{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:6RALGGXQNMEP3R74NRDHXBX5OP","short_pith_number":"pith:6RALGGXQ","canonical_record":{"source":{"id":"0907.2663","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2009-07-15T17:25:43Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"15b6de69ee91f4a841cec644356c61734ad9cf78afb93edd1497e3a830e3a824","abstract_canon_sha256":"7b6937c1d9c6807d02d6dc7557dcb581dccb161a6100584ee7d196487826b627"},"schema_version":"1.0"},"canonical_sha256":"f440b31af06b08fdc7fc6c467b86fd73d351161bf482fc1c7e270c08434af34d","source":{"kind":"arxiv","id":"0907.2663","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.2663","created_at":"2026-05-18T04:12:56Z"},{"alias_kind":"arxiv_version","alias_value":"0907.2663v2","created_at":"2026-05-18T04:12:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.2663","created_at":"2026-05-18T04:12:56Z"},{"alias_kind":"pith_short_12","alias_value":"6RALGGXQNMEP","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"6RALGGXQNMEP3R74","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"6RALGGXQ","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:6RALGGXQNMEP3R74NRDHXBX5OP","target":"record","payload":{"canonical_record":{"source":{"id":"0907.2663","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2009-07-15T17:25:43Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"15b6de69ee91f4a841cec644356c61734ad9cf78afb93edd1497e3a830e3a824","abstract_canon_sha256":"7b6937c1d9c6807d02d6dc7557dcb581dccb161a6100584ee7d196487826b627"},"schema_version":"1.0"},"canonical_sha256":"f440b31af06b08fdc7fc6c467b86fd73d351161bf482fc1c7e270c08434af34d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:56.560503Z","signature_b64":"azkxviHdDKRd7lZx4g6xeuRVid0diuQ4uizhfptu9M9vW9AyYHsrLsCiCbs+d68+psg8OG/gS0k7Tn5gcQoGCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f440b31af06b08fdc7fc6c467b86fd73d351161bf482fc1c7e270c08434af34d","last_reissued_at":"2026-05-18T04:12:56.559988Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:56.559988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0907.2663","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":"2026-05-18T04:12:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"evn+O7uDI+e4UQa4qxAnpGgqEKRDSCDUOt7pLHOVXeotrtFl2erMRsOFk9EizMfYjbx27dH8NUTJaEMc3ZVTBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T03:45:03.769363Z"},"content_sha256":"d3ad6cbb01073e590cb1e6a55473127579e493dd251f454d954f9548715f8ec1","schema_version":"1.0","event_id":"sha256:d3ad6cbb01073e590cb1e6a55473127579e493dd251f454d954f9548715f8ec1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:6RALGGXQNMEP3R74NRDHXBX5OP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Complexity of Approximating Bounded-Degree Boolean \\sharp CSP","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"cs.CC","authors_text":"David Richerby, Leslie Ann Goldberg, Markus Jalsenius, Martin Dyer","submitted_at":"2009-07-15T17:25:43Z","abstract_excerpt":"The degree of a CSP instance is the maximum number of times that any variable appears in the scopes of constraints. We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum allowed degree is large enough (at least 6) we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial-time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.2663","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":"2026-05-18T04:12:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sIT4VVRycaWdamlO5FpCEGAZvczTzKLyg0lw+baHxBRV18CjpY4LVQR02gOt4qYPoqhz3Cx859qebPp4pj20Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T03:45:03.769720Z"},"content_sha256":"b521d3d872a90b90baecf85a5752259ac7cc5d656e7ad966377180ed9e1c8bf7","schema_version":"1.0","event_id":"sha256:b521d3d872a90b90baecf85a5752259ac7cc5d656e7ad966377180ed9e1c8bf7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6RALGGXQNMEP3R74NRDHXBX5OP/bundle.json","state_url":"https://pith.science/pith/6RALGGXQNMEP3R74NRDHXBX5OP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6RALGGXQNMEP3R74NRDHXBX5OP/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":"2026-05-31T03:45:03Z","links":{"resolver":"https://pith.science/pith/6RALGGXQNMEP3R74NRDHXBX5OP","bundle":"https://pith.science/pith/6RALGGXQNMEP3R74NRDHXBX5OP/bundle.json","state":"https://pith.science/pith/6RALGGXQNMEP3R74NRDHXBX5OP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6RALGGXQNMEP3R74NRDHXBX5OP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:6RALGGXQNMEP3R74NRDHXBX5OP","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":"7b6937c1d9c6807d02d6dc7557dcb581dccb161a6100584ee7d196487826b627","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2009-07-15T17:25:43Z","title_canon_sha256":"15b6de69ee91f4a841cec644356c61734ad9cf78afb93edd1497e3a830e3a824"},"schema_version":"1.0","source":{"id":"0907.2663","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.2663","created_at":"2026-05-18T04:12:56Z"},{"alias_kind":"arxiv_version","alias_value":"0907.2663v2","created_at":"2026-05-18T04:12:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.2663","created_at":"2026-05-18T04:12:56Z"},{"alias_kind":"pith_short_12","alias_value":"6RALGGXQNMEP","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"6RALGGXQNMEP3R74","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"6RALGGXQ","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:b521d3d872a90b90baecf85a5752259ac7cc5d656e7ad966377180ed9e1c8bf7","target":"graph","created_at":"2026-05-18T04:12:56Z","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":"The degree of a CSP instance is the maximum number of times that any variable appears in the scopes of constraints. We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum allowed degree is large enough (at least 6) we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial-time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bi","authors_text":"David Richerby, Leslie Ann Goldberg, Markus Jalsenius, Martin Dyer","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2009-07-15T17:25:43Z","title":"The Complexity of Approximating Bounded-Degree Boolean \\sharp CSP"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.2663","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:d3ad6cbb01073e590cb1e6a55473127579e493dd251f454d954f9548715f8ec1","target":"record","created_at":"2026-05-18T04:12:56Z","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":"7b6937c1d9c6807d02d6dc7557dcb581dccb161a6100584ee7d196487826b627","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2009-07-15T17:25:43Z","title_canon_sha256":"15b6de69ee91f4a841cec644356c61734ad9cf78afb93edd1497e3a830e3a824"},"schema_version":"1.0","source":{"id":"0907.2663","kind":"arxiv","version":2}},"canonical_sha256":"f440b31af06b08fdc7fc6c467b86fd73d351161bf482fc1c7e270c08434af34d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f440b31af06b08fdc7fc6c467b86fd73d351161bf482fc1c7e270c08434af34d","first_computed_at":"2026-05-18T04:12:56.559988Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:12:56.559988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"azkxviHdDKRd7lZx4g6xeuRVid0diuQ4uizhfptu9M9vW9AyYHsrLsCiCbs+d68+psg8OG/gS0k7Tn5gcQoGCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:12:56.560503Z","signed_message":"canonical_sha256_bytes"},"source_id":"0907.2663","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d3ad6cbb01073e590cb1e6a55473127579e493dd251f454d954f9548715f8ec1","sha256:b521d3d872a90b90baecf85a5752259ac7cc5d656e7ad966377180ed9e1c8bf7"],"state_sha256":"9e61e74d59c72036541b0fb5489e2c26d55d596a2cf5f8f5fcc05dd04a4f33ff"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TPIEL/habNR+snV+MWRzV3lO8RtW01gk6dhOrvlgQqdHtpruLi9P1hleD/p+x6GwXkyIPbAIKihwBRHM6dGxCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T03:45:03.771700Z","bundle_sha256":"9648d8b89b087329b9500daee030094c665a388f973f59e0df034c06a860278d"}}