{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:U5JYOEKKCXEKGGXERD2DPAJ22N","short_pith_number":"pith:U5JYOEKK","canonical_record":{"source":{"id":"1509.01866","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-09-06T23:21:39Z","cross_cats_sorted":["cs.CC","cs.DM","math.CO"],"title_canon_sha256":"e8189f06544fe48ea4c3978dd49ccfeec12ae69beabc94be22621ae0aa1bcffc","abstract_canon_sha256":"e4235a53670b19ee575a5250335581e5fcfe0178dc3823210cfaf270fdfd5d9f"},"schema_version":"1.0"},"canonical_sha256":"a75387114a15c8a31ae488f437813ad357299acf409e3b25fe31bab923e8c681","source":{"kind":"arxiv","id":"1509.01866","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.01866","created_at":"2026-05-18T01:14:17Z"},{"alias_kind":"arxiv_version","alias_value":"1509.01866v2","created_at":"2026-05-18T01:14:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.01866","created_at":"2026-05-18T01:14:17Z"},{"alias_kind":"pith_short_12","alias_value":"U5JYOEKKCXEK","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"U5JYOEKKCXEKGGXE","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"U5JYOEKK","created_at":"2026-05-18T12:29:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:U5JYOEKKCXEKGGXERD2DPAJ22N","target":"record","payload":{"canonical_record":{"source":{"id":"1509.01866","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-09-06T23:21:39Z","cross_cats_sorted":["cs.CC","cs.DM","math.CO"],"title_canon_sha256":"e8189f06544fe48ea4c3978dd49ccfeec12ae69beabc94be22621ae0aa1bcffc","abstract_canon_sha256":"e4235a53670b19ee575a5250335581e5fcfe0178dc3823210cfaf270fdfd5d9f"},"schema_version":"1.0"},"canonical_sha256":"a75387114a15c8a31ae488f437813ad357299acf409e3b25fe31bab923e8c681","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:17.849693Z","signature_b64":"pJIzYf3+A/pmXdvvb9/vp9gH+6f/IRrtqVL1YqQDq79IfHIJf0K8TyvMEi/VWXm2K/oV+5TXPzvptt19cCDeBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a75387114a15c8a31ae488f437813ad357299acf409e3b25fe31bab923e8c681","last_reissued_at":"2026-05-18T01:14:17.849230Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:17.849230Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.01866","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:14:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2RgrRYATp4lauIdxN5Y7ZGkgEczp3S78EPC7UsO3AzLBVTjKzsTvSHGhK+xySuuhToKFmgFcxH6246HKOwIpDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T09:07:26.794391Z"},"content_sha256":"436a31f8fc015b41e62a32b71d37dd5a7d661d6bb37efbebb7a306eb07e1fdd5","schema_version":"1.0","event_id":"sha256:436a31f8fc015b41e62a32b71d37dd5a7d661d6bb37efbebb7a306eb07e1fdd5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:U5JYOEKKCXEKGGXERD2DPAJ22N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximation of the Quadratic Knapsack Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.DM","math.CO"],"primary_cat":"cs.DS","authors_text":"Richard Taylor","submitted_at":"2015-09-06T23:21:39Z","abstract_excerpt":"For any given $\\epsilon>0$ we provide an algorithm for the Quadratic Knapsack Problem that has an approximation ratio within $O(n^{2/5+\\epsilon})$ and a run time within $O(n^{9/\\epsilon})$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01866","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:14:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wk4sZ6SFcghbYT5zjXkQB2Ibnr7jKH8uaIu9ESQM51fkIYWl7xd6LtaDKQGC2Zkp7ea5QMRCLDajATzcnB/GDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T09:07:26.794747Z"},"content_sha256":"331f16b10f66f2e53f89a4cebcb4faa20831ec947581cd2c8edd52e322921aa7","schema_version":"1.0","event_id":"sha256:331f16b10f66f2e53f89a4cebcb4faa20831ec947581cd2c8edd52e322921aa7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U5JYOEKKCXEKGGXERD2DPAJ22N/bundle.json","state_url":"https://pith.science/pith/U5JYOEKKCXEKGGXERD2DPAJ22N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U5JYOEKKCXEKGGXERD2DPAJ22N/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":"2026-06-11T09:07:26Z","links":{"resolver":"https://pith.science/pith/U5JYOEKKCXEKGGXERD2DPAJ22N","bundle":"https://pith.science/pith/U5JYOEKKCXEKGGXERD2DPAJ22N/bundle.json","state":"https://pith.science/pith/U5JYOEKKCXEKGGXERD2DPAJ22N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U5JYOEKKCXEKGGXERD2DPAJ22N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:U5JYOEKKCXEKGGXERD2DPAJ22N","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":"e4235a53670b19ee575a5250335581e5fcfe0178dc3823210cfaf270fdfd5d9f","cross_cats_sorted":["cs.CC","cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-09-06T23:21:39Z","title_canon_sha256":"e8189f06544fe48ea4c3978dd49ccfeec12ae69beabc94be22621ae0aa1bcffc"},"schema_version":"1.0","source":{"id":"1509.01866","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.01866","created_at":"2026-05-18T01:14:17Z"},{"alias_kind":"arxiv_version","alias_value":"1509.01866v2","created_at":"2026-05-18T01:14:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.01866","created_at":"2026-05-18T01:14:17Z"},{"alias_kind":"pith_short_12","alias_value":"U5JYOEKKCXEK","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"U5JYOEKKCXEKGGXE","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"U5JYOEKK","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:331f16b10f66f2e53f89a4cebcb4faa20831ec947581cd2c8edd52e322921aa7","target":"graph","created_at":"2026-05-18T01:14:17Z","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":"For any given $\\epsilon>0$ we provide an algorithm for the Quadratic Knapsack Problem that has an approximation ratio within $O(n^{2/5+\\epsilon})$ and a run time within $O(n^{9/\\epsilon})$.","authors_text":"Richard Taylor","cross_cats":["cs.CC","cs.DM","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-09-06T23:21:39Z","title":"Approximation of the Quadratic Knapsack Problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01866","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":"sha256:436a31f8fc015b41e62a32b71d37dd5a7d661d6bb37efbebb7a306eb07e1fdd5","target":"record","created_at":"2026-05-18T01:14:17Z","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":"e4235a53670b19ee575a5250335581e5fcfe0178dc3823210cfaf270fdfd5d9f","cross_cats_sorted":["cs.CC","cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-09-06T23:21:39Z","title_canon_sha256":"e8189f06544fe48ea4c3978dd49ccfeec12ae69beabc94be22621ae0aa1bcffc"},"schema_version":"1.0","source":{"id":"1509.01866","kind":"arxiv","version":2}},"canonical_sha256":"a75387114a15c8a31ae488f437813ad357299acf409e3b25fe31bab923e8c681","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a75387114a15c8a31ae488f437813ad357299acf409e3b25fe31bab923e8c681","first_computed_at":"2026-05-18T01:14:17.849230Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:14:17.849230Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pJIzYf3+A/pmXdvvb9/vp9gH+6f/IRrtqVL1YqQDq79IfHIJf0K8TyvMEi/VWXm2K/oV+5TXPzvptt19cCDeBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:14:17.849693Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.01866","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:436a31f8fc015b41e62a32b71d37dd5a7d661d6bb37efbebb7a306eb07e1fdd5","sha256:331f16b10f66f2e53f89a4cebcb4faa20831ec947581cd2c8edd52e322921aa7"],"state_sha256":"39e99fdedf20f0102da68cfeaa68c019dd1dd4ceea11f9db675b0a6032777885"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0bL+3BMd4ly9v3K4I/gGnz7QK96PpDdc+BV6fYKiHKLJd8x8vl6jElaZOMvoXxw+6AI6tuY62fd06Mkp2lFQAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T09:07:26.796691Z","bundle_sha256":"1980acbc01bea02129f04851c93697db196bdd2f42c67361937805dc008952f8"}}