{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:BRK4XG6ALZDA35ZWAO4Y4Y6XUX","short_pith_number":"pith:BRK4XG6A","canonical_record":{"source":{"id":"2606.10345","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-06-09T02:54:27Z","cross_cats_sorted":[],"title_canon_sha256":"475e8d5dbec10322bb15c9585662d780d823649d9224aa73f316da6db6d33238","abstract_canon_sha256":"94ffa86c333d10c1179cb2c88108aaf2ed843a6e00e5273e139beeea7ae653fe"},"schema_version":"1.0"},"canonical_sha256":"0c55cb9bc05e460df73603b98e63d7a5d6c71f5e830d319a59b92cf59a0f4d2f","source":{"kind":"arxiv","id":"2606.10345","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.10345","created_at":"2026-06-10T01:10:12Z"},{"alias_kind":"arxiv_version","alias_value":"2606.10345v1","created_at":"2026-06-10T01:10:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.10345","created_at":"2026-06-10T01:10:12Z"},{"alias_kind":"pith_short_12","alias_value":"BRK4XG6ALZDA","created_at":"2026-06-10T01:10:12Z"},{"alias_kind":"pith_short_16","alias_value":"BRK4XG6ALZDA35ZW","created_at":"2026-06-10T01:10:12Z"},{"alias_kind":"pith_short_8","alias_value":"BRK4XG6A","created_at":"2026-06-10T01:10:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:BRK4XG6ALZDA35ZWAO4Y4Y6XUX","target":"record","payload":{"canonical_record":{"source":{"id":"2606.10345","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-06-09T02:54:27Z","cross_cats_sorted":[],"title_canon_sha256":"475e8d5dbec10322bb15c9585662d780d823649d9224aa73f316da6db6d33238","abstract_canon_sha256":"94ffa86c333d10c1179cb2c88108aaf2ed843a6e00e5273e139beeea7ae653fe"},"schema_version":"1.0"},"canonical_sha256":"0c55cb9bc05e460df73603b98e63d7a5d6c71f5e830d319a59b92cf59a0f4d2f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-10T01:10:12.853877Z","signature_b64":"QNQ48Gt+U/kbRX+U6x+jhareAC0QC6xlzSBQLHbM2Prw49FtbcYjMulUABmuO3ckgQkDRqUkVZnGlxz7agz7Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c55cb9bc05e460df73603b98e63d7a5d6c71f5e830d319a59b92cf59a0f4d2f","last_reissued_at":"2026-06-10T01:10:12.853022Z","signature_status":"signed_v1","first_computed_at":"2026-06-10T01:10:12.853022Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.10345","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":"2026-06-10T01:10:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n1WHryyu5UNuF6Ihii/sAYPdMjkjzPvLAIYMgqMZvmBZQYDAxCaafidjJ3ZGVqFBItqccvuLRsp6TRjHdai4BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T11:40:59.530508Z"},"content_sha256":"98fbff6bfa162c5841cf0dd1d3de823fae03e2d0b38d2e47228658073ba538a7","schema_version":"1.0","event_id":"sha256:98fbff6bfa162c5841cf0dd1d3de823fae03e2d0b38d2e47228658073ba538a7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:BRK4XG6ALZDA35ZWAO4Y4Y6XUX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Supermodularity and Submodularity in Network Interdiction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Bo Zhou, Ruiwei Jiang, Siqian Shen","submitted_at":"2026-06-09T02:54:27Z","abstract_excerpt":"We study a bilevel network interdiction problem, with an attacker interdicting (attacking) certain components of a network and a defender optimizing operations over the ensuing network. We study when the defender's optimal objective is submodular or supermodular with respect to the attacker's interdiction decisions, for optimizing the bilevel integer program more efficiently. We first consider the min-cost flow (MinCF) interdiction problem and derive necessary and sufficient conditions for the supermodularity or submodularity to hold under three types of attacks, respectively on supplies/deman"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.10345","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.10345/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":"2026-06-10T01:10:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ti2NQo2/T6uEI8tFwHvCBSn0RPAQWvryKkXGisKxHWmDKZq4tIkqIFk7jwq4Af3AcAhGZdSnfZwAKmIBi2y4BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T11:40:59.531002Z"},"content_sha256":"c56686f0a3636784b08382dd0e654ecd531a8848bd0b66ed71f46dcc9396b6c8","schema_version":"1.0","event_id":"sha256:c56686f0a3636784b08382dd0e654ecd531a8848bd0b66ed71f46dcc9396b6c8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BRK4XG6ALZDA35ZWAO4Y4Y6XUX/bundle.json","state_url":"https://pith.science/pith/BRK4XG6ALZDA35ZWAO4Y4Y6XUX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BRK4XG6ALZDA35ZWAO4Y4Y6XUX/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":"2026-08-14T11:40:59Z","links":{"resolver":"https://pith.science/pith/BRK4XG6ALZDA35ZWAO4Y4Y6XUX","bundle":"https://pith.science/pith/BRK4XG6ALZDA35ZWAO4Y4Y6XUX/bundle.json","state":"https://pith.science/pith/BRK4XG6ALZDA35ZWAO4Y4Y6XUX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BRK4XG6ALZDA35ZWAO4Y4Y6XUX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:BRK4XG6ALZDA35ZWAO4Y4Y6XUX","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":"94ffa86c333d10c1179cb2c88108aaf2ed843a6e00e5273e139beeea7ae653fe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-06-09T02:54:27Z","title_canon_sha256":"475e8d5dbec10322bb15c9585662d780d823649d9224aa73f316da6db6d33238"},"schema_version":"1.0","source":{"id":"2606.10345","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.10345","created_at":"2026-06-10T01:10:12Z"},{"alias_kind":"arxiv_version","alias_value":"2606.10345v1","created_at":"2026-06-10T01:10:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.10345","created_at":"2026-06-10T01:10:12Z"},{"alias_kind":"pith_short_12","alias_value":"BRK4XG6ALZDA","created_at":"2026-06-10T01:10:12Z"},{"alias_kind":"pith_short_16","alias_value":"BRK4XG6ALZDA35ZW","created_at":"2026-06-10T01:10:12Z"},{"alias_kind":"pith_short_8","alias_value":"BRK4XG6A","created_at":"2026-06-10T01:10:12Z"}],"graph_snapshots":[{"event_id":"sha256:c56686f0a3636784b08382dd0e654ecd531a8848bd0b66ed71f46dcc9396b6c8","target":"graph","created_at":"2026-06-10T01:10:12Z","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/2606.10345/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study a bilevel network interdiction problem, with an attacker interdicting (attacking) certain components of a network and a defender optimizing operations over the ensuing network. We study when the defender's optimal objective is submodular or supermodular with respect to the attacker's interdiction decisions, for optimizing the bilevel integer program more efficiently. We first consider the min-cost flow (MinCF) interdiction problem and derive necessary and sufficient conditions for the supermodularity or submodularity to hold under three types of attacks, respectively on supplies/deman","authors_text":"Bo Zhou, Ruiwei Jiang, Siqian Shen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-06-09T02:54:27Z","title":"Supermodularity and Submodularity in Network Interdiction"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.10345","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":"sha256:98fbff6bfa162c5841cf0dd1d3de823fae03e2d0b38d2e47228658073ba538a7","target":"record","created_at":"2026-06-10T01:10:12Z","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":"94ffa86c333d10c1179cb2c88108aaf2ed843a6e00e5273e139beeea7ae653fe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-06-09T02:54:27Z","title_canon_sha256":"475e8d5dbec10322bb15c9585662d780d823649d9224aa73f316da6db6d33238"},"schema_version":"1.0","source":{"id":"2606.10345","kind":"arxiv","version":1}},"canonical_sha256":"0c55cb9bc05e460df73603b98e63d7a5d6c71f5e830d319a59b92cf59a0f4d2f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0c55cb9bc05e460df73603b98e63d7a5d6c71f5e830d319a59b92cf59a0f4d2f","first_computed_at":"2026-06-10T01:10:12.853022Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-10T01:10:12.853022Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QNQ48Gt+U/kbRX+U6x+jhareAC0QC6xlzSBQLHbM2Prw49FtbcYjMulUABmuO3ckgQkDRqUkVZnGlxz7agz7Bw==","signature_status":"signed_v1","signed_at":"2026-06-10T01:10:12.853877Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.10345","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:98fbff6bfa162c5841cf0dd1d3de823fae03e2d0b38d2e47228658073ba538a7","sha256:c56686f0a3636784b08382dd0e654ecd531a8848bd0b66ed71f46dcc9396b6c8"],"state_sha256":"06eef06d8046ddb9d9a461c4e75fd4bb9473de16b965436af0a7607e1623a103"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8KCrtmgZz+SdeWKOhijF5NUX5nqo8Uj/xduMmeQyC3jE2QIev9TTNNyfgQgz0vtIpvL794dJeUiY9/vhMhAvCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T11:40:59.534778Z","bundle_sha256":"1ddc76039799528ea9fb6a38149e144f83c4ecbc3cc73718d911e15eaa6eaca2"}}