{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:OZVULYXSZRDC4RJSN2AQ2C6GFU","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":"396536173e56a591fad7a3b13949ca8f0d952ada7e32917c8a56e7e58887d46b","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2025-02-18T21:23:03Z","title_canon_sha256":"4b0c73ceb45fc0efb97098553d0042f2f262739d181ad784c75a304a82308242"},"schema_version":"1.0","source":{"id":"2502.13292","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.13292","created_at":"2026-07-05T10:16:45Z"},{"alias_kind":"arxiv_version","alias_value":"2502.13292v1","created_at":"2026-07-05T10:16:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.13292","created_at":"2026-07-05T10:16:45Z"},{"alias_kind":"pith_short_12","alias_value":"OZVULYXSZRDC","created_at":"2026-07-05T10:16:45Z"},{"alias_kind":"pith_short_16","alias_value":"OZVULYXSZRDC4RJS","created_at":"2026-07-05T10:16:45Z"},{"alias_kind":"pith_short_8","alias_value":"OZVULYXS","created_at":"2026-07-05T10:16:45Z"}],"graph_snapshots":[{"event_id":"sha256:7fd9c5fda8d83f57100e675fba4b3855eee624ec3ca6a9dc8c2b1be6b0df1dde","target":"graph","created_at":"2026-07-05T10:16:45Z","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/2502.13292/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"These notes give a self-contained exposition of Karlin, Mathieu and Nguyen's tight estimate of the integrality gap of the sum-of-squares semidefinite program for solving the knapsack problem. They are based on a sequence of three lectures in CMU course on Advanced Approximation Algorithms in Fall'21 that used the KMN result to introduce the Sum-of-Squares method for algorithm design. The treatment in these notes uses the pseudo-distribution view of solutions to the sum-of-squares SDPs and only rely on a few basic, reusable results about pseudo-distributions.","authors_text":"Pravesh K. Kothari, Sherry Sarkar","cross_cats":["math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2025-02-18T21:23:03Z","title":"Sum-Of-Squares To Approximate Knapsack"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.13292","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":"sha256:371940f1a9e9d7b18803479110b32fc702c715c8e917ebe0f3b7d231f9d39af1","target":"record","created_at":"2026-07-05T10:16:45Z","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":"396536173e56a591fad7a3b13949ca8f0d952ada7e32917c8a56e7e58887d46b","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2025-02-18T21:23:03Z","title_canon_sha256":"4b0c73ceb45fc0efb97098553d0042f2f262739d181ad784c75a304a82308242"},"schema_version":"1.0","source":{"id":"2502.13292","kind":"arxiv","version":1}},"canonical_sha256":"766b45e2f2cc462e45326e810d0bc62d39af2ce6d35bac64860f680577fa4cc7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"766b45e2f2cc462e45326e810d0bc62d39af2ce6d35bac64860f680577fa4cc7","first_computed_at":"2026-07-05T10:16:45.066435Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:16:45.066435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1c6603T+Ddo1wKMtULZM1hLitkQchmnUZBntsWsk+bRLsMGFyPmMMggOvgmEKX53jQaYYsHBLknPg3e2l6+hDg==","signature_status":"signed_v1","signed_at":"2026-07-05T10:16:45.066849Z","signed_message":"canonical_sha256_bytes"},"source_id":"2502.13292","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:371940f1a9e9d7b18803479110b32fc702c715c8e917ebe0f3b7d231f9d39af1","sha256:7fd9c5fda8d83f57100e675fba4b3855eee624ec3ca6a9dc8c2b1be6b0df1dde"],"state_sha256":"6f5f7bd71f2e3e20d5010c2efeb1388cf4b63729d6882a9cc4e5870b8390fe4f"}