{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:R3DNZHJTXKEEUGFHLKJCKSO4HX","short_pith_number":"pith:R3DNZHJT","canonical_record":{"source":{"id":"cs/0702032","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"cs.DS","submitted_at":"2007-02-05T19:29:38Z","cross_cats_sorted":[],"title_canon_sha256":"cf55a7dce8792c939bafc409529d18cf346f4e62929ab34edf9a7486b084b84d","abstract_canon_sha256":"ffb41876d99bdfcf270c0c0500fad57265615993d6d5f94ce112a0cc1ab4df14"},"schema_version":"1.0"},"canonical_sha256":"8ec6dc9d33ba884a18a75a922549dc3dd70aa13b0f6e0288870338434c127789","source":{"kind":"arxiv","id":"cs/0702032","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"cs/0702032","created_at":"2026-07-04T14:59:14Z"},{"alias_kind":"arxiv_version","alias_value":"cs/0702032v1","created_at":"2026-07-04T14:59:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.cs/0702032","created_at":"2026-07-04T14:59:14Z"},{"alias_kind":"pith_short_12","alias_value":"R3DNZHJTXKEE","created_at":"2026-07-04T14:59:14Z"},{"alias_kind":"pith_short_16","alias_value":"R3DNZHJTXKEEUGFH","created_at":"2026-07-04T14:59:14Z"},{"alias_kind":"pith_short_8","alias_value":"R3DNZHJT","created_at":"2026-07-04T14:59:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:R3DNZHJTXKEEUGFHLKJCKSO4HX","target":"record","payload":{"canonical_record":{"source":{"id":"cs/0702032","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"cs.DS","submitted_at":"2007-02-05T19:29:38Z","cross_cats_sorted":[],"title_canon_sha256":"cf55a7dce8792c939bafc409529d18cf346f4e62929ab34edf9a7486b084b84d","abstract_canon_sha256":"ffb41876d99bdfcf270c0c0500fad57265615993d6d5f94ce112a0cc1ab4df14"},"schema_version":"1.0"},"canonical_sha256":"8ec6dc9d33ba884a18a75a922549dc3dd70aa13b0f6e0288870338434c127789","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:59:14.551919Z","signature_b64":"7w2JfNIYglqd9vBmreTRkOtsvpNvpGwMg5Y/PVLk8TjIQl7fvAg7JvG76DRapFfMI9/bNgvGMwa+nP5QIHZMCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ec6dc9d33ba884a18a75a922549dc3dd70aa13b0f6e0288870338434c127789","last_reissued_at":"2026-07-04T14:59:14.551575Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:59:14.551575Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"cs/0702032","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T14:59:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qXROVVrUFSLtfQZe5bTbsKVxWXRLXLkIm98CKskQHO6VF74exvl78J9idw5SZc3/pVXiAZAhMHYwsRjOnjMMCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T07:23:09.999227Z"},"content_sha256":"b2ada978aa75873b8bd959746713bf7d60a5a6c0a94db6927967da2242c9095d","schema_version":"1.0","event_id":"sha256:b2ada978aa75873b8bd959746713bf7d60a5a6c0a94db6927967da2242c9095d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:R3DNZHJTXKEEUGFHLKJCKSO4HX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Finding large and small dense subgraphs","license":"","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Reid Andersen","submitted_at":"2007-02-05T19:29:38Z","abstract_excerpt":"We consider two optimization problems related to finding dense subgraphs. The densest at-least-k-subgraph problem (DalkS) is to find an induced subgraph of highest average degree among all subgraphs with at least k vertices, and the densest at-most-k-subgraph problem (DamkS) is defined similarly. These problems are related to the well-known densest k-subgraph problem (DkS), which is to find the densest subgraph on exactly k vertices. We show that DalkS can be approximated efficiently, while DamkS is nearly as hard to approximate as the densest k-subgraph problem."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"cs/0702032","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/cs/0702032/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T14:59:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y9+3E/KbEdvToGU1yLb7o9IQr1doRSwY77cxz0O4GFP5QUfXdKNMUzyVvSwlvL9jhtQawUdzdblvQivXOW0nCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T07:23:09.999739Z"},"content_sha256":"0e74ec826f5de7b6c016a5c09a4196fb97b47721a39086a9006d8eba978a9ebb","schema_version":"1.0","event_id":"sha256:0e74ec826f5de7b6c016a5c09a4196fb97b47721a39086a9006d8eba978a9ebb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/R3DNZHJTXKEEUGFHLKJCKSO4HX/bundle.json","state_url":"https://pith.science/pith/R3DNZHJTXKEEUGFHLKJCKSO4HX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/R3DNZHJTXKEEUGFHLKJCKSO4HX/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":"2026-08-04T07:23:10Z","links":{"resolver":"https://pith.science/pith/R3DNZHJTXKEEUGFHLKJCKSO4HX","bundle":"https://pith.science/pith/R3DNZHJTXKEEUGFHLKJCKSO4HX/bundle.json","state":"https://pith.science/pith/R3DNZHJTXKEEUGFHLKJCKSO4HX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/R3DNZHJTXKEEUGFHLKJCKSO4HX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:R3DNZHJTXKEEUGFHLKJCKSO4HX","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":"ffb41876d99bdfcf270c0c0500fad57265615993d6d5f94ce112a0cc1ab4df14","cross_cats_sorted":[],"license":"","primary_cat":"cs.DS","submitted_at":"2007-02-05T19:29:38Z","title_canon_sha256":"cf55a7dce8792c939bafc409529d18cf346f4e62929ab34edf9a7486b084b84d"},"schema_version":"1.0","source":{"id":"cs/0702032","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"cs/0702032","created_at":"2026-07-04T14:59:14Z"},{"alias_kind":"arxiv_version","alias_value":"cs/0702032v1","created_at":"2026-07-04T14:59:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.cs/0702032","created_at":"2026-07-04T14:59:14Z"},{"alias_kind":"pith_short_12","alias_value":"R3DNZHJTXKEE","created_at":"2026-07-04T14:59:14Z"},{"alias_kind":"pith_short_16","alias_value":"R3DNZHJTXKEEUGFH","created_at":"2026-07-04T14:59:14Z"},{"alias_kind":"pith_short_8","alias_value":"R3DNZHJT","created_at":"2026-07-04T14:59:14Z"}],"graph_snapshots":[{"event_id":"sha256:0e74ec826f5de7b6c016a5c09a4196fb97b47721a39086a9006d8eba978a9ebb","target":"graph","created_at":"2026-07-04T14:59:14Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/cs/0702032/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider two optimization problems related to finding dense subgraphs. The densest at-least-k-subgraph problem (DalkS) is to find an induced subgraph of highest average degree among all subgraphs with at least k vertices, and the densest at-most-k-subgraph problem (DamkS) is defined similarly. These problems are related to the well-known densest k-subgraph problem (DkS), which is to find the densest subgraph on exactly k vertices. We show that DalkS can be approximated efficiently, while DamkS is nearly as hard to approximate as the densest k-subgraph problem.","authors_text":"Reid Andersen","cross_cats":[],"headline":"","license":"","primary_cat":"cs.DS","submitted_at":"2007-02-05T19:29:38Z","title":"Finding large and small dense subgraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"cs/0702032","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:b2ada978aa75873b8bd959746713bf7d60a5a6c0a94db6927967da2242c9095d","target":"record","created_at":"2026-07-04T14:59:14Z","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":"ffb41876d99bdfcf270c0c0500fad57265615993d6d5f94ce112a0cc1ab4df14","cross_cats_sorted":[],"license":"","primary_cat":"cs.DS","submitted_at":"2007-02-05T19:29:38Z","title_canon_sha256":"cf55a7dce8792c939bafc409529d18cf346f4e62929ab34edf9a7486b084b84d"},"schema_version":"1.0","source":{"id":"cs/0702032","kind":"arxiv","version":1}},"canonical_sha256":"8ec6dc9d33ba884a18a75a922549dc3dd70aa13b0f6e0288870338434c127789","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8ec6dc9d33ba884a18a75a922549dc3dd70aa13b0f6e0288870338434c127789","first_computed_at":"2026-07-04T14:59:14.551575Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:59:14.551575Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7w2JfNIYglqd9vBmreTRkOtsvpNvpGwMg5Y/PVLk8TjIQl7fvAg7JvG76DRapFfMI9/bNgvGMwa+nP5QIHZMCA==","signature_status":"signed_v1","signed_at":"2026-07-04T14:59:14.551919Z","signed_message":"canonical_sha256_bytes"},"source_id":"cs/0702032","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b2ada978aa75873b8bd959746713bf7d60a5a6c0a94db6927967da2242c9095d","sha256:0e74ec826f5de7b6c016a5c09a4196fb97b47721a39086a9006d8eba978a9ebb"],"state_sha256":"c0b6a488ade9d1d2c8845e33a333c21c0653ab3845c9d725754e386d53658958"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JjSeNFq+ACQqnZzT9pqwl0HBTkgx+0L4/hP9rIT9PfWZ5dTx8AIontm+aVcr4G+h15coPLMycv6nKnFPzyj8Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T07:23:10.005670Z","bundle_sha256":"2279844f99cb3c83d4d3b0305e3d3de0e41274f9121c3cdd7d5a73c5d94a119e"}}