{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:XL2MVCRXGJOOMUCIGBKPVD7KCW","short_pith_number":"pith:XL2MVCRX","canonical_record":{"source":{"id":"2106.02353","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2021-06-04T09:06:35Z","cross_cats_sorted":[],"title_canon_sha256":"1de576a98296e4453820dec83ca1ea64680a0b80d5b03593fa844917e3faa5b5","abstract_canon_sha256":"8e4667db33c5261051e9df4c34d6485f04090995f3c1019f9296b9a4bfaf34c7"},"schema_version":"1.0"},"canonical_sha256":"baf4ca8a37325ce650483054fa8fea15af759ae9105f859cf6ef4ff323339d9d","source":{"kind":"arxiv","id":"2106.02353","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.02353","created_at":"2026-07-05T02:46:18Z"},{"alias_kind":"arxiv_version","alias_value":"2106.02353v1","created_at":"2026-07-05T02:46:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.02353","created_at":"2026-07-05T02:46:18Z"},{"alias_kind":"pith_short_12","alias_value":"XL2MVCRXGJOO","created_at":"2026-07-05T02:46:18Z"},{"alias_kind":"pith_short_16","alias_value":"XL2MVCRXGJOOMUCI","created_at":"2026-07-05T02:46:18Z"},{"alias_kind":"pith_short_8","alias_value":"XL2MVCRX","created_at":"2026-07-05T02:46:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:XL2MVCRXGJOOMUCIGBKPVD7KCW","target":"record","payload":{"canonical_record":{"source":{"id":"2106.02353","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2021-06-04T09:06:35Z","cross_cats_sorted":[],"title_canon_sha256":"1de576a98296e4453820dec83ca1ea64680a0b80d5b03593fa844917e3faa5b5","abstract_canon_sha256":"8e4667db33c5261051e9df4c34d6485f04090995f3c1019f9296b9a4bfaf34c7"},"schema_version":"1.0"},"canonical_sha256":"baf4ca8a37325ce650483054fa8fea15af759ae9105f859cf6ef4ff323339d9d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:46:18.491383Z","signature_b64":"KjlJDHvCXcx84P7gkDOwebcdiDXswxSTRUh+LRfK8+iQ0GGmeIWSZS//CV2fkskUBJY/Bh0RGqJJAdkwz8uxAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"baf4ca8a37325ce650483054fa8fea15af759ae9105f859cf6ef4ff323339d9d","last_reissued_at":"2026-07-05T02:46:18.490991Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:46:18.490991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2106.02353","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":"2026-07-05T02:46:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FFb6pS1usUwJLsngQAyvhyxABwS/M2azc3nvBSHCjZQ3tNb+tHf+3c/MW4eHYBjgJY/lkN4fDh0snPp0VeA2Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T08:19:24.984560Z"},"content_sha256":"9481ed65dbcf034700f4a89f8b414582e8e7ec85db370d86ac9097350a002fa7","schema_version":"1.0","event_id":"sha256:9481ed65dbcf034700f4a89f8b414582e8e7ec85db370d86ac9097350a002fa7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:XL2MVCRXGJOOMUCIGBKPVD7KCW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectral Hypergraph Sparsifiers of Nearly Linear Size","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Jakab Tardos, Michael Kapralov, Robert Krauthgamer, Yuichi Yoshida","submitted_at":"2021-06-04T09:06:35Z","abstract_excerpt":"Graph sparsification has been studied extensively over the past two decades, culminating in spectral sparsifiers of optimal size (up to constant factors). Spectral hypergraph sparsification is a natural analogue of this problem, for which optimal bounds on the sparsifier size are not known, mainly because the hypergraph Laplacian is non-linear, and thus lacks the linear-algebraic structure and tools that have been so effective for graphs.\n  Our main contribution is the first algorithm for constructing $\\epsilon$-spectral sparsifiers for hypergraphs with $O^*(n)$ hyperedges, where $O^*$ suppres"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.02353","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/2106.02353/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":"2026-07-05T02:46:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X5kc+/2fhgkTMQ8gH1BSULuzBgSygvN5iPP73G2dlz3S8K8+H74HkijOUqewByru4HMN/bdgAvjfAI5BPT3lCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T08:19:24.985050Z"},"content_sha256":"9baeb1d9bc127774923ff63608058b26598786d59643ec46e8cab26db62c4b51","schema_version":"1.0","event_id":"sha256:9baeb1d9bc127774923ff63608058b26598786d59643ec46e8cab26db62c4b51"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XL2MVCRXGJOOMUCIGBKPVD7KCW/bundle.json","state_url":"https://pith.science/pith/XL2MVCRXGJOOMUCIGBKPVD7KCW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XL2MVCRXGJOOMUCIGBKPVD7KCW/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":"2026-08-23T08:19:24Z","links":{"resolver":"https://pith.science/pith/XL2MVCRXGJOOMUCIGBKPVD7KCW","bundle":"https://pith.science/pith/XL2MVCRXGJOOMUCIGBKPVD7KCW/bundle.json","state":"https://pith.science/pith/XL2MVCRXGJOOMUCIGBKPVD7KCW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XL2MVCRXGJOOMUCIGBKPVD7KCW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:XL2MVCRXGJOOMUCIGBKPVD7KCW","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":"8e4667db33c5261051e9df4c34d6485f04090995f3c1019f9296b9a4bfaf34c7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2021-06-04T09:06:35Z","title_canon_sha256":"1de576a98296e4453820dec83ca1ea64680a0b80d5b03593fa844917e3faa5b5"},"schema_version":"1.0","source":{"id":"2106.02353","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.02353","created_at":"2026-07-05T02:46:18Z"},{"alias_kind":"arxiv_version","alias_value":"2106.02353v1","created_at":"2026-07-05T02:46:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.02353","created_at":"2026-07-05T02:46:18Z"},{"alias_kind":"pith_short_12","alias_value":"XL2MVCRXGJOO","created_at":"2026-07-05T02:46:18Z"},{"alias_kind":"pith_short_16","alias_value":"XL2MVCRXGJOOMUCI","created_at":"2026-07-05T02:46:18Z"},{"alias_kind":"pith_short_8","alias_value":"XL2MVCRX","created_at":"2026-07-05T02:46:18Z"}],"graph_snapshots":[{"event_id":"sha256:9baeb1d9bc127774923ff63608058b26598786d59643ec46e8cab26db62c4b51","target":"graph","created_at":"2026-07-05T02:46:18Z","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/2106.02353/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Graph sparsification has been studied extensively over the past two decades, culminating in spectral sparsifiers of optimal size (up to constant factors). Spectral hypergraph sparsification is a natural analogue of this problem, for which optimal bounds on the sparsifier size are not known, mainly because the hypergraph Laplacian is non-linear, and thus lacks the linear-algebraic structure and tools that have been so effective for graphs.\n  Our main contribution is the first algorithm for constructing $\\epsilon$-spectral sparsifiers for hypergraphs with $O^*(n)$ hyperedges, where $O^*$ suppres","authors_text":"Jakab Tardos, Michael Kapralov, Robert Krauthgamer, Yuichi Yoshida","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2021-06-04T09:06:35Z","title":"Spectral Hypergraph Sparsifiers of Nearly Linear Size"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.02353","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":"sha256:9481ed65dbcf034700f4a89f8b414582e8e7ec85db370d86ac9097350a002fa7","target":"record","created_at":"2026-07-05T02:46:18Z","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":"8e4667db33c5261051e9df4c34d6485f04090995f3c1019f9296b9a4bfaf34c7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2021-06-04T09:06:35Z","title_canon_sha256":"1de576a98296e4453820dec83ca1ea64680a0b80d5b03593fa844917e3faa5b5"},"schema_version":"1.0","source":{"id":"2106.02353","kind":"arxiv","version":1}},"canonical_sha256":"baf4ca8a37325ce650483054fa8fea15af759ae9105f859cf6ef4ff323339d9d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"baf4ca8a37325ce650483054fa8fea15af759ae9105f859cf6ef4ff323339d9d","first_computed_at":"2026-07-05T02:46:18.490991Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:46:18.490991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KjlJDHvCXcx84P7gkDOwebcdiDXswxSTRUh+LRfK8+iQ0GGmeIWSZS//CV2fkskUBJY/Bh0RGqJJAdkwz8uxAA==","signature_status":"signed_v1","signed_at":"2026-07-05T02:46:18.491383Z","signed_message":"canonical_sha256_bytes"},"source_id":"2106.02353","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9481ed65dbcf034700f4a89f8b414582e8e7ec85db370d86ac9097350a002fa7","sha256:9baeb1d9bc127774923ff63608058b26598786d59643ec46e8cab26db62c4b51"],"state_sha256":"1d943b78443557ba553cfde881f1cbae0681714db364760b2da70a59475a5a23"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e8o45R0NHOOnVChZJChEHOpFYn5ZwmQmySLOsJ/tHMkO0KhX0hWwHpL1Ev2O4mKmLEEQ7q4BVn5QFn6MU8jjAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-23T08:19:24.989498Z","bundle_sha256":"55e453f48b212957529964a3ebc3f4425759c44b3910e11f67c0552ee852215f"}}