{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:GF7SWB5PFCRIZ7DMPR6DPFG4J6","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":"33fb7385c5aded2dd89b63f4c79dc1d531dc7f76e1bb41301c864156ce7c6812","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2014-02-21T23:34:31Z","title_canon_sha256":"79c0f51c53941a9b0d413054d9ab66b1c5924bdda53e8b7e3eaa088a63c2eafa"},"schema_version":"1.0","source":{"id":"1402.5449","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.5449","created_at":"2026-05-18T02:58:21Z"},{"alias_kind":"arxiv_version","alias_value":"1402.5449v1","created_at":"2026-05-18T02:58:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.5449","created_at":"2026-05-18T02:58:21Z"},{"alias_kind":"pith_short_12","alias_value":"GF7SWB5PFCRI","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"GF7SWB5PFCRIZ7DM","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"GF7SWB5P","created_at":"2026-05-18T12:28:30Z"}],"graph_snapshots":[{"event_id":"sha256:61478a17b0d053bc72a828d07fa266a4ba107b7b4362f273b983ad1cecc3d76f","target":"graph","created_at":"2026-05-18T02:58:21Z","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":"Given two sets $A$ and $B$ of integers, we consider the problem of finding a set $S \\subseteq A$ of the smallest possible cardinality such the greatest common divisor of the elements of $S \\cup B$ equals that of those of $A \\cup B$. The particular cases of $B = \\emptyset$ and $\\#B = 1$ are of special interest and have some links with graph theory. We also consider the corresponding question for the least common multiple of the elements. We establish NP-completeness and approximation results for these problems by relating them to the Minimum Cover Problem.","authors_text":"Igor E. Shparlinski, Joachim von zur Gathen","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2014-02-21T23:34:31Z","title":"Circulant graphs and GCD and LCM of Subsets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.5449","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":"sha256:4b30b2bb009fca1d1a293c9bb6c06ec6548cc1bec603fc7ea1a522257f8b5b41","target":"record","created_at":"2026-05-18T02:58:21Z","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":"33fb7385c5aded2dd89b63f4c79dc1d531dc7f76e1bb41301c864156ce7c6812","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2014-02-21T23:34:31Z","title_canon_sha256":"79c0f51c53941a9b0d413054d9ab66b1c5924bdda53e8b7e3eaa088a63c2eafa"},"schema_version":"1.0","source":{"id":"1402.5449","kind":"arxiv","version":1}},"canonical_sha256":"317f2b07af28a28cfc6c7c7c3794dc4f8e24d512da2c26a67f9c9dae6d7059d6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"317f2b07af28a28cfc6c7c7c3794dc4f8e24d512da2c26a67f9c9dae6d7059d6","first_computed_at":"2026-05-18T02:58:21.113499Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:21.113499Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PPEhZ1X/T3vgfMQ0Jawo4bk0kpGhqm1ni8EfPMi/X61Iv1kB8+L8W8w4csGUdRHnCQ9eaBKsIdCs/94sNkxTDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:21.114017Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.5449","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4b30b2bb009fca1d1a293c9bb6c06ec6548cc1bec603fc7ea1a522257f8b5b41","sha256:61478a17b0d053bc72a828d07fa266a4ba107b7b4362f273b983ad1cecc3d76f"],"state_sha256":"00262c1a3e8ac71a4875b2855cab15f9c52ddc7579d23a037cf4772ce21eda08"}