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A black pebbling is a sequence $P_0,\\ldots, P_t \\subseteq V$ of sets of nodes which must satisfy the following properties: $P_0 = \\emptyset$ (we start off with no pebbles on $G$), $\\mathsf{sinks}(G) \\subseteq \\bigcup_{j \\leq t} P_j$ (every sink node was pebbled at some point) and $\\mathsf{parents}\\big(P_{i+1}\\backslash P_i\\big) \\subseteq P_i$ (we can only place a new pebble on a node $v$ if all of $v$'s parents had a pebble during the last round). The cumulative cost of a"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1609.04449","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CR","submitted_at":"2016-09-14T21:22:51Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"b25eba1807d2b564cce62a85c75c147d35839b1e170463f29cd5b92ef0cd9f44","abstract_canon_sha256":"670103053d8def470e195404f7c33ae3989d9c6521c163010756350352a137d5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:25:27.191750Z","signature_b64":"GjSqW/MRqe8KTWTD3buTsuLClH7SGi5BMOL95kw1itfFgE3y3GT5tRUqlh3qzbAmi+HK5aJA3JU2cqt2Ol7UDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"01c4d65d0be4beeb9f34ea78bb1a36090502772ae486f77c386fddfe41ee9cbc","last_reissued_at":"2026-05-18T00:25:27.191195Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:25:27.191195Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Computational Complexity of Minimal Cumulative Cost Graph Pebbling","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"cs.CR","authors_text":"Jeremiah Blocki, Samson Zhou","submitted_at":"2016-09-14T21:22:51Z","abstract_excerpt":"We consider the computational complexity of finding a legal black pebbling of a DAG $G=(V,E)$ with minimum cumulative cost. 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