{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:ESO4S4ZTCYVA2BUHAE6XLFFOG7","short_pith_number":"pith:ESO4S4ZT","canonical_record":{"source":{"id":"1207.5215","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-07-22T11:01:33Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"e1e402a532ec6ee833ea159d5952b29a7278576bf77b139236707cf37e9d47ae","abstract_canon_sha256":"8ee2e84070e368761faa1e56de37adcec732817796b2dd5470d354a0a7642914"},"schema_version":"1.0"},"canonical_sha256":"249dc97333162a0d0687013d7594ae37c787e35e4cba26ad4b719c819816ee0d","source":{"kind":"arxiv","id":"1207.5215","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.5215","created_at":"2026-05-18T03:49:57Z"},{"alias_kind":"arxiv_version","alias_value":"1207.5215v2","created_at":"2026-05-18T03:49:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.5215","created_at":"2026-05-18T03:49:57Z"},{"alias_kind":"pith_short_12","alias_value":"ESO4S4ZTCYVA","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"ESO4S4ZTCYVA2BUH","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"ESO4S4ZT","created_at":"2026-05-18T12:27:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:ESO4S4ZTCYVA2BUHAE6XLFFOG7","target":"record","payload":{"canonical_record":{"source":{"id":"1207.5215","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-07-22T11:01:33Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"e1e402a532ec6ee833ea159d5952b29a7278576bf77b139236707cf37e9d47ae","abstract_canon_sha256":"8ee2e84070e368761faa1e56de37adcec732817796b2dd5470d354a0a7642914"},"schema_version":"1.0"},"canonical_sha256":"249dc97333162a0d0687013d7594ae37c787e35e4cba26ad4b719c819816ee0d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:57.694586Z","signature_b64":"TLbbopc3QAdPiHs55Fl/FKnxr9l9ZLNUKi7QHVhXacPjH+Ux/e65ZudiLavg7DVktqhS4gRZb5+FmgoqHUVVAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"249dc97333162a0d0687013d7594ae37c787e35e4cba26ad4b719c819816ee0d","last_reissued_at":"2026-05-18T03:49:57.693934Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:57.693934Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.5215","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":"2026-05-18T03:49:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"14XewnAPQ2Wv4w/hg+yujqxAkJwS9bhiyDD3K4PICXoPupIHYni7/aPbbMS6ZLWKVf8ESt9XCJtgquSOoAirCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T21:18:01.362839Z"},"content_sha256":"e374b3268634ca648945724915dbc4022eb03207df2d6ac9481a866555be4252","schema_version":"1.0","event_id":"sha256:e374b3268634ca648945724915dbc4022eb03207df2d6ac9481a866555be4252"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:ESO4S4ZTCYVA2BUHAE6XLFFOG7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Density Functions subject to a Co-Matroid Constraint","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"cs.DS","authors_text":"Natwar Modani, Sambuddha Roy, Sivaramakrishnan R. Natarajan, Venkatesan T. Chakaravarthy, Yogish Sabharwal","submitted_at":"2012-07-22T11:01:33Z","abstract_excerpt":"In this paper we consider the problem of finding the {\\em densest} subset subject to {\\em co-matroid constraints}. We are given a {\\em monotone supermodular} set function $f$ defined over a universe $U$, and the density of a subset $S$ is defined to be $f(S)/\\crd{S}$. This generalizes the concept of graph density. Co-matroid constraints are the following: given matroid $\\calM$ a set $S$ is feasible, iff the complement of $S$ is {\\em independent} in the matroid. Under such constraints, the problem becomes $\\np$-hard. The specific case of graph density has been considered in literature under spe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.5215","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":"2026-05-18T03:49:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HPQRCY6nJDv2Tt09Yo8DM/ZG99v5VHUOy0h7yMFWGfziWtBEMEHt4U5njmp/6evmXRC95sWnL49fQ3leJ8XoAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T21:18:01.363309Z"},"content_sha256":"aab925bc7d8c82d8a70e1590efe91e1b14f1b23df0ff84483a26221a65c04bc5","schema_version":"1.0","event_id":"sha256:aab925bc7d8c82d8a70e1590efe91e1b14f1b23df0ff84483a26221a65c04bc5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ESO4S4ZTCYVA2BUHAE6XLFFOG7/bundle.json","state_url":"https://pith.science/pith/ESO4S4ZTCYVA2BUHAE6XLFFOG7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ESO4S4ZTCYVA2BUHAE6XLFFOG7/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":"2026-05-27T21:18:01Z","links":{"resolver":"https://pith.science/pith/ESO4S4ZTCYVA2BUHAE6XLFFOG7","bundle":"https://pith.science/pith/ESO4S4ZTCYVA2BUHAE6XLFFOG7/bundle.json","state":"https://pith.science/pith/ESO4S4ZTCYVA2BUHAE6XLFFOG7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ESO4S4ZTCYVA2BUHAE6XLFFOG7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:ESO4S4ZTCYVA2BUHAE6XLFFOG7","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":"8ee2e84070e368761faa1e56de37adcec732817796b2dd5470d354a0a7642914","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-07-22T11:01:33Z","title_canon_sha256":"e1e402a532ec6ee833ea159d5952b29a7278576bf77b139236707cf37e9d47ae"},"schema_version":"1.0","source":{"id":"1207.5215","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.5215","created_at":"2026-05-18T03:49:57Z"},{"alias_kind":"arxiv_version","alias_value":"1207.5215v2","created_at":"2026-05-18T03:49:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.5215","created_at":"2026-05-18T03:49:57Z"},{"alias_kind":"pith_short_12","alias_value":"ESO4S4ZTCYVA","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"ESO4S4ZTCYVA2BUH","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"ESO4S4ZT","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:aab925bc7d8c82d8a70e1590efe91e1b14f1b23df0ff84483a26221a65c04bc5","target":"graph","created_at":"2026-05-18T03:49:57Z","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":"In this paper we consider the problem of finding the {\\em densest} subset subject to {\\em co-matroid constraints}. We are given a {\\em monotone supermodular} set function $f$ defined over a universe $U$, and the density of a subset $S$ is defined to be $f(S)/\\crd{S}$. This generalizes the concept of graph density. Co-matroid constraints are the following: given matroid $\\calM$ a set $S$ is feasible, iff the complement of $S$ is {\\em independent} in the matroid. Under such constraints, the problem becomes $\\np$-hard. The specific case of graph density has been considered in literature under spe","authors_text":"Natwar Modani, Sambuddha Roy, Sivaramakrishnan R. Natarajan, Venkatesan T. Chakaravarthy, Yogish Sabharwal","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-07-22T11:01:33Z","title":"Density Functions subject to a Co-Matroid Constraint"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.5215","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:e374b3268634ca648945724915dbc4022eb03207df2d6ac9481a866555be4252","target":"record","created_at":"2026-05-18T03:49:57Z","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":"8ee2e84070e368761faa1e56de37adcec732817796b2dd5470d354a0a7642914","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-07-22T11:01:33Z","title_canon_sha256":"e1e402a532ec6ee833ea159d5952b29a7278576bf77b139236707cf37e9d47ae"},"schema_version":"1.0","source":{"id":"1207.5215","kind":"arxiv","version":2}},"canonical_sha256":"249dc97333162a0d0687013d7594ae37c787e35e4cba26ad4b719c819816ee0d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"249dc97333162a0d0687013d7594ae37c787e35e4cba26ad4b719c819816ee0d","first_computed_at":"2026-05-18T03:49:57.693934Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:49:57.693934Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TLbbopc3QAdPiHs55Fl/FKnxr9l9ZLNUKi7QHVhXacPjH+Ux/e65ZudiLavg7DVktqhS4gRZb5+FmgoqHUVVAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:49:57.694586Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.5215","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e374b3268634ca648945724915dbc4022eb03207df2d6ac9481a866555be4252","sha256:aab925bc7d8c82d8a70e1590efe91e1b14f1b23df0ff84483a26221a65c04bc5"],"state_sha256":"1c658c963332fb6909db2630905b5b855a346a07a28a194f7aafcb67f97a9645"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ENpfraEORxH0H8L6V/dFZjBrsTxl7v6moAYNCEPxiHpn6diFlwp+AQ3HRkVtwDMWEXKqItA+p2k8402yXaQ7Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T21:18:01.366947Z","bundle_sha256":"db0537411d8d21c3bfe2fdcd84d18dd73b185c2e28dba4dad0e745e291058f00"}}