{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:XIIYB7YV73ZB7CWLDSHBEHKEZW","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":"874f199c3875a9014188c881c22e4e3266f129ed788cc886f46a79db6df21f65","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2013-06-18T23:00:35Z","title_canon_sha256":"40c137e0f96f4031aeae00c41ec7570c4cd8e58f4b01662d80755fa110476783"},"schema_version":"1.0","source":{"id":"1306.4384","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.4384","created_at":"2026-05-18T03:11:09Z"},{"alias_kind":"arxiv_version","alias_value":"1306.4384v2","created_at":"2026-05-18T03:11:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.4384","created_at":"2026-05-18T03:11:09Z"},{"alias_kind":"pith_short_12","alias_value":"XIIYB7YV73ZB","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XIIYB7YV73ZB7CWL","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XIIYB7YV","created_at":"2026-05-18T12:28:06Z"}],"graph_snapshots":[{"event_id":"sha256:c81d62a7a282fdb2fbd0f30be5b2342b05ad36da22d4c2e0653f7a55db5ff46e","target":"graph","created_at":"2026-05-18T03:11:09Z","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":"Given a graph $G$, the sparsest-cut problem asks to find the set of vertices $S$ which has the least expansion defined as $$\\phi_G(S) := \\frac{w(E(S,\\bar{S}))}{\\min \\set{w(S), w(\\bar{S})}}, $$ where $w$ is the total edge weight of a subset. Here we study the natural generalization of this problem: given an integer $k$, compute a $k$-partition $\\set{P_1, \\ldots, P_k}$ of the vertex set so as to minimize $$ \\phi_k(\\set{P_1, \\ldots, P_k}) := \\max_i \\phi_G(P_i). $$ Our main result is a polynomial time bi-criteria approximation algorithm which outputs a $(1 - \\e)k$-partition of the vertex set such ","authors_text":"Anand Louis, Konstantin Makarychev","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2013-06-18T23:00:35Z","title":"Approximation Algorithm for Sparsest k-Partitioning"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.4384","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:cb29036b182d6c660a3a03a4b8ab45f1f27942256b72ab026c06a3a9011b6778","target":"record","created_at":"2026-05-18T03:11:09Z","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":"874f199c3875a9014188c881c22e4e3266f129ed788cc886f46a79db6df21f65","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2013-06-18T23:00:35Z","title_canon_sha256":"40c137e0f96f4031aeae00c41ec7570c4cd8e58f4b01662d80755fa110476783"},"schema_version":"1.0","source":{"id":"1306.4384","kind":"arxiv","version":2}},"canonical_sha256":"ba1180ff15fef21f8acb1c8e121d44cda1ef407b112ad2420abe4e2998c307d9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ba1180ff15fef21f8acb1c8e121d44cda1ef407b112ad2420abe4e2998c307d9","first_computed_at":"2026-05-18T03:11:09.106514Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:11:09.106514Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"92/3MUW+6YmlMBr49c58PhuosIKetbyQAOozPJ9kal3M0Xm4oz+nkAlogvZr8zDzHGN6RICX0TrCL4AzPjXNAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:11:09.107303Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.4384","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cb29036b182d6c660a3a03a4b8ab45f1f27942256b72ab026c06a3a9011b6778","sha256:c81d62a7a282fdb2fbd0f30be5b2342b05ad36da22d4c2e0653f7a55db5ff46e"],"state_sha256":"76d8ae21e2a0a775b6dad62ba4ff7fe38375ebeae9132380a8ecf48a3230a0e0"}