{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:32T6SPOGEI532SYEXYAZFIUHIV","short_pith_number":"pith:32T6SPOG","schema_version":"1.0","canonical_sha256":"dea7e93dc6223bbd4b04be0192a287456a673c10264db2c87910d553705be9c9","source":{"kind":"arxiv","id":"1604.03084","version":2},"attestation_state":"computed","paper":{"title":"A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Aaron Potechin, Ankur Moitra, Boaz Barak, Jonathan Kelner, Pravesh K. Kothari, Samuel B. Hopkins","submitted_at":"2016-04-11T19:49:03Z","abstract_excerpt":"We prove that with high probability over the choice of a random graph $G$ from the Erd\\H{o}s-R\\'enyi distribution $G(n,1/2)$, the $n^{O(d)}$-time degree $d$ Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least $n^{1/2-c(d/\\log n)^{1/2}}$ for some constant $c>0$. This yields a nearly tight $n^{1/2 - o(1)}$ bound on the value of this program for any degree $d = o(\\log n)$. Moreover we introduce a new framework that we call \\emph{pseudo-calibration} to construct Sum of Squares lower bounds. This framework is inspired by taking a computational ana"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1604.03084","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-04-11T19:49:03Z","cross_cats_sorted":[],"title_canon_sha256":"292f8b6cc16b27ffe0199c437dae07c4e9a581e2c2b76a51423fc5af5ca8d185","abstract_canon_sha256":"608baa23ea45dd773f9c60d09413821b0145e68bcdbabde54a648ef1b494cfa2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:16.499480Z","signature_b64":"/k8WCWvviJG0iUeTJ/r7pRlGxTuWJ1U+eeGXXMa5RG22qMB/nX661upoM49caJulz2AXSXMHUYfSCHE7dkXVBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dea7e93dc6223bbd4b04be0192a287456a673c10264db2c87910d553705be9c9","last_reissued_at":"2026-05-18T01:17:16.498387Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:16.498387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Aaron Potechin, Ankur Moitra, Boaz Barak, Jonathan Kelner, Pravesh K. Kothari, Samuel B. Hopkins","submitted_at":"2016-04-11T19:49:03Z","abstract_excerpt":"We prove that with high probability over the choice of a random graph $G$ from the Erd\\H{o}s-R\\'enyi distribution $G(n,1/2)$, the $n^{O(d)}$-time degree $d$ Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least $n^{1/2-c(d/\\log n)^{1/2}}$ for some constant $c>0$. This yields a nearly tight $n^{1/2 - o(1)}$ bound on the value of this program for any degree $d = o(\\log n)$. Moreover we introduce a new framework that we call \\emph{pseudo-calibration} to construct Sum of Squares lower bounds. This framework is inspired by taking a computational ana"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.03084","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1604.03084","created_at":"2026-05-18T01:17:16.498516+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.03084v2","created_at":"2026-05-18T01:17:16.498516+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.03084","created_at":"2026-05-18T01:17:16.498516+00:00"},{"alias_kind":"pith_short_12","alias_value":"32T6SPOGEI53","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"32T6SPOGEI532SYE","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"32T6SPOG","created_at":"2026-05-18T12:29:55.572404+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/32T6SPOGEI532SYEXYAZFIUHIV","json":"https://pith.science/pith/32T6SPOGEI532SYEXYAZFIUHIV.json","graph_json":"https://pith.science/api/pith-number/32T6SPOGEI532SYEXYAZFIUHIV/graph.json","events_json":"https://pith.science/api/pith-number/32T6SPOGEI532SYEXYAZFIUHIV/events.json","paper":"https://pith.science/paper/32T6SPOG"},"agent_actions":{"view_html":"https://pith.science/pith/32T6SPOGEI532SYEXYAZFIUHIV","download_json":"https://pith.science/pith/32T6SPOGEI532SYEXYAZFIUHIV.json","view_paper":"https://pith.science/paper/32T6SPOG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.03084&json=true","fetch_graph":"https://pith.science/api/pith-number/32T6SPOGEI532SYEXYAZFIUHIV/graph.json","fetch_events":"https://pith.science/api/pith-number/32T6SPOGEI532SYEXYAZFIUHIV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/32T6SPOGEI532SYEXYAZFIUHIV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/32T6SPOGEI532SYEXYAZFIUHIV/action/storage_attestation","attest_author":"https://pith.science/pith/32T6SPOGEI532SYEXYAZFIUHIV/action/author_attestation","sign_citation":"https://pith.science/pith/32T6SPOGEI532SYEXYAZFIUHIV/action/citation_signature","submit_replication":"https://pith.science/pith/32T6SPOGEI532SYEXYAZFIUHIV/action/replication_record"}},"created_at":"2026-05-18T01:17:16.498516+00:00","updated_at":"2026-05-18T01:17:16.498516+00:00"}