{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:MQAQ6C2EPDDPKWKMYGKAEZPRIM","short_pith_number":"pith:MQAQ6C2E","canonical_record":{"source":{"id":"1509.05322","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2015-09-17T16:33:23Z","cross_cats_sorted":[],"title_canon_sha256":"15a5fdb40435436136de97c70a979684a108eae2978fb9bc896839edb1a35537","abstract_canon_sha256":"40e1c77fdf570bf3652924e41c9fdf120b0d496314ccd05bb34a5e95c007f12f"},"schema_version":"1.0"},"canonical_sha256":"64010f0b4478c6f5594cc1940265f14327f7f4e22d0534da818fa7c0cc6bc7af","source":{"kind":"arxiv","id":"1509.05322","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.05322","created_at":"2026-05-18T01:32:44Z"},{"alias_kind":"arxiv_version","alias_value":"1509.05322v1","created_at":"2026-05-18T01:32:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05322","created_at":"2026-05-18T01:32:44Z"},{"alias_kind":"pith_short_12","alias_value":"MQAQ6C2EPDDP","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"MQAQ6C2EPDDPKWKM","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"MQAQ6C2E","created_at":"2026-05-18T12:29:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:MQAQ6C2EPDDPKWKMYGKAEZPRIM","target":"record","payload":{"canonical_record":{"source":{"id":"1509.05322","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2015-09-17T16:33:23Z","cross_cats_sorted":[],"title_canon_sha256":"15a5fdb40435436136de97c70a979684a108eae2978fb9bc896839edb1a35537","abstract_canon_sha256":"40e1c77fdf570bf3652924e41c9fdf120b0d496314ccd05bb34a5e95c007f12f"},"schema_version":"1.0"},"canonical_sha256":"64010f0b4478c6f5594cc1940265f14327f7f4e22d0534da818fa7c0cc6bc7af","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:44.173616Z","signature_b64":"vY5N+tOESOLbgt4ASg5xCF4i5P4ebe5Jau+MSgfGP0Gtkcabd8bplUHvvq0ev0cPn5+Eu0VS6J91vBMb+aBvDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64010f0b4478c6f5594cc1940265f14327f7f4e22d0534da818fa7c0cc6bc7af","last_reissued_at":"2026-05-18T01:32:44.173136Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:44.173136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.05322","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":"2026-05-18T01:32:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M8B0C0EyxtEvwjwWCtVXkHJrRA+woruj3sQ4IcsTKg1zpCKr3SOVIOGLZ1UPLyJ1MwJndtSIOl4n3QMhPPBADg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T07:31:47.718284Z"},"content_sha256":"f7e3e3b3bb3c2f008e55bcf50147c711d0a67edbbe76210cb6831ce019a63e3e","schema_version":"1.0","event_id":"sha256:f7e3e3b3bb3c2f008e55bcf50147c711d0a67edbbe76210cb6831ce019a63e3e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:MQAQ6C2EPDDPKWKMYGKAEZPRIM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Computing stable outcomes in symmetric additively-separable hedonic games","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.GT","authors_text":"Martin Gairing, Rahul Savani","submitted_at":"2015-09-17T16:33:23Z","abstract_excerpt":"We study the computational complexity of finding stable outcomes in hedonic games, which are a class of coalition formation games. We restrict our attention to symmetric additively-separable hedonic games, which are a nontrivial subclass of such games that are guaranteed to possess stable outcomes. These games are specified by an undirected edge- weighted graph: nodes are players, an outcome of the game is a partition of the nodes into coalitions, and the utility of a node is the sum of incident edge weights in the same coalition. We consider several stability requirements defined in the liter"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05322","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":"2026-05-18T01:32:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"71nz0i3jQsxkH+lTxSkjOFHei9E8PYaIhIgh7MUwNGgZ7fre5GViKP1JiDBWFDjcYZoCM1wyO2eW8FdYwmmfBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T07:31:47.718972Z"},"content_sha256":"8899500fa228f38800ec790ddecb0c2ffe2ee7d7b694b89f7705fe4de37910db","schema_version":"1.0","event_id":"sha256:8899500fa228f38800ec790ddecb0c2ffe2ee7d7b694b89f7705fe4de37910db"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MQAQ6C2EPDDPKWKMYGKAEZPRIM/bundle.json","state_url":"https://pith.science/pith/MQAQ6C2EPDDPKWKMYGKAEZPRIM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MQAQ6C2EPDDPKWKMYGKAEZPRIM/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":"2026-06-07T07:31:47Z","links":{"resolver":"https://pith.science/pith/MQAQ6C2EPDDPKWKMYGKAEZPRIM","bundle":"https://pith.science/pith/MQAQ6C2EPDDPKWKMYGKAEZPRIM/bundle.json","state":"https://pith.science/pith/MQAQ6C2EPDDPKWKMYGKAEZPRIM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MQAQ6C2EPDDPKWKMYGKAEZPRIM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:MQAQ6C2EPDDPKWKMYGKAEZPRIM","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":"40e1c77fdf570bf3652924e41c9fdf120b0d496314ccd05bb34a5e95c007f12f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2015-09-17T16:33:23Z","title_canon_sha256":"15a5fdb40435436136de97c70a979684a108eae2978fb9bc896839edb1a35537"},"schema_version":"1.0","source":{"id":"1509.05322","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.05322","created_at":"2026-05-18T01:32:44Z"},{"alias_kind":"arxiv_version","alias_value":"1509.05322v1","created_at":"2026-05-18T01:32:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05322","created_at":"2026-05-18T01:32:44Z"},{"alias_kind":"pith_short_12","alias_value":"MQAQ6C2EPDDP","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"MQAQ6C2EPDDPKWKM","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"MQAQ6C2E","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:8899500fa228f38800ec790ddecb0c2ffe2ee7d7b694b89f7705fe4de37910db","target":"graph","created_at":"2026-05-18T01:32:44Z","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 study the computational complexity of finding stable outcomes in hedonic games, which are a class of coalition formation games. We restrict our attention to symmetric additively-separable hedonic games, which are a nontrivial subclass of such games that are guaranteed to possess stable outcomes. These games are specified by an undirected edge- weighted graph: nodes are players, an outcome of the game is a partition of the nodes into coalitions, and the utility of a node is the sum of incident edge weights in the same coalition. We consider several stability requirements defined in the liter","authors_text":"Martin Gairing, Rahul Savani","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2015-09-17T16:33:23Z","title":"Computing stable outcomes in symmetric additively-separable hedonic games"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05322","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:f7e3e3b3bb3c2f008e55bcf50147c711d0a67edbbe76210cb6831ce019a63e3e","target":"record","created_at":"2026-05-18T01:32:44Z","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":"40e1c77fdf570bf3652924e41c9fdf120b0d496314ccd05bb34a5e95c007f12f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2015-09-17T16:33:23Z","title_canon_sha256":"15a5fdb40435436136de97c70a979684a108eae2978fb9bc896839edb1a35537"},"schema_version":"1.0","source":{"id":"1509.05322","kind":"arxiv","version":1}},"canonical_sha256":"64010f0b4478c6f5594cc1940265f14327f7f4e22d0534da818fa7c0cc6bc7af","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"64010f0b4478c6f5594cc1940265f14327f7f4e22d0534da818fa7c0cc6bc7af","first_computed_at":"2026-05-18T01:32:44.173136Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:32:44.173136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vY5N+tOESOLbgt4ASg5xCF4i5P4ebe5Jau+MSgfGP0Gtkcabd8bplUHvvq0ev0cPn5+Eu0VS6J91vBMb+aBvDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:32:44.173616Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.05322","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f7e3e3b3bb3c2f008e55bcf50147c711d0a67edbbe76210cb6831ce019a63e3e","sha256:8899500fa228f38800ec790ddecb0c2ffe2ee7d7b694b89f7705fe4de37910db"],"state_sha256":"c2af9565bae5b3d9d6370e17bb733c688fbe3b5947abd579d4ec06a199b1a78c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tnAz0BZ4lZ+zjIBR+nvHprRw817/Ua9kKa4MUBJEmvU9qfj5wICpFP/egHANUPn/qBKPZbM+cQLM25eIc3vFCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T07:31:47.722702Z","bundle_sha256":"f57b9146fa2b8de8dd728efaa30c64268afba31fde37305a19068326a9e2b88b"}}