{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:6E3HL33UN55WL7KWNW6H4N4SXA","short_pith_number":"pith:6E3HL33U","canonical_record":{"source":{"id":"1410.6400","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-10-23T15:33:01Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"85c159f31fbb714f8cfff22fff643a9c07023469140f91ee9a425b3926ad9e61","abstract_canon_sha256":"eb55c677f5fd67d3a90ffb45556e726955c443b7730f56751222685e194847bd"},"schema_version":"1.0"},"canonical_sha256":"f13675ef746f7b65fd566dbc7e3792b81fa144c00f117814e611dceccc2c1dc5","source":{"kind":"arxiv","id":"1410.6400","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.6400","created_at":"2026-05-18T02:39:30Z"},{"alias_kind":"arxiv_version","alias_value":"1410.6400v1","created_at":"2026-05-18T02:39:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.6400","created_at":"2026-05-18T02:39:30Z"},{"alias_kind":"pith_short_12","alias_value":"6E3HL33UN55W","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"6E3HL33UN55WL7KW","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"6E3HL33U","created_at":"2026-05-18T12:28:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:6E3HL33UN55WL7KWNW6H4N4SXA","target":"record","payload":{"canonical_record":{"source":{"id":"1410.6400","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-10-23T15:33:01Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"85c159f31fbb714f8cfff22fff643a9c07023469140f91ee9a425b3926ad9e61","abstract_canon_sha256":"eb55c677f5fd67d3a90ffb45556e726955c443b7730f56751222685e194847bd"},"schema_version":"1.0"},"canonical_sha256":"f13675ef746f7b65fd566dbc7e3792b81fa144c00f117814e611dceccc2c1dc5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:30.986992Z","signature_b64":"Ht71jSHbCyCQqzAfnkOgVjxWlN+tsoWRl/PxFW2eqQiRKqiz5ADjbUt2pd6WeBMm1PbGme5v83VjvT0bwN5bBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f13675ef746f7b65fd566dbc7e3792b81fa144c00f117814e611dceccc2c1dc5","last_reissued_at":"2026-05-18T02:39:30.986411Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:30.986411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.6400","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":"2026-05-18T02:39:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+bvrpJUqiP5/IXoN+jzYhNcVRHWCmrOvPveqIJHHkTF02jBwtB8kTB/XUclSfM0PgZt1SlmBTxSiQJFE8p8nBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T14:12:53.342223Z"},"content_sha256":"973cf3b92cbe69f17b0ba8a7c605b60ce1a26f5b8e8e2c243e9175b6a4addc48","schema_version":"1.0","event_id":"sha256:973cf3b92cbe69f17b0ba8a7c605b60ce1a26f5b8e8e2c243e9175b6a4addc48"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:6E3HL33UN55WL7KWNW6H4N4SXA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Average-case Complexity of Parameterized Clique","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"cs.DS","authors_text":"Danny Hermelin, Nikolaos Fountoulakis, Tobias Friedrich","submitted_at":"2014-10-23T15:33:01Z","abstract_excerpt":"The k-Clique problem is a fundamental combinatorial problem that plays a prominent role in classical as well as in parameterized complexity theory. It is among the most well-known NP-complete and W[1]-complete problems. Moreover, its average-case complexity analysis has created a long thread of research already since the 1970s. Here, we continue this line of research by studying the dependence of the average-case complexity of the k-Clique problem on the parameter k. To this end, we define two natural parameterized analogs of efficient average-case algorithms. We then show that k-Clique admits"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.6400","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":""},"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":"2026-05-18T02:39:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Koawnf2NdPdc1HQrME1ddPZlk0hXGISbgSpdXocYCbMXSfPuLtfT+k30irhGm8emhsIRrtQhdQIwPiYN529VAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T14:12:53.342605Z"},"content_sha256":"3aef532b2e60d9523d5f4491d16ba0eeae50a841112ec094c0b4fd5903158798","schema_version":"1.0","event_id":"sha256:3aef532b2e60d9523d5f4491d16ba0eeae50a841112ec094c0b4fd5903158798"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6E3HL33UN55WL7KWNW6H4N4SXA/bundle.json","state_url":"https://pith.science/pith/6E3HL33UN55WL7KWNW6H4N4SXA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6E3HL33UN55WL7KWNW6H4N4SXA/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":"2026-06-24T14:12:53Z","links":{"resolver":"https://pith.science/pith/6E3HL33UN55WL7KWNW6H4N4SXA","bundle":"https://pith.science/pith/6E3HL33UN55WL7KWNW6H4N4SXA/bundle.json","state":"https://pith.science/pith/6E3HL33UN55WL7KWNW6H4N4SXA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6E3HL33UN55WL7KWNW6H4N4SXA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:6E3HL33UN55WL7KWNW6H4N4SXA","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":"eb55c677f5fd67d3a90ffb45556e726955c443b7730f56751222685e194847bd","cross_cats_sorted":["cs.CC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-10-23T15:33:01Z","title_canon_sha256":"85c159f31fbb714f8cfff22fff643a9c07023469140f91ee9a425b3926ad9e61"},"schema_version":"1.0","source":{"id":"1410.6400","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.6400","created_at":"2026-05-18T02:39:30Z"},{"alias_kind":"arxiv_version","alias_value":"1410.6400v1","created_at":"2026-05-18T02:39:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.6400","created_at":"2026-05-18T02:39:30Z"},{"alias_kind":"pith_short_12","alias_value":"6E3HL33UN55W","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"6E3HL33UN55WL7KW","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"6E3HL33U","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:3aef532b2e60d9523d5f4491d16ba0eeae50a841112ec094c0b4fd5903158798","target":"graph","created_at":"2026-05-18T02:39:30Z","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"},"paper":{"abstract_excerpt":"The k-Clique problem is a fundamental combinatorial problem that plays a prominent role in classical as well as in parameterized complexity theory. It is among the most well-known NP-complete and W[1]-complete problems. Moreover, its average-case complexity analysis has created a long thread of research already since the 1970s. Here, we continue this line of research by studying the dependence of the average-case complexity of the k-Clique problem on the parameter k. To this end, we define two natural parameterized analogs of efficient average-case algorithms. We then show that k-Clique admits","authors_text":"Danny Hermelin, Nikolaos Fountoulakis, Tobias Friedrich","cross_cats":["cs.CC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-10-23T15:33:01Z","title":"On the Average-case Complexity of Parameterized Clique"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.6400","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":"sha256:973cf3b92cbe69f17b0ba8a7c605b60ce1a26f5b8e8e2c243e9175b6a4addc48","target":"record","created_at":"2026-05-18T02:39:30Z","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":"eb55c677f5fd67d3a90ffb45556e726955c443b7730f56751222685e194847bd","cross_cats_sorted":["cs.CC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-10-23T15:33:01Z","title_canon_sha256":"85c159f31fbb714f8cfff22fff643a9c07023469140f91ee9a425b3926ad9e61"},"schema_version":"1.0","source":{"id":"1410.6400","kind":"arxiv","version":1}},"canonical_sha256":"f13675ef746f7b65fd566dbc7e3792b81fa144c00f117814e611dceccc2c1dc5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f13675ef746f7b65fd566dbc7e3792b81fa144c00f117814e611dceccc2c1dc5","first_computed_at":"2026-05-18T02:39:30.986411Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:39:30.986411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ht71jSHbCyCQqzAfnkOgVjxWlN+tsoWRl/PxFW2eqQiRKqiz5ADjbUt2pd6WeBMm1PbGme5v83VjvT0bwN5bBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:39:30.986992Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.6400","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:973cf3b92cbe69f17b0ba8a7c605b60ce1a26f5b8e8e2c243e9175b6a4addc48","sha256:3aef532b2e60d9523d5f4491d16ba0eeae50a841112ec094c0b4fd5903158798"],"state_sha256":"3b21385e8292272ad37457d4e752d21ba1b6c0cbf70a28acab782ac3d4d44627"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gzYKnNwKSUUBIOINDB7mK6pvz6vnrrdZ5bDueDUUGdOJrdaePLpoOJHfqXTWirMnBTfOLf9LsI8i4Bagm1YTAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T14:12:53.345109Z","bundle_sha256":"9d48b035a3b0da8d5fe855b07d5fe8c41012a5c0c3705b9b8335d6fda46981e5"}}