{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:GSBX6NS5WYHCENOGLCYGLTYBRH","short_pith_number":"pith:GSBX6NS5","schema_version":"1.0","canonical_sha256":"34837f365db60e2235c658b065cf0189ec36d8884aab938e5e4c733bab1b7284","source":{"kind":"arxiv","id":"1408.6509","version":2},"attestation_state":"computed","paper":{"title":"Knapsack problems in products of groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"math.GR","authors_text":"Alexander Ushakov, Andrey Nikolaev, Elizaveta Frenkel","submitted_at":"2014-08-27T19:46:14Z","abstract_excerpt":"The classic knapsack and related problems have natural generalizations to arbitrary (non-commutative) groups, collectively called knapsack-type problems in groups. We study the effect of free and direct products on their time complexity. We show that free products in certain sense preserve time complexity of knapsack-type problems, while direct products may amplify it. Our methods allow to obtain complexity results for rational subset membership problem in amalgamated free products over finite subgroups."},"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":"1408.6509","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-08-27T19:46:14Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"2318f3612132235700361f741f691f363d23ff232ce0b0eed7334b4b89fd15fc","abstract_canon_sha256":"4b493a7c2e4ddc19874421541bd4882eb5f22d05f21563a2c67d0f897b86d1d8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:35.330732Z","signature_b64":"bDFvgxYv5TTcEKfwPiJjCg4VXQEFt6YQCC8igZ/sPqoT+5voxCuRJb2Z0vFJ7SIs6cFMlSYXR3NMXjOIvZUuCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"34837f365db60e2235c658b065cf0189ec36d8884aab938e5e4c733bab1b7284","last_reissued_at":"2026-05-18T01:35:35.329676Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:35.329676Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Knapsack problems in products of groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"math.GR","authors_text":"Alexander Ushakov, Andrey Nikolaev, Elizaveta Frenkel","submitted_at":"2014-08-27T19:46:14Z","abstract_excerpt":"The classic knapsack and related problems have natural generalizations to arbitrary (non-commutative) groups, collectively called knapsack-type problems in groups. We study the effect of free and direct products on their time complexity. We show that free products in certain sense preserve time complexity of knapsack-type problems, while direct products may amplify it. Our methods allow to obtain complexity results for rational subset membership problem in amalgamated free products over finite subgroups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.6509","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":"1408.6509","created_at":"2026-05-18T01:35:35.329782+00:00"},{"alias_kind":"arxiv_version","alias_value":"1408.6509v2","created_at":"2026-05-18T01:35:35.329782+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.6509","created_at":"2026-05-18T01:35:35.329782+00:00"},{"alias_kind":"pith_short_12","alias_value":"GSBX6NS5WYHC","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_16","alias_value":"GSBX6NS5WYHCENOG","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_8","alias_value":"GSBX6NS5","created_at":"2026-05-18T12:28:30.664211+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/GSBX6NS5WYHCENOGLCYGLTYBRH","json":"https://pith.science/pith/GSBX6NS5WYHCENOGLCYGLTYBRH.json","graph_json":"https://pith.science/api/pith-number/GSBX6NS5WYHCENOGLCYGLTYBRH/graph.json","events_json":"https://pith.science/api/pith-number/GSBX6NS5WYHCENOGLCYGLTYBRH/events.json","paper":"https://pith.science/paper/GSBX6NS5"},"agent_actions":{"view_html":"https://pith.science/pith/GSBX6NS5WYHCENOGLCYGLTYBRH","download_json":"https://pith.science/pith/GSBX6NS5WYHCENOGLCYGLTYBRH.json","view_paper":"https://pith.science/paper/GSBX6NS5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1408.6509&json=true","fetch_graph":"https://pith.science/api/pith-number/GSBX6NS5WYHCENOGLCYGLTYBRH/graph.json","fetch_events":"https://pith.science/api/pith-number/GSBX6NS5WYHCENOGLCYGLTYBRH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GSBX6NS5WYHCENOGLCYGLTYBRH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GSBX6NS5WYHCENOGLCYGLTYBRH/action/storage_attestation","attest_author":"https://pith.science/pith/GSBX6NS5WYHCENOGLCYGLTYBRH/action/author_attestation","sign_citation":"https://pith.science/pith/GSBX6NS5WYHCENOGLCYGLTYBRH/action/citation_signature","submit_replication":"https://pith.science/pith/GSBX6NS5WYHCENOGLCYGLTYBRH/action/replication_record"}},"created_at":"2026-05-18T01:35:35.329782+00:00","updated_at":"2026-05-18T01:35:35.329782+00:00"}