{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:EPISVXKHEE2GSF4XIC6DWP6AKW","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":"3009537f065eac54e6124210661041e1f6da8585093eee2083ebe17eaa160760","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2026-07-16T07:21:14Z","title_canon_sha256":"0cdd0eb304a95ac3a42abbaa4ade9645e3b1a5d13ef52c156e1a92ff42994249"},"schema_version":"1.0","source":{"id":"2607.14654","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.14654","created_at":"2026-07-17T01:21:23Z"},{"alias_kind":"arxiv_version","alias_value":"2607.14654v1","created_at":"2026-07-17T01:21:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.14654","created_at":"2026-07-17T01:21:23Z"},{"alias_kind":"pith_short_12","alias_value":"EPISVXKHEE2G","created_at":"2026-07-17T01:21:23Z"},{"alias_kind":"pith_short_16","alias_value":"EPISVXKHEE2GSF4X","created_at":"2026-07-17T01:21:23Z"},{"alias_kind":"pith_short_8","alias_value":"EPISVXKH","created_at":"2026-07-17T01:21:23Z"}],"graph_snapshots":[{"event_id":"sha256:5e8419f076ab38251f684d1c05cae8ac76dfbe2ae5f8c770d26506188bfd7312","target":"graph","created_at":"2026-07-17T01:21:23Z","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/2607.14654/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We give a new dual fitting algorithm which gives improved approximation ratios of $3+\\ln 2 + \\epsilon\\ (\\approx 3.694)$ and $4.9+\\epsilon$ for $k$-Means in (high-dimensional) Euclidean and general metrics respectively, improving upon the previously known ratios of $4+\\epsilon$ [Charikar, Cohen-Addad, Gao, Grandoni, Lee, and van Wijland STOC'26] and $5+\\epsilon$ [Byrka, Guo, Hu, Li, Wan, Wang FOCS'26], resp. In particular, our result for Euclidean $k$-Means breaks the hardness barrier of $1+8/e\\approx 3.94$ for Metric $k$-Means. Prior to our work, no such separation between general and Euclidea","authors_text":"Aditya Anand, Amatya Sharma, Ernest van Wijland, Euiwoong Lee, Fabrizio Grandoni, Moses Charikar, Ruiquan Gao, Vincent Cohen-Addad","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2026-07-16T07:21:14Z","title":"Spectral Dual Fitting for $k$-Means"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.14654","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:e0e8ed2f17f70cc48d244b91d6700f7ee49dce32f05fc825fa78bd54fd7a24cb","target":"record","created_at":"2026-07-17T01:21:23Z","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":"3009537f065eac54e6124210661041e1f6da8585093eee2083ebe17eaa160760","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2026-07-16T07:21:14Z","title_canon_sha256":"0cdd0eb304a95ac3a42abbaa4ade9645e3b1a5d13ef52c156e1a92ff42994249"},"schema_version":"1.0","source":{"id":"2607.14654","kind":"arxiv","version":1}},"canonical_sha256":"23d12add47213469179740bc3b3fc0558a164506c8bc93e34ff1c52ab9d3b027","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"23d12add47213469179740bc3b3fc0558a164506c8bc93e34ff1c52ab9d3b027","first_computed_at":"2026-07-17T01:21:23.236337Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-17T01:21:23.236337Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZFN1w653YnpZjKp5LzHmFhVhTPLCA3bE/t6/iIMP44spxjRjNBd0IN4tp6Vi0J3m/O0kJqMVzm7tqVDJHEKiAQ==","signature_status":"signed_v1","signed_at":"2026-07-17T01:21:23.237184Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.14654","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e0e8ed2f17f70cc48d244b91d6700f7ee49dce32f05fc825fa78bd54fd7a24cb","sha256:5e8419f076ab38251f684d1c05cae8ac76dfbe2ae5f8c770d26506188bfd7312"],"state_sha256":"47a5a067d93537c1fcc0ea8d55a8c18b510982a177ea7387a795895fa1edda2f"}