{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:Y3RIFJSAZLGARNCC7EJEZTNGBU","short_pith_number":"pith:Y3RIFJSA","canonical_record":{"source":{"id":"2303.17506","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2023-03-30T16:15:53Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"2ce38acf53d64e6d699a5fdb97f210293e7719a5b7e4e5261737ed53b8c21b75","abstract_canon_sha256":"f9ffaaf926c5f39b6cfd380c7130d4d61db6e7d6b3e1cba562db55f93a966a57"},"schema_version":"1.0"},"canonical_sha256":"c6e282a640cacc08b442f9124ccda60d0876a446abd3034b12fab95b7c601186","source":{"kind":"arxiv","id":"2303.17506","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.17506","created_at":"2026-07-05T05:56:35Z"},{"alias_kind":"arxiv_version","alias_value":"2303.17506v1","created_at":"2026-07-05T05:56:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.17506","created_at":"2026-07-05T05:56:35Z"},{"alias_kind":"pith_short_12","alias_value":"Y3RIFJSAZLGA","created_at":"2026-07-05T05:56:35Z"},{"alias_kind":"pith_short_16","alias_value":"Y3RIFJSAZLGARNCC","created_at":"2026-07-05T05:56:35Z"},{"alias_kind":"pith_short_8","alias_value":"Y3RIFJSA","created_at":"2026-07-05T05:56:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:Y3RIFJSAZLGARNCC7EJEZTNGBU","target":"record","payload":{"canonical_record":{"source":{"id":"2303.17506","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2023-03-30T16:15:53Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"2ce38acf53d64e6d699a5fdb97f210293e7719a5b7e4e5261737ed53b8c21b75","abstract_canon_sha256":"f9ffaaf926c5f39b6cfd380c7130d4d61db6e7d6b3e1cba562db55f93a966a57"},"schema_version":"1.0"},"canonical_sha256":"c6e282a640cacc08b442f9124ccda60d0876a446abd3034b12fab95b7c601186","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:56:35.127295Z","signature_b64":"rmWpD1uGow2gs8ZTUs3SB4E1SxUbVe322uehKfzCh6YGSSBU6uFnKFqATRp6puRsdosO7GoVUGMY39gMBIqIBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c6e282a640cacc08b442f9124ccda60d0876a446abd3034b12fab95b7c601186","last_reissued_at":"2026-07-05T05:56:35.126840Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:56:35.126840Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2303.17506","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":"2026-07-05T05:56:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PkZxTD1v01zdfvRrPbpw4kuF0Zu77ywwO1xO3iJl5QoDet29hulTXnuFopdlfaNkMGWs4OIc+Ur2mgQnYzFgDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T21:45:49.571723Z"},"content_sha256":"7221e64dc4bfd39b388540e8e566866e188417f7f7a8885228a5172719cac315","schema_version":"1.0","event_id":"sha256:7221e64dc4bfd39b388540e8e566866e188417f7f7a8885228a5172719cac315"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:Y3RIFJSAZLGARNCC7EJEZTNGBU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sum-of-Squares Lower Bounds for Densest $k$-Subgraph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.CC","authors_text":"Aaron Potechin, Chris Jones, Goutham Rajendran, Jeff Xu","submitted_at":"2023-03-30T16:15:53Z","abstract_excerpt":"Given a graph and an integer $k$, Densest $k$-Subgraph is the algorithmic task of finding the subgraph on $k$ vertices with the maximum number of edges. This is a fundamental problem that has been subject to intense study for decades, with applications spanning a wide variety of fields. The state-of-the-art algorithm is an $O(n^{1/4 + \\epsilon})$-factor approximation (for any $\\epsilon > 0$) due to Bhaskara et al. [STOC '10]. Moreover, the so-called log-density framework predicts that this is optimal, i.e. it is impossible for an efficient algorithm to achieve an $O(n^{1/4 - \\epsilon})$-factor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.17506","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/2303.17506/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":"2026-07-05T05:56:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R9pnISGw/iGGkVwz764llJchcBknCoo1eX9fDsMMrALWuRClYy4isJhUuplGdY7w7hLj+TVQKircWoXA+NlVAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T21:45:49.572587Z"},"content_sha256":"854457efe2ea0b293f6b9f777a55cdb17f71db3881f11a4d03623f1a1ba5165b","schema_version":"1.0","event_id":"sha256:854457efe2ea0b293f6b9f777a55cdb17f71db3881f11a4d03623f1a1ba5165b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y3RIFJSAZLGARNCC7EJEZTNGBU/bundle.json","state_url":"https://pith.science/pith/Y3RIFJSAZLGARNCC7EJEZTNGBU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y3RIFJSAZLGARNCC7EJEZTNGBU/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":"2026-08-21T21:45:49Z","links":{"resolver":"https://pith.science/pith/Y3RIFJSAZLGARNCC7EJEZTNGBU","bundle":"https://pith.science/pith/Y3RIFJSAZLGARNCC7EJEZTNGBU/bundle.json","state":"https://pith.science/pith/Y3RIFJSAZLGARNCC7EJEZTNGBU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y3RIFJSAZLGARNCC7EJEZTNGBU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:Y3RIFJSAZLGARNCC7EJEZTNGBU","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":"f9ffaaf926c5f39b6cfd380c7130d4d61db6e7d6b3e1cba562db55f93a966a57","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2023-03-30T16:15:53Z","title_canon_sha256":"2ce38acf53d64e6d699a5fdb97f210293e7719a5b7e4e5261737ed53b8c21b75"},"schema_version":"1.0","source":{"id":"2303.17506","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.17506","created_at":"2026-07-05T05:56:35Z"},{"alias_kind":"arxiv_version","alias_value":"2303.17506v1","created_at":"2026-07-05T05:56:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.17506","created_at":"2026-07-05T05:56:35Z"},{"alias_kind":"pith_short_12","alias_value":"Y3RIFJSAZLGA","created_at":"2026-07-05T05:56:35Z"},{"alias_kind":"pith_short_16","alias_value":"Y3RIFJSAZLGARNCC","created_at":"2026-07-05T05:56:35Z"},{"alias_kind":"pith_short_8","alias_value":"Y3RIFJSA","created_at":"2026-07-05T05:56:35Z"}],"graph_snapshots":[{"event_id":"sha256:854457efe2ea0b293f6b9f777a55cdb17f71db3881f11a4d03623f1a1ba5165b","target":"graph","created_at":"2026-07-05T05:56:35Z","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/2303.17506/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given a graph and an integer $k$, Densest $k$-Subgraph is the algorithmic task of finding the subgraph on $k$ vertices with the maximum number of edges. This is a fundamental problem that has been subject to intense study for decades, with applications spanning a wide variety of fields. The state-of-the-art algorithm is an $O(n^{1/4 + \\epsilon})$-factor approximation (for any $\\epsilon > 0$) due to Bhaskara et al. [STOC '10]. Moreover, the so-called log-density framework predicts that this is optimal, i.e. it is impossible for an efficient algorithm to achieve an $O(n^{1/4 - \\epsilon})$-factor","authors_text":"Aaron Potechin, Chris Jones, Goutham Rajendran, Jeff Xu","cross_cats":["cs.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2023-03-30T16:15:53Z","title":"Sum-of-Squares Lower Bounds for Densest $k$-Subgraph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.17506","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":"sha256:7221e64dc4bfd39b388540e8e566866e188417f7f7a8885228a5172719cac315","target":"record","created_at":"2026-07-05T05:56:35Z","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":"f9ffaaf926c5f39b6cfd380c7130d4d61db6e7d6b3e1cba562db55f93a966a57","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2023-03-30T16:15:53Z","title_canon_sha256":"2ce38acf53d64e6d699a5fdb97f210293e7719a5b7e4e5261737ed53b8c21b75"},"schema_version":"1.0","source":{"id":"2303.17506","kind":"arxiv","version":1}},"canonical_sha256":"c6e282a640cacc08b442f9124ccda60d0876a446abd3034b12fab95b7c601186","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c6e282a640cacc08b442f9124ccda60d0876a446abd3034b12fab95b7c601186","first_computed_at":"2026-07-05T05:56:35.126840Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:56:35.126840Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rmWpD1uGow2gs8ZTUs3SB4E1SxUbVe322uehKfzCh6YGSSBU6uFnKFqATRp6puRsdosO7GoVUGMY39gMBIqIBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T05:56:35.127295Z","signed_message":"canonical_sha256_bytes"},"source_id":"2303.17506","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7221e64dc4bfd39b388540e8e566866e188417f7f7a8885228a5172719cac315","sha256:854457efe2ea0b293f6b9f777a55cdb17f71db3881f11a4d03623f1a1ba5165b"],"state_sha256":"fb683ffde4bc6b8c89b17913454d2aa31ea4f213e526dbce06cddb1e38334167"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+DIhDi26u9QRW04zVH9mXqPwNP4JBFG7ADEk/3fLxz3SzaRfsptSwCtV+taFawMJQnrA1JWXI7efbRhQ1/6jBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T21:45:49.577932Z","bundle_sha256":"4747f73ddc8faf160cf2293ba895af1f057b52e225f0bdae27eea034e42c37ef"}}