{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:HXK2UM3E5RU374KWWU37J4GOY2","short_pith_number":"pith:HXK2UM3E","canonical_record":{"source":{"id":"1602.03985","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-02-12T09:14:23Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ba97d86373fabc911a2bb6a9ce28b4409f18c968f1d49801ed18fe27a1e4cb72","abstract_canon_sha256":"868517208012840936d767527f509beab64758a1b0aa85271c085af42d5e5e81"},"schema_version":"1.0"},"canonical_sha256":"3dd5aa3364ec69bff156b537f4f0cec692d68e607e701a2395298d6a1f4b6e52","source":{"kind":"arxiv","id":"1602.03985","version":6},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.03985","created_at":"2026-05-18T00:53:20Z"},{"alias_kind":"arxiv_version","alias_value":"1602.03985v6","created_at":"2026-05-18T00:53:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.03985","created_at":"2026-05-18T00:53:20Z"},{"alias_kind":"pith_short_12","alias_value":"HXK2UM3E5RU3","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"HXK2UM3E5RU374KW","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"HXK2UM3E","created_at":"2026-05-18T12:30:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:HXK2UM3E5RU374KWWU37J4GOY2","target":"record","payload":{"canonical_record":{"source":{"id":"1602.03985","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-02-12T09:14:23Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ba97d86373fabc911a2bb6a9ce28b4409f18c968f1d49801ed18fe27a1e4cb72","abstract_canon_sha256":"868517208012840936d767527f509beab64758a1b0aa85271c085af42d5e5e81"},"schema_version":"1.0"},"canonical_sha256":"3dd5aa3364ec69bff156b537f4f0cec692d68e607e701a2395298d6a1f4b6e52","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:20.996128Z","signature_b64":"bt1ihHteIhNSojdrQGAJxiMvCgesqF+G8tb65M1UPrbA06srHMbXDm1fPNGxZxW35CKmr3CNPqjiXw8rvXWKAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3dd5aa3364ec69bff156b537f4f0cec692d68e607e701a2395298d6a1f4b6e52","last_reissued_at":"2026-05-18T00:53:20.995710Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:20.995710Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.03985","source_version":6,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:53:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ImJitRG9xz8UEORXrpD6gsx9r17TcbkdU0IC4WzcbDfmyiXMXn2e8paJbpNXfKDwcqtBWD2hXfujhRvw+q11Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T16:54:12.363180Z"},"content_sha256":"57a4907fa09a81e4da381d78643f100bdc429c885f94888132573e5689b49524","schema_version":"1.0","event_id":"sha256:57a4907fa09a81e4da381d78643f100bdc429c885f94888132573e5689b49524"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:HXK2UM3E5RU374KWWU37J4GOY2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A complexity trichotomy for approximately counting list H-colourings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CC","authors_text":"Andreas Galanis, Leslie Ann Goldberg, Mark Jerrum","submitted_at":"2016-02-12T09:14:23Z","abstract_excerpt":"We examine the computational complexity of approximately counting the list H-colourings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H. If H is an irreflexive bipartite graph or a reflexive complete graph then counting list H-colourings is trivially in polynomial time. Otherwise, if H is an irreflexive bipartite permutation graph or a reflexive proper interval graph then approximately counting list H-colourings is equivalent to #BIS, the problem of approximately counting independent sets in a bipartite graph. This is a well-studied problem wh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.03985","kind":"arxiv","version":6},"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":"2026-05-18T00:53:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A4vi4QzTzm2v8uQbpeey4czXec3By9BHRe1t7sz3EVb+TA+2ykUIBiTaFNLUpmEwUYZn3eLZEB3Ex7bvAzcpBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T16:54:12.363542Z"},"content_sha256":"3057eb1fdc1334263aa0f230ebdae03e8606fc641cb72cc080c8ef2cd7be8566","schema_version":"1.0","event_id":"sha256:3057eb1fdc1334263aa0f230ebdae03e8606fc641cb72cc080c8ef2cd7be8566"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HXK2UM3E5RU374KWWU37J4GOY2/bundle.json","state_url":"https://pith.science/pith/HXK2UM3E5RU374KWWU37J4GOY2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HXK2UM3E5RU374KWWU37J4GOY2/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":"2026-06-28T16:54:12Z","links":{"resolver":"https://pith.science/pith/HXK2UM3E5RU374KWWU37J4GOY2","bundle":"https://pith.science/pith/HXK2UM3E5RU374KWWU37J4GOY2/bundle.json","state":"https://pith.science/pith/HXK2UM3E5RU374KWWU37J4GOY2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HXK2UM3E5RU374KWWU37J4GOY2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:HXK2UM3E5RU374KWWU37J4GOY2","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":"868517208012840936d767527f509beab64758a1b0aa85271c085af42d5e5e81","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-02-12T09:14:23Z","title_canon_sha256":"ba97d86373fabc911a2bb6a9ce28b4409f18c968f1d49801ed18fe27a1e4cb72"},"schema_version":"1.0","source":{"id":"1602.03985","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.03985","created_at":"2026-05-18T00:53:20Z"},{"alias_kind":"arxiv_version","alias_value":"1602.03985v6","created_at":"2026-05-18T00:53:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.03985","created_at":"2026-05-18T00:53:20Z"},{"alias_kind":"pith_short_12","alias_value":"HXK2UM3E5RU3","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"HXK2UM3E5RU374KW","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"HXK2UM3E","created_at":"2026-05-18T12:30:22Z"}],"graph_snapshots":[{"event_id":"sha256:3057eb1fdc1334263aa0f230ebdae03e8606fc641cb72cc080c8ef2cd7be8566","target":"graph","created_at":"2026-05-18T00:53:20Z","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 examine the computational complexity of approximately counting the list H-colourings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H. If H is an irreflexive bipartite graph or a reflexive complete graph then counting list H-colourings is trivially in polynomial time. Otherwise, if H is an irreflexive bipartite permutation graph or a reflexive proper interval graph then approximately counting list H-colourings is equivalent to #BIS, the problem of approximately counting independent sets in a bipartite graph. This is a well-studied problem wh","authors_text":"Andreas Galanis, Leslie Ann Goldberg, Mark Jerrum","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-02-12T09:14:23Z","title":"A complexity trichotomy for approximately counting list H-colourings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.03985","kind":"arxiv","version":6},"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":"sha256:57a4907fa09a81e4da381d78643f100bdc429c885f94888132573e5689b49524","target":"record","created_at":"2026-05-18T00:53:20Z","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":"868517208012840936d767527f509beab64758a1b0aa85271c085af42d5e5e81","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-02-12T09:14:23Z","title_canon_sha256":"ba97d86373fabc911a2bb6a9ce28b4409f18c968f1d49801ed18fe27a1e4cb72"},"schema_version":"1.0","source":{"id":"1602.03985","kind":"arxiv","version":6}},"canonical_sha256":"3dd5aa3364ec69bff156b537f4f0cec692d68e607e701a2395298d6a1f4b6e52","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3dd5aa3364ec69bff156b537f4f0cec692d68e607e701a2395298d6a1f4b6e52","first_computed_at":"2026-05-18T00:53:20.995710Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:20.995710Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bt1ihHteIhNSojdrQGAJxiMvCgesqF+G8tb65M1UPrbA06srHMbXDm1fPNGxZxW35CKmr3CNPqjiXw8rvXWKAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:20.996128Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.03985","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:57a4907fa09a81e4da381d78643f100bdc429c885f94888132573e5689b49524","sha256:3057eb1fdc1334263aa0f230ebdae03e8606fc641cb72cc080c8ef2cd7be8566"],"state_sha256":"97b945d82f0ec7506d505f0254be9b72faf05dda37c6d7dfa21a92d9da4eae8c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Fm9bNulnhdzEiLjwlbCxr9t355eYHHphksarM2irMro/k7/gJDuqBM99f/Ir6W3GC2lQAMxVRB69hhSEEjkXCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T16:54:12.365737Z","bundle_sha256":"0eeab17ce0fe7d6326b95170c853c9bb09bccdb3e9ac2e7a265e1c3472965b18"}}