{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:2RHMMJN6BJDU52ZNX3JU5ILDT2","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":"0d32e87200b2ff12b79f953e418a0ca24d8f886613ffabe5e948216f440667ad","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-07-28T07:52:01Z","title_canon_sha256":"aa1b7ba3f4a022386cf51f124a59284150086919aec58190df008409305a5da5"},"schema_version":"1.0","source":{"id":"1607.08338","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.08338","created_at":"2026-05-18T01:10:19Z"},{"alias_kind":"arxiv_version","alias_value":"1607.08338v1","created_at":"2026-05-18T01:10:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.08338","created_at":"2026-05-18T01:10:19Z"},{"alias_kind":"pith_short_12","alias_value":"2RHMMJN6BJDU","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"2RHMMJN6BJDU52ZN","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"2RHMMJN6","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:d6d3d310848b12563f49947648ff9c46895572b4502181f6030ae77493dfab21","target":"graph","created_at":"2026-05-18T01:10:19Z","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":"We consider a variant of the knapsack problem, where items are available with different possible weights. Using a separate budget for these item improvements, the question is: Which items should be improved to which degree such that the resulting classic knapsack problem yields maximum profit?\n  We present a detailed analysis for several cases of improvable knapsack problems, presenting constant factor approximation algorithms and two PTAS.","authors_text":"Anita Sch\\\"obel, Marc Goerigk, Sandeep Sen, Yogish Sabharwal","cross_cats":["cs.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-07-28T07:52:01Z","title":"Improvable Knapsack Problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.08338","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":"sha256:0ea42e21980f6aac13029679d5c69df6beff52ad307e117b2c35df4f0da76fbe","target":"record","created_at":"2026-05-18T01:10:19Z","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":"0d32e87200b2ff12b79f953e418a0ca24d8f886613ffabe5e948216f440667ad","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-07-28T07:52:01Z","title_canon_sha256":"aa1b7ba3f4a022386cf51f124a59284150086919aec58190df008409305a5da5"},"schema_version":"1.0","source":{"id":"1607.08338","kind":"arxiv","version":1}},"canonical_sha256":"d44ec625be0a474eeb2dbed34ea1639eb22ce19bac907aec9aca676929e23256","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d44ec625be0a474eeb2dbed34ea1639eb22ce19bac907aec9aca676929e23256","first_computed_at":"2026-05-18T01:10:19.397597Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:10:19.397597Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2oH8JMRZCJhwLQ3LFOwBWiYImNvu6r4b4iBLtsCcserJw6O2FKoZ3zlqmT5Cf+C8e0ZCYOqAL+rJgKFXKCXUCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:10:19.398250Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.08338","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0ea42e21980f6aac13029679d5c69df6beff52ad307e117b2c35df4f0da76fbe","sha256:d6d3d310848b12563f49947648ff9c46895572b4502181f6030ae77493dfab21"],"state_sha256":"a933e4bb4c0b431aa4dd592dc1f7f53e91f3e0b52232741dd9dd9b927d1dc43c"}