{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:LBZH5DPS6UCDJWRW7SXWAWI57Q","short_pith_number":"pith:LBZH5DPS","canonical_record":{"source":{"id":"1102.1161","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2011-02-06T15:46:36Z","cross_cats_sorted":["cs.CC","cs.DS"],"title_canon_sha256":"3e58d01694b4d8564af802ff635b57672e27c186edf48ecba80cc208fd400bb7","abstract_canon_sha256":"5ccf617087b6df4dc59c7adce282e9fb7153e8e80630305f7a5b482e51288c7f"},"schema_version":"1.0"},"canonical_sha256":"58727e8df2f50434da36fcaf60591dfc284ac767afcf04932faa5cc9cd070cf0","source":{"kind":"arxiv","id":"1102.1161","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.1161","created_at":"2026-05-18T04:29:54Z"},{"alias_kind":"arxiv_version","alias_value":"1102.1161v1","created_at":"2026-05-18T04:29:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.1161","created_at":"2026-05-18T04:29:54Z"},{"alias_kind":"pith_short_12","alias_value":"LBZH5DPS6UCD","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"LBZH5DPS6UCDJWRW","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"LBZH5DPS","created_at":"2026-05-18T12:26:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:LBZH5DPS6UCDJWRW7SXWAWI57Q","target":"record","payload":{"canonical_record":{"source":{"id":"1102.1161","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2011-02-06T15:46:36Z","cross_cats_sorted":["cs.CC","cs.DS"],"title_canon_sha256":"3e58d01694b4d8564af802ff635b57672e27c186edf48ecba80cc208fd400bb7","abstract_canon_sha256":"5ccf617087b6df4dc59c7adce282e9fb7153e8e80630305f7a5b482e51288c7f"},"schema_version":"1.0"},"canonical_sha256":"58727e8df2f50434da36fcaf60591dfc284ac767afcf04932faa5cc9cd070cf0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:29:54.888207Z","signature_b64":"IwOD0zyguDfESMOIlTulb864aK3eZkDsNg55WJmhSauN1eRbKQC59CtQqYiMhHlL3jwdStIdny8uIQAhjo/tCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"58727e8df2f50434da36fcaf60591dfc284ac767afcf04932faa5cc9cd070cf0","last_reissued_at":"2026-05-18T04:29:54.887644Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:29:54.887644Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1102.1161","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":"2026-05-18T04:29:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rRXa9YUp7BPHKF6YFh4KRv8iEo4Bxiwpb3TAWGbP1DiTv33cHBWQKvD0UbJc24iUev1lyzCE9HS0ckSlSUJBCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T15:30:57.336686Z"},"content_sha256":"d568290ee82ccc9c5eb406ce1f498ff8ac8f3391fadeed7155a1a85e059421c3","schema_version":"1.0","event_id":"sha256:d568290ee82ccc9c5eb406ce1f498ff8ac8f3391fadeed7155a1a85e059421c3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:LBZH5DPS6UCDJWRW7SXWAWI57Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The complexity of approximate Nash equilibrium in congestion games with negative delays","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.DS"],"primary_cat":"cs.GT","authors_text":"Frederic Magniez, Michel de Rougemont, Miklos Santha, Xavier Zeitoun","submitted_at":"2011-02-06T15:46:36Z","abstract_excerpt":"We extend the study of the complexity of finding an $\\eps$-approximate Nash equilibrium in congestion games from the case of positive delay functions to delays of arbitrary sign. We first prove that in symmetric games with $\\alpha$-bounded jump the $\\eps$-Nash dynamic converges in polynomial time when all delay functions are negative, similarly to the case of positive delays. We then establish a hardness result for symmetric games with $\\alpha$-bounded jump and with arbitrary delay functions: in that case finding an $\\eps$-Nash equilibrium becomes $\\PLS$-complete."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.1161","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":""},"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":"2026-05-18T04:29:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yAXx5epCAUBFtt/zAHYREqW0HC3I6VgM0km4hFq8lqeMQ8I98vRulZnKKQ5oKN2iEQjnrooaz1sThw049smuDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T15:30:57.337356Z"},"content_sha256":"b2596d888572253ec66077d7469b58f8d7fd086e583e8e49633759880ab20e04","schema_version":"1.0","event_id":"sha256:b2596d888572253ec66077d7469b58f8d7fd086e583e8e49633759880ab20e04"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LBZH5DPS6UCDJWRW7SXWAWI57Q/bundle.json","state_url":"https://pith.science/pith/LBZH5DPS6UCDJWRW7SXWAWI57Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LBZH5DPS6UCDJWRW7SXWAWI57Q/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":"2026-06-07T15:30:57Z","links":{"resolver":"https://pith.science/pith/LBZH5DPS6UCDJWRW7SXWAWI57Q","bundle":"https://pith.science/pith/LBZH5DPS6UCDJWRW7SXWAWI57Q/bundle.json","state":"https://pith.science/pith/LBZH5DPS6UCDJWRW7SXWAWI57Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LBZH5DPS6UCDJWRW7SXWAWI57Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:LBZH5DPS6UCDJWRW7SXWAWI57Q","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":"5ccf617087b6df4dc59c7adce282e9fb7153e8e80630305f7a5b482e51288c7f","cross_cats_sorted":["cs.CC","cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2011-02-06T15:46:36Z","title_canon_sha256":"3e58d01694b4d8564af802ff635b57672e27c186edf48ecba80cc208fd400bb7"},"schema_version":"1.0","source":{"id":"1102.1161","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.1161","created_at":"2026-05-18T04:29:54Z"},{"alias_kind":"arxiv_version","alias_value":"1102.1161v1","created_at":"2026-05-18T04:29:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.1161","created_at":"2026-05-18T04:29:54Z"},{"alias_kind":"pith_short_12","alias_value":"LBZH5DPS6UCD","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"LBZH5DPS6UCDJWRW","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"LBZH5DPS","created_at":"2026-05-18T12:26:34Z"}],"graph_snapshots":[{"event_id":"sha256:b2596d888572253ec66077d7469b58f8d7fd086e583e8e49633759880ab20e04","target":"graph","created_at":"2026-05-18T04:29:54Z","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 extend the study of the complexity of finding an $\\eps$-approximate Nash equilibrium in congestion games from the case of positive delay functions to delays of arbitrary sign. We first prove that in symmetric games with $\\alpha$-bounded jump the $\\eps$-Nash dynamic converges in polynomial time when all delay functions are negative, similarly to the case of positive delays. We then establish a hardness result for symmetric games with $\\alpha$-bounded jump and with arbitrary delay functions: in that case finding an $\\eps$-Nash equilibrium becomes $\\PLS$-complete.","authors_text":"Frederic Magniez, Michel de Rougemont, Miklos Santha, Xavier Zeitoun","cross_cats":["cs.CC","cs.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2011-02-06T15:46:36Z","title":"The complexity of approximate Nash equilibrium in congestion games with negative delays"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.1161","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:d568290ee82ccc9c5eb406ce1f498ff8ac8f3391fadeed7155a1a85e059421c3","target":"record","created_at":"2026-05-18T04:29:54Z","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":"5ccf617087b6df4dc59c7adce282e9fb7153e8e80630305f7a5b482e51288c7f","cross_cats_sorted":["cs.CC","cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2011-02-06T15:46:36Z","title_canon_sha256":"3e58d01694b4d8564af802ff635b57672e27c186edf48ecba80cc208fd400bb7"},"schema_version":"1.0","source":{"id":"1102.1161","kind":"arxiv","version":1}},"canonical_sha256":"58727e8df2f50434da36fcaf60591dfc284ac767afcf04932faa5cc9cd070cf0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"58727e8df2f50434da36fcaf60591dfc284ac767afcf04932faa5cc9cd070cf0","first_computed_at":"2026-05-18T04:29:54.887644Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:29:54.887644Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IwOD0zyguDfESMOIlTulb864aK3eZkDsNg55WJmhSauN1eRbKQC59CtQqYiMhHlL3jwdStIdny8uIQAhjo/tCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:29:54.888207Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.1161","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d568290ee82ccc9c5eb406ce1f498ff8ac8f3391fadeed7155a1a85e059421c3","sha256:b2596d888572253ec66077d7469b58f8d7fd086e583e8e49633759880ab20e04"],"state_sha256":"0627f61ff1d9a8447cdaa2dca30a4a1df94fd9f11c5c46ba02c4a6b170d3ac85"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FKl6tcgE0UTxwhY1UaAhXn44Kwde7X/RMVPyk2sIoDGPoq+4WiKm6f531gvnNlNcF8lDaOhEc6A+2KSTI6woAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T15:30:57.340687Z","bundle_sha256":"c491a5e76ad7a87f6d39126f95374afca0266484342c40d1907869ff5aa4d20f"}}