{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:GSBX6NS5WYHCENOGLCYGLTYBRH","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":"4b493a7c2e4ddc19874421541bd4882eb5f22d05f21563a2c67d0f897b86d1d8","cross_cats_sorted":["cs.CC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-08-27T19:46:14Z","title_canon_sha256":"2318f3612132235700361f741f691f363d23ff232ce0b0eed7334b4b89fd15fc"},"schema_version":"1.0","source":{"id":"1408.6509","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.6509","created_at":"2026-05-18T01:35:35Z"},{"alias_kind":"arxiv_version","alias_value":"1408.6509v2","created_at":"2026-05-18T01:35:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.6509","created_at":"2026-05-18T01:35:35Z"},{"alias_kind":"pith_short_12","alias_value":"GSBX6NS5WYHC","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"GSBX6NS5WYHCENOG","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"GSBX6NS5","created_at":"2026-05-18T12:28:30Z"}],"graph_snapshots":[{"event_id":"sha256:1578a7c66a8520aec14557ff821ec8e49926a33eda31673b6c87a826737e7485","target":"graph","created_at":"2026-05-18T01:35:35Z","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"},"paper":{"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.","authors_text":"Alexander Ushakov, Andrey Nikolaev, Elizaveta Frenkel","cross_cats":["cs.CC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-08-27T19:46:14Z","title":"Knapsack problems in products of groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.6509","kind":"arxiv","version":2},"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":"sha256:08a4ae46ee9c8e50ed6a344b0b0a1a7b7fbf32fae0dfeee7e1b8c61d890b8a40","target":"record","created_at":"2026-05-18T01:35:35Z","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":"4b493a7c2e4ddc19874421541bd4882eb5f22d05f21563a2c67d0f897b86d1d8","cross_cats_sorted":["cs.CC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-08-27T19:46:14Z","title_canon_sha256":"2318f3612132235700361f741f691f363d23ff232ce0b0eed7334b4b89fd15fc"},"schema_version":"1.0","source":{"id":"1408.6509","kind":"arxiv","version":2}},"canonical_sha256":"34837f365db60e2235c658b065cf0189ec36d8884aab938e5e4c733bab1b7284","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"34837f365db60e2235c658b065cf0189ec36d8884aab938e5e4c733bab1b7284","first_computed_at":"2026-05-18T01:35:35.329676Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:35:35.329676Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bDFvgxYv5TTcEKfwPiJjCg4VXQEFt6YQCC8igZ/sPqoT+5voxCuRJb2Z0vFJ7SIs6cFMlSYXR3NMXjOIvZUuCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:35:35.330732Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.6509","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:08a4ae46ee9c8e50ed6a344b0b0a1a7b7fbf32fae0dfeee7e1b8c61d890b8a40","sha256:1578a7c66a8520aec14557ff821ec8e49926a33eda31673b6c87a826737e7485"],"state_sha256":"78bc71f695f88ee695fdb805ff2dcb55f01da18dda48df6be5b3dc6512b146d2"}