{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:PBPKDZINNSU4A35STV7V2BYKOY","short_pith_number":"pith:PBPKDZIN","canonical_record":{"source":{"id":"1502.01335","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-02-04T20:57:56Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"cd95f87dc0a8af6577e158921d29562c82f147ac9724f0ce5455f19e57ae58db","abstract_canon_sha256":"619552ee3788ba3816f65386b64c53d6855285ef4d20a9348e69670ba18f6c8a"},"schema_version":"1.0"},"canonical_sha256":"785ea1e50d6ca9c06fb29d7f5d070a7621b1ca81595fdf474c9e26f286ceec66","source":{"kind":"arxiv","id":"1502.01335","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.01335","created_at":"2026-05-18T01:13:31Z"},{"alias_kind":"arxiv_version","alias_value":"1502.01335v2","created_at":"2026-05-18T01:13:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.01335","created_at":"2026-05-18T01:13:31Z"},{"alias_kind":"pith_short_12","alias_value":"PBPKDZINNSU4","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PBPKDZINNSU4A35S","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PBPKDZIN","created_at":"2026-05-18T12:29:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:PBPKDZINNSU4A35STV7V2BYKOY","target":"record","payload":{"canonical_record":{"source":{"id":"1502.01335","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-02-04T20:57:56Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"cd95f87dc0a8af6577e158921d29562c82f147ac9724f0ce5455f19e57ae58db","abstract_canon_sha256":"619552ee3788ba3816f65386b64c53d6855285ef4d20a9348e69670ba18f6c8a"},"schema_version":"1.0"},"canonical_sha256":"785ea1e50d6ca9c06fb29d7f5d070a7621b1ca81595fdf474c9e26f286ceec66","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:13:31.068221Z","signature_b64":"FU7X1wKSZvEuMh/zClVLZHKuE2mrXY+bcq5Qjmjz+k8Iuop4ITAIIGLiPvO7fz/TGaGTRxRrJhWPV9b6VIdvBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"785ea1e50d6ca9c06fb29d7f5d070a7621b1ca81595fdf474c9e26f286ceec66","last_reissued_at":"2026-05-18T01:13:31.067492Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:13:31.067492Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1502.01335","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":"2026-05-18T01:13:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/JDrcbdx5sPc/6vfri/lMbG8YCc+A9WAO1NvB//2+G8EERClbBa+a8ebNhIx3hOiKFuexNeV/iUoYR28ULxtAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T09:35:15.456720Z"},"content_sha256":"5ce456316f6e7dd888e541e852e5ea7857e6d4516476330ac7d5fa1570b48bb6","schema_version":"1.0","event_id":"sha256:5ce456316f6e7dd888e541e852e5ea7857e6d4516476330ac7d5fa1570b48bb6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:PBPKDZINNSU4A35STV7V2BYKOY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximately Counting H-Colourings is #BIS-Hard","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"cs.CC","authors_text":"Andreas Galanis, Leslie Ann Goldberg, Mark Jerrum","submitted_at":"2015-02-04T20:57:56Z","abstract_excerpt":"We consider the problem of counting H-colourings from an input graph G to a target graph H. We show that if H is any fixed graph without trivial components, then the problem is as hard as the well-known problem #BIS, which is the problem of (approximately) counting independent sets in a bipartite graph. #BIS is a complete problem in an important complexity class for approximate counting, and is believed not to have an FPRAS. If this is so, then our result shows that for every graph H without trivial components, the H-colouring counting problem has no FPRAS. This problem was studied a decade ag"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01335","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":"2026-05-18T01:13:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lW6k/97IiKUGUJED0jL3dJimzRBKVLi+ELVwJUzBBloUSMZPHXj/ulMgX+MAw/GWGdGS0KCPM8Gm1v2G7zrhCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T09:35:15.457083Z"},"content_sha256":"3383b4bbc6a6c94d9585be13dbd294ba8888a8d9d9e9a8dc73bd56a59946e6d1","schema_version":"1.0","event_id":"sha256:3383b4bbc6a6c94d9585be13dbd294ba8888a8d9d9e9a8dc73bd56a59946e6d1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PBPKDZINNSU4A35STV7V2BYKOY/bundle.json","state_url":"https://pith.science/pith/PBPKDZINNSU4A35STV7V2BYKOY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PBPKDZINNSU4A35STV7V2BYKOY/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":"2026-06-12T09:35:15Z","links":{"resolver":"https://pith.science/pith/PBPKDZINNSU4A35STV7V2BYKOY","bundle":"https://pith.science/pith/PBPKDZINNSU4A35STV7V2BYKOY/bundle.json","state":"https://pith.science/pith/PBPKDZINNSU4A35STV7V2BYKOY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PBPKDZINNSU4A35STV7V2BYKOY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:PBPKDZINNSU4A35STV7V2BYKOY","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":"619552ee3788ba3816f65386b64c53d6855285ef4d20a9348e69670ba18f6c8a","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-02-04T20:57:56Z","title_canon_sha256":"cd95f87dc0a8af6577e158921d29562c82f147ac9724f0ce5455f19e57ae58db"},"schema_version":"1.0","source":{"id":"1502.01335","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.01335","created_at":"2026-05-18T01:13:31Z"},{"alias_kind":"arxiv_version","alias_value":"1502.01335v2","created_at":"2026-05-18T01:13:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.01335","created_at":"2026-05-18T01:13:31Z"},{"alias_kind":"pith_short_12","alias_value":"PBPKDZINNSU4","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PBPKDZINNSU4A35S","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PBPKDZIN","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:3383b4bbc6a6c94d9585be13dbd294ba8888a8d9d9e9a8dc73bd56a59946e6d1","target":"graph","created_at":"2026-05-18T01:13:31Z","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 consider the problem of counting H-colourings from an input graph G to a target graph H. We show that if H is any fixed graph without trivial components, then the problem is as hard as the well-known problem #BIS, which is the problem of (approximately) counting independent sets in a bipartite graph. #BIS is a complete problem in an important complexity class for approximate counting, and is believed not to have an FPRAS. If this is so, then our result shows that for every graph H without trivial components, the H-colouring counting problem has no FPRAS. This problem was studied a decade ag","authors_text":"Andreas Galanis, Leslie Ann Goldberg, Mark Jerrum","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-02-04T20:57:56Z","title":"Approximately Counting H-Colourings is #BIS-Hard"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01335","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":"sha256:5ce456316f6e7dd888e541e852e5ea7857e6d4516476330ac7d5fa1570b48bb6","target":"record","created_at":"2026-05-18T01:13:31Z","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":"619552ee3788ba3816f65386b64c53d6855285ef4d20a9348e69670ba18f6c8a","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-02-04T20:57:56Z","title_canon_sha256":"cd95f87dc0a8af6577e158921d29562c82f147ac9724f0ce5455f19e57ae58db"},"schema_version":"1.0","source":{"id":"1502.01335","kind":"arxiv","version":2}},"canonical_sha256":"785ea1e50d6ca9c06fb29d7f5d070a7621b1ca81595fdf474c9e26f286ceec66","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"785ea1e50d6ca9c06fb29d7f5d070a7621b1ca81595fdf474c9e26f286ceec66","first_computed_at":"2026-05-18T01:13:31.067492Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:13:31.067492Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FU7X1wKSZvEuMh/zClVLZHKuE2mrXY+bcq5Qjmjz+k8Iuop4ITAIIGLiPvO7fz/TGaGTRxRrJhWPV9b6VIdvBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:13:31.068221Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.01335","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5ce456316f6e7dd888e541e852e5ea7857e6d4516476330ac7d5fa1570b48bb6","sha256:3383b4bbc6a6c94d9585be13dbd294ba8888a8d9d9e9a8dc73bd56a59946e6d1"],"state_sha256":"0930346656ea9fd9805fd64ff721402a94ce09c05bc86968fd8730274b5e9305"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7j/livk/VCaCyFsEaSUjlwemmGpdTeC3y74kkbu2fcVtdIF3tjUtxd6m7Fk3Ughi6RC8KhyOWW+3ExvsFIDPAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T09:35:15.459141Z","bundle_sha256":"205879caa8c03308a13a7c672fc352c66982d0d6a762a1b81e84c8dc748d3ac5"}}