{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:IKDXPJTOQDHC25FCAIZUQ5Z7FR","short_pith_number":"pith:IKDXPJTO","canonical_record":{"source":{"id":"1612.02788","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2016-12-08T20:09:55Z","cross_cats_sorted":["cs.CC","cs.CR"],"title_canon_sha256":"3b9d1cac3b39f0db72a6c4372c8a79394fa04f9f70a96dbbf5ac7edacea8c96d","abstract_canon_sha256":"09b93848d396296a7087dad5300591b1e4cc452c0ea3acce9ef7fcafaf77d1c3"},"schema_version":"1.0"},"canonical_sha256":"428777a66e80ce2d74a2023348773f2c51ceac1e22a77946a631bcc259b61f36","source":{"kind":"arxiv","id":"1612.02788","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.02788","created_at":"2026-05-18T00:41:47Z"},{"alias_kind":"arxiv_version","alias_value":"1612.02788v2","created_at":"2026-05-18T00:41:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.02788","created_at":"2026-05-18T00:41:47Z"},{"alias_kind":"pith_short_12","alias_value":"IKDXPJTOQDHC","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"IKDXPJTOQDHC25FC","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"IKDXPJTO","created_at":"2026-05-18T12:30:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:IKDXPJTOQDHC25FCAIZUQ5Z7FR","target":"record","payload":{"canonical_record":{"source":{"id":"1612.02788","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2016-12-08T20:09:55Z","cross_cats_sorted":["cs.CC","cs.CR"],"title_canon_sha256":"3b9d1cac3b39f0db72a6c4372c8a79394fa04f9f70a96dbbf5ac7edacea8c96d","abstract_canon_sha256":"09b93848d396296a7087dad5300591b1e4cc452c0ea3acce9ef7fcafaf77d1c3"},"schema_version":"1.0"},"canonical_sha256":"428777a66e80ce2d74a2023348773f2c51ceac1e22a77946a631bcc259b61f36","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:41:47.834632Z","signature_b64":"++l3Iz2+zAlgdEoK/MKNDdzfjpgDRupgXK52VmSo4YVHWWSXF/Hbt8NgBTLIZdwMter4o3QMsaTqv4VwQepoAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"428777a66e80ce2d74a2023348773f2c51ceac1e22a77946a631bcc259b61f36","last_reissued_at":"2026-05-18T00:41:47.833856Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:41:47.833856Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.02788","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":"2026-05-18T00:41:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iTzd7JxqiZEBXsox51L2SlR/KfjZhlgDPuG1Lauymur4zs5adn9HYd+a95bQ5NDJ56Pq0fDBQfpQ6ydwi0MWAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T20:29:45.887616Z"},"content_sha256":"ff4b0353dd81b84cf85229393c61c566ec2a399684833b2f717803b4d0f0f1b5","schema_version":"1.0","event_id":"sha256:ff4b0353dd81b84cf85229393c61c566ec2a399684833b2f717803b4d0f0f1b5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:IKDXPJTOQDHC25FCAIZUQ5Z7FR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Faster Space-Efficient Algorithms for Subset Sum, k-Sum and Related Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.CR"],"primary_cat":"cs.DS","authors_text":"Jesper Nederlof, Nikhil Bansal, Nikhil Vyas, Shashwat Garg","submitted_at":"2016-12-08T20:09:55Z","abstract_excerpt":"We present space efficient Monte Carlo algorithms that solve Subset Sum and Knapsack instances with $n$ items using $O^*(2^{0.86n})$ time and polynomial space, where the $O^*(\\cdot)$ notation suppresses factors polynomial in the input size. Both algorithms assume random read-only access to random bits. Modulo this mild assumption, this resolves a long-standing open problem in exact algorithms for NP-hard problems. These results can be extended to solve Binary Linear Programming on $n$ variables with few constraints in a similar running time. We also show that for any constant $k\\geq 2$, random"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.02788","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":"2026-05-18T00:41:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2Bq9EOZfXVdm+sSIi2MTYH4Tj+7P2hrgI2fTJYrxhOY6XWPG+qo1OkZYdEDUvqCsM22jrrcN9/9QknaTQqymBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T20:29:45.887974Z"},"content_sha256":"0cd7b9df66195a1e9dc83ecfa76403ed74b9e30690289b4ef69cb8fb8fe7849d","schema_version":"1.0","event_id":"sha256:0cd7b9df66195a1e9dc83ecfa76403ed74b9e30690289b4ef69cb8fb8fe7849d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IKDXPJTOQDHC25FCAIZUQ5Z7FR/bundle.json","state_url":"https://pith.science/pith/IKDXPJTOQDHC25FCAIZUQ5Z7FR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IKDXPJTOQDHC25FCAIZUQ5Z7FR/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":"2026-05-30T20:29:45Z","links":{"resolver":"https://pith.science/pith/IKDXPJTOQDHC25FCAIZUQ5Z7FR","bundle":"https://pith.science/pith/IKDXPJTOQDHC25FCAIZUQ5Z7FR/bundle.json","state":"https://pith.science/pith/IKDXPJTOQDHC25FCAIZUQ5Z7FR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IKDXPJTOQDHC25FCAIZUQ5Z7FR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:IKDXPJTOQDHC25FCAIZUQ5Z7FR","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":"09b93848d396296a7087dad5300591b1e4cc452c0ea3acce9ef7fcafaf77d1c3","cross_cats_sorted":["cs.CC","cs.CR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2016-12-08T20:09:55Z","title_canon_sha256":"3b9d1cac3b39f0db72a6c4372c8a79394fa04f9f70a96dbbf5ac7edacea8c96d"},"schema_version":"1.0","source":{"id":"1612.02788","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.02788","created_at":"2026-05-18T00:41:47Z"},{"alias_kind":"arxiv_version","alias_value":"1612.02788v2","created_at":"2026-05-18T00:41:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.02788","created_at":"2026-05-18T00:41:47Z"},{"alias_kind":"pith_short_12","alias_value":"IKDXPJTOQDHC","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"IKDXPJTOQDHC25FC","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"IKDXPJTO","created_at":"2026-05-18T12:30:22Z"}],"graph_snapshots":[{"event_id":"sha256:0cd7b9df66195a1e9dc83ecfa76403ed74b9e30690289b4ef69cb8fb8fe7849d","target":"graph","created_at":"2026-05-18T00:41:47Z","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 present space efficient Monte Carlo algorithms that solve Subset Sum and Knapsack instances with $n$ items using $O^*(2^{0.86n})$ time and polynomial space, where the $O^*(\\cdot)$ notation suppresses factors polynomial in the input size. Both algorithms assume random read-only access to random bits. Modulo this mild assumption, this resolves a long-standing open problem in exact algorithms for NP-hard problems. These results can be extended to solve Binary Linear Programming on $n$ variables with few constraints in a similar running time. We also show that for any constant $k\\geq 2$, random","authors_text":"Jesper Nederlof, Nikhil Bansal, Nikhil Vyas, Shashwat Garg","cross_cats":["cs.CC","cs.CR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2016-12-08T20:09:55Z","title":"Faster Space-Efficient Algorithms for Subset Sum, k-Sum and Related Problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.02788","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:ff4b0353dd81b84cf85229393c61c566ec2a399684833b2f717803b4d0f0f1b5","target":"record","created_at":"2026-05-18T00:41:47Z","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":"09b93848d396296a7087dad5300591b1e4cc452c0ea3acce9ef7fcafaf77d1c3","cross_cats_sorted":["cs.CC","cs.CR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2016-12-08T20:09:55Z","title_canon_sha256":"3b9d1cac3b39f0db72a6c4372c8a79394fa04f9f70a96dbbf5ac7edacea8c96d"},"schema_version":"1.0","source":{"id":"1612.02788","kind":"arxiv","version":2}},"canonical_sha256":"428777a66e80ce2d74a2023348773f2c51ceac1e22a77946a631bcc259b61f36","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"428777a66e80ce2d74a2023348773f2c51ceac1e22a77946a631bcc259b61f36","first_computed_at":"2026-05-18T00:41:47.833856Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:41:47.833856Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"++l3Iz2+zAlgdEoK/MKNDdzfjpgDRupgXK52VmSo4YVHWWSXF/Hbt8NgBTLIZdwMter4o3QMsaTqv4VwQepoAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:41:47.834632Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.02788","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ff4b0353dd81b84cf85229393c61c566ec2a399684833b2f717803b4d0f0f1b5","sha256:0cd7b9df66195a1e9dc83ecfa76403ed74b9e30690289b4ef69cb8fb8fe7849d"],"state_sha256":"58fa61cee834e5c7f477723c69ee9b5853122f1ad8d7bdc547c6cc25d65994f1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y8RFYE3N3c3CB5ivf5bkslQgGaAlCLE+Iwl9auEP3QYMb3CW/xpm9M8GzBKAHVdySnUk57zJqISqtB+OlQe1BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T20:29:45.890026Z","bundle_sha256":"c430b65ca376cf38b5fd89cde2ff5d2d05694b5bc1d08f5503d5455e8fadd9fd"}}