":"2507.12304v1","created_at":"2026-07-05T11:38:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.12304","created_at":"2026-07-05T11:38:18Z"},{"alias_kind":"pith_short_12","alias_value":"L4DI6PTFWFLY","created_at":"2026-07-05T11:38:18Z"},{"alias_kind":"pith_short_16","alias_value":"L4DI6PTFWFLYIX2E","created_at":"2026-07-05T11:38:18Z"},{"alias_kind":"pith_short_8","alias_value":"L4DI6PTF","created_at":"2026-07-05T11:38:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:L4DI6PTFWFLYIX2EZUDWADSFBI","target":"record","payload":{"canonical_record":{"source":{"id":"2507.12304","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.DS","submitted_at":"2025-07-16T15:04:26Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"99c548e0eb6b480180dad1b2041dff7e8c76c5070eefa3c8695deef5c80cedc6","abstract_canon_sha256":"6c8db20a4a3c9e9addc6625785afc9c176e68077b3608c5d6f805f97364ea59a"},"schema_version":"1.0"},"canonical_sha256":"5f068f3e65b157845f44cd07600e450a2ff4bf1945f8d616989aec504281b1b1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:38:18.052568Z","signature_b64":"O7PERzFm8IHee+FFwjS1QQHCQAbofpyAtkID3jE6Lo8MkEL9NfmrlYpVkegZRxKxHdUD4K4E3Cy3rV+w4o0KBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f068f3e65b157845f44cd07600e450a2ff4bf1945f8d616989aec504281b1b1","last_reissued_at":"2026-07-05T11:38:18.052099Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:38:18.052099Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.12304","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":"2026-07-05T11:38:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Du69NF8pPK/XEMFfuKzhTg0CvxgJ/pJFkR6TL4TjoRVLa2WjAFWaY7zrXIRH2B065rWUIWYBhFgFPvPd1No/Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T03:26:20.555288Z"},"content_sha256":"d43f5301483b49fa8ea1831c35e6b2ce6f857c42d1cfc96e17f75bf07cd28aa3","schema_version":"1.0","event_id":"sha256:d43f5301483b49fa8ea1831c35e6b2ce6f857c42d1cfc96e17f75bf07cd28aa3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:L4DI6PTFWFLYIX2EZUDWADSFBI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A near-complete resolution of the exponential-time complexity of k-opt for the traveling salesman problem","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"cs.DS","authors_text":"Hung P. Hoang, Sophia Heimann, Stefan Hougardy","submitted_at":"2025-07-16T15:04:26Z","abstract_excerpt":"The $k$-opt algorithm is one of the simplest and most widely used heuristics for solving the traveling salesman problem. Starting from an arbitrary tour, the $k$-opt algorithm improves the current tour in each iteration by exchanging up to $k$ edges. The algorithm continues until no further improvement of this kind is possible. For a long time, it remained an open question how many iterations the $k$-opt algorithm might require for small values of $k$, assuming the use of an optimal pivot rule. In this paper, we resolve this question for the cases $k = 3$ and $k = 4$ by proving that in both th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.12304","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/2507.12304/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":"2026-07-05T11:38:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vU42PFlPibGQmrRkg6elIcko6wFbKUXiZd75o962FuNkWsvRMgI2S/lUh5PdiX5qjDgAJScIUnMfTuEPhhwVCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T03:26:20.555665Z"},"content_sha256":"42cdacb4e1be4245779a4cce40ae4833c3c33d1cfbb0010138710fea281854eb","schema_version":"1.0","event_id":"sha256:42cdacb4e1be4245779a4cce40ae4833c3c33d1cfbb0010138710fea281854eb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L4DI6PTFWFLYIX2EZUDWADSFBI/bundle.json","state_url":"https://pith.science/pith/L4DI6PTFWFLYIX2EZUDWADSFBI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L4DI6PTFWFLYIX2EZUDWADSFBI/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":"2026-08-07T03:26:20Z","links":{"resolver":"https://pith.science/pith/L4DI6PTFWFLYIX2EZUDWADSFBI","bundle":"https://pith.science/pith/L4DI6PTFWFLYIX2EZUDWADSFBI/bundle.json","state":"https://pith.science/pith/L4DI6PTFWFLYIX2EZUDWADSFBI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L4DI6PTFWFLYIX2EZUDWADSFBI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:L4DI6PTFWFLYIX2EZUDWADSFBI","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":"6c8db20a4a3c9e9addc6625785afc9c176e68077b3608c5d6f805f97364ea59a","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.DS","submitted_at":"2025-07-16T15:04:26Z","title_canon_sha256":"99c548e0eb6b480180dad1b2041dff7e8c76c5070eefa3c8695deef5c80cedc6"},"schema_version":"1.0","source":{"id":"2507.12304","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.12304","created_at":"2026-07-05T11:38:18Z"},{"alias_kind":"arxiv_version","alias_value":"2507.12304v1","created_at":"2026-07-05T11:38:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.12304","created_at":"2026-07-05T11:38:18Z"},{"alias_kind":"pith_short_12","alias_value":"L4DI6PTFWFLY","created_at":"2026-07-05T11:38:18Z"},{"alias_kind":"pith_short_16","alias_value":"L4DI6PTFWFLYIX2E","created_at":"2026-07-05T11:38:18Z"},{"alias_kind":"pith_short_8","alias_value":"L4DI6PTF","created_at":"2026-07-05T11:38:18Z"}],"graph_snapshots":[{"event_id":"sha256:42cdacb4e1be4245779a4cce40ae4833c3c33d1cfbb0010138710fea281854eb","target":"graph","created_at":"2026-07-05T11:38:18Z","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/2507.12304/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The $k$-opt algorithm is one of the simplest and most widely used heuristics for solving the traveling salesman problem. Starting from an arbitrary tour, the $k$-opt algorithm improves the current tour in each iteration by exchanging up to $k$ edges. The algorithm continues until no further improvement of this kind is possible. For a long time, it remained an open question how many iterations the $k$-opt algorithm might require for small values of $k$, assuming the use of an optimal pivot rule. In this paper, we resolve this question for the cases $k = 3$ and $k = 4$ by proving that in both th","authors_text":"Hung P. Hoang, Sophia Heimann, Stefan Hougardy","cross_cats":["cs.DM"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.DS","submitted_at":"2025-07-16T15:04:26Z","title":"A near-complete resolution of the exponential-time complexity of k-opt for the traveling salesman problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.12304","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:d43f5301483b49fa8ea1831c35e6b2ce6f857c42d1cfc96e17f75bf07cd28aa3","target":"record","created_at":"2026-07-05T11:38:18Z","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":"6c8db20a4a3c9e9addc6625785afc9c176e68077b3608c5d6f805f97364ea59a","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.DS","submitted_at":"2025-07-16T15:04:26Z","title_canon_sha256":"99c548e0eb6b480180dad1b2041dff7e8c76c5070eefa3c8695deef5c80cedc6"},"schema_version":"1.0","source":{"id":"2507.12304","kind":"arxiv","version":1}},"canonical_sha256":"5f068f3e65b157845f44cd07600e450a2ff4bf1945f8d616989aec504281b1b1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5f068f3e65b157845f44cd07600e450a2ff4bf1945f8d616989aec504281b1b1","first_computed_at":"2026-07-05T11:38:18.052099Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:38:18.052099Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"O7PERzFm8IHee+FFwjS1QQHCQAbofpyAtkID3jE6Lo8MkEL9NfmrlYpVkegZRxKxHdUD4K4E3Cy3rV+w4o0KBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:38:18.052568Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.12304","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d43f5301483b49fa8ea1831c35e6b2ce6f857c42d1cfc96e17f75bf07cd28aa3","sha256:42cdacb4e1be4245779a4cce40ae4833c3c33d1cfbb0010138710fea281854eb"],"state_sha256":"a4459b38a40d8eada9105ce03b7bd4b78717f5bcb85dec2459394fdb8ec25f95"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dfSKGoOFHQ/izJcHSm6EOyOHC2NgBGKPvV6nvvZKkF1Ez6WKzk7+EFXbnVL0Fn0m/Bt4tGEuuWNVGgirPiF1Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T03:26:20.558369Z","bundle_sha256":"3a6cd6a0e9a96745a1b173a099b09275942b51f917b064251074da0c4ad5fd7a"}}