{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:UQ2Q7K5P6BENPIVECL2WAJTMWR","short_pith_number":"pith:UQ2Q7K5P","canonical_record":{"source":{"id":"1910.07568","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2019-10-16T18:52:28Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"904f51cbe09c0746fa6073ca802ecbb6c9caf0bd95a993d8f66ac25f1a21c05c","abstract_canon_sha256":"12cf14c9444ecd37bd800c8b7f8a68246ded614e84350f9335f1a5f4bb9841db"},"schema_version":"1.0"},"canonical_sha256":"a4350fabaff048d7a2a412f560266cb477fd556b9b6a5f858af1d447351e1d77","source":{"kind":"arxiv","id":"1910.07568","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1910.07568","created_at":"2026-07-05T03:54:53Z"},{"alias_kind":"arxiv_version","alias_value":"1910.07568v3","created_at":"2026-07-05T03:54:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1910.07568","created_at":"2026-07-05T03:54:53Z"},{"alias_kind":"pith_short_12","alias_value":"UQ2Q7K5P6BEN","created_at":"2026-07-05T03:54:53Z"},{"alias_kind":"pith_short_16","alias_value":"UQ2Q7K5P6BENPIVE","created_at":"2026-07-05T03:54:53Z"},{"alias_kind":"pith_short_8","alias_value":"UQ2Q7K5P","created_at":"2026-07-05T03:54:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:UQ2Q7K5P6BENPIVECL2WAJTMWR","target":"record","payload":{"canonical_record":{"source":{"id":"1910.07568","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2019-10-16T18:52:28Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"904f51cbe09c0746fa6073ca802ecbb6c9caf0bd95a993d8f66ac25f1a21c05c","abstract_canon_sha256":"12cf14c9444ecd37bd800c8b7f8a68246ded614e84350f9335f1a5f4bb9841db"},"schema_version":"1.0"},"canonical_sha256":"a4350fabaff048d7a2a412f560266cb477fd556b9b6a5f858af1d447351e1d77","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:54:53.450100Z","signature_b64":"o+y6u6zCRns182k7k8boggxsrhElR99wk+ewu65EKN5e0I2/MeXHoobX8jG8Hv0lSOIlsFmfYyffw0pB7sBHBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a4350fabaff048d7a2a412f560266cb477fd556b9b6a5f858af1d447351e1d77","last_reissued_at":"2026-07-05T03:54:53.449691Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:54:53.449691Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1910.07568","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:54:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QlpN0yp2j/vUT30VjDaBAXV+N9qHvL7Zj6WJ/9yuIc9JPh7n6aZo0ssZ1JvXG+vkP2JAdlIMgZyuOQCl9CQhCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T09:23:54.214171Z"},"content_sha256":"3153bcf8a8828fa2772becb3cefbb2725c19eb1ac7a9ef0d108a7cdd6f229a76","schema_version":"1.0","event_id":"sha256:3153bcf8a8828fa2772becb3cefbb2725c19eb1ac7a9ef0d108a7cdd6f229a76"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:UQ2Q7K5P6BENPIVECL2WAJTMWR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Computational Complexity of Finding a Sparse Wasserstein Barycenter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.OC","authors_text":"Steffen Borgwardt, Stephan Patterson","submitted_at":"2019-10-16T18:52:28Z","abstract_excerpt":"The discrete Wasserstein barycenter problem is a minimum-cost mass transport problem for a set of probability measures with finite support. In this paper, we show that finding a barycenter of sparse support is hard, even in dimension 2 and for only 3 measures. We prove this claim by showing that a special case of an intimately related decision problem SCMP -- does there exist a measure with a non-mass-splitting transport cost and support size below prescribed bounds? -- is NP-hard for all rational data. Our proof is based on a reduction from planar 3-dimensional matching and follows a strategy"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.07568","kind":"arxiv","version":3},"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/1910.07568/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":"2026-07-05T03:54:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5GPnX/zfIeNHzM2afOIGIIJXnmwnWgdPQz+lxmrXCJ/rrhWTqQuKTXgM0HS9C6XA85iOV4QSckPQIRZGMZZNDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T09:23:54.215071Z"},"content_sha256":"5e490ccdbf67ddac74de1826ff10e7339e59f678ab2567463ab870d95310d827","schema_version":"1.0","event_id":"sha256:5e490ccdbf67ddac74de1826ff10e7339e59f678ab2567463ab870d95310d827"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UQ2Q7K5P6BENPIVECL2WAJTMWR/bundle.json","state_url":"https://pith.science/pith/UQ2Q7K5P6BENPIVECL2WAJTMWR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UQ2Q7K5P6BENPIVECL2WAJTMWR/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":"2026-08-11T09:23:54Z","links":{"resolver":"https://pith.science/pith/UQ2Q7K5P6BENPIVECL2WAJTMWR","bundle":"https://pith.science/pith/UQ2Q7K5P6BENPIVECL2WAJTMWR/bundle.json","state":"https://pith.science/pith/UQ2Q7K5P6BENPIVECL2WAJTMWR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UQ2Q7K5P6BENPIVECL2WAJTMWR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:UQ2Q7K5P6BENPIVECL2WAJTMWR","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":"12cf14c9444ecd37bd800c8b7f8a68246ded614e84350f9335f1a5f4bb9841db","cross_cats_sorted":["cs.CG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2019-10-16T18:52:28Z","title_canon_sha256":"904f51cbe09c0746fa6073ca802ecbb6c9caf0bd95a993d8f66ac25f1a21c05c"},"schema_version":"1.0","source":{"id":"1910.07568","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1910.07568","created_at":"2026-07-05T03:54:53Z"},{"alias_kind":"arxiv_version","alias_value":"1910.07568v3","created_at":"2026-07-05T03:54:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1910.07568","created_at":"2026-07-05T03:54:53Z"},{"alias_kind":"pith_short_12","alias_value":"UQ2Q7K5P6BEN","created_at":"2026-07-05T03:54:53Z"},{"alias_kind":"pith_short_16","alias_value":"UQ2Q7K5P6BENPIVE","created_at":"2026-07-05T03:54:53Z"},{"alias_kind":"pith_short_8","alias_value":"UQ2Q7K5P","created_at":"2026-07-05T03:54:53Z"}],"graph_snapshots":[{"event_id":"sha256:5e490ccdbf67ddac74de1826ff10e7339e59f678ab2567463ab870d95310d827","target":"graph","created_at":"2026-07-05T03:54:53Z","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/1910.07568/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The discrete Wasserstein barycenter problem is a minimum-cost mass transport problem for a set of probability measures with finite support. In this paper, we show that finding a barycenter of sparse support is hard, even in dimension 2 and for only 3 measures. We prove this claim by showing that a special case of an intimately related decision problem SCMP -- does there exist a measure with a non-mass-splitting transport cost and support size below prescribed bounds? -- is NP-hard for all rational data. Our proof is based on a reduction from planar 3-dimensional matching and follows a strategy","authors_text":"Steffen Borgwardt, Stephan Patterson","cross_cats":["cs.CG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2019-10-16T18:52:28Z","title":"On the Computational Complexity of Finding a Sparse Wasserstein Barycenter"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.07568","kind":"arxiv","version":3},"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":"sha256:3153bcf8a8828fa2772becb3cefbb2725c19eb1ac7a9ef0d108a7cdd6f229a76","target":"record","created_at":"2026-07-05T03:54:53Z","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":"12cf14c9444ecd37bd800c8b7f8a68246ded614e84350f9335f1a5f4bb9841db","cross_cats_sorted":["cs.CG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2019-10-16T18:52:28Z","title_canon_sha256":"904f51cbe09c0746fa6073ca802ecbb6c9caf0bd95a993d8f66ac25f1a21c05c"},"schema_version":"1.0","source":{"id":"1910.07568","kind":"arxiv","version":3}},"canonical_sha256":"a4350fabaff048d7a2a412f560266cb477fd556b9b6a5f858af1d447351e1d77","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a4350fabaff048d7a2a412f560266cb477fd556b9b6a5f858af1d447351e1d77","first_computed_at":"2026-07-05T03:54:53.449691Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:54:53.449691Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"o+y6u6zCRns182k7k8boggxsrhElR99wk+ewu65EKN5e0I2/MeXHoobX8jG8Hv0lSOIlsFmfYyffw0pB7sBHBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:54:53.450100Z","signed_message":"canonical_sha256_bytes"},"source_id":"1910.07568","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3153bcf8a8828fa2772becb3cefbb2725c19eb1ac7a9ef0d108a7cdd6f229a76","sha256:5e490ccdbf67ddac74de1826ff10e7339e59f678ab2567463ab870d95310d827"],"state_sha256":"782711ebd8cdcdb6ed5d8556262b1e3f91afa76132f14efddbe0a2d2f55e6c82"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jd5HwIgRclu80TY0GKolna8Jeo0UzVjHuTD2NpF3H2olD90JSRigxnhX32ZTMfSN8c7T00hHhJrCNojtzyAzDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T09:23:54.220051Z","bundle_sha256":"da6dcb3c5ba55d75e17266dfbf5f6aa0e4f2efd9902b3d82ed21813400ea122b"}}