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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 "},"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":"1306.4384","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2013-06-18T23:00:35Z","cross_cats_sorted":[],"title_canon_sha256":"40c137e0f96f4031aeae00c41ec7570c4cd8e58f4b01662d80755fa110476783","abstract_canon_sha256":"874f199c3875a9014188c881c22e4e3266f129ed788cc886f46a79db6df21f65"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:11:09.107303Z","signature_b64":"92/3MUW+6YmlMBr49c58PhuosIKetbyQAOozPJ9kal3M0Xm4oz+nkAlogvZr8zDzHGN6RICX0TrCL4AzPjXNAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ba1180ff15fef21f8acb1c8e121d44cda1ef407b112ad2420abe4e2998c307d9","last_reissued_at":"2026-05-18T03:11:09.106514Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:11:09.106514Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Approximation Algorithm for Sparsest k-Partitioning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Anand Louis, Konstantin Makarychev","submitted_at":"2013-06-18T23:00:35Z","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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.4384","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1306.4384","created_at":"2026-05-18T03:11:09.106642+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.4384v2","created_at":"2026-05-18T03:11:09.106642+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.4384","created_at":"2026-05-18T03:11:09.106642+00:00"},{"alias_kind":"pith_short_12","alias_value":"XIIYB7YV73ZB","created_at":"2026-05-18T12:28:06.772260+00:00"},{"alias_kind":"pith_short_16","alias_value":"XIIYB7YV73ZB7CWL","created_at":"2026-05-18T12:28:06.772260+00:00"},{"alias_kind":"pith_short_8","alias_value":"XIIYB7YV","created_at":"2026-05-18T12:28:06.772260+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XIIYB7YV73ZB7CWLDSHBEHKEZW","json":"https://pith.science/pith/XIIYB7YV73ZB7CWLDSHBEHKEZW.json","graph_json":"https://pith.science/api/pith-number/XIIYB7YV73ZB7CWLDSHBEHKEZW/graph.json","events_json":"https://pith.science/api/pith-number/XIIYB7YV73ZB7CWLDSHBEHKEZW/events.json","paper":"https://pith.science/paper/XIIYB7YV"},"agent_actions":{"view_html":"https://pith.science/pith/XIIYB7YV73ZB7CWLDSHBEHKEZW","download_json":"https://pith.science/pith/XIIYB7YV73ZB7CWLDSHBEHKEZW.json","view_paper":"https://pith.science/paper/XIIYB7YV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.4384&json=true","fetch_graph":"https://pith.science/api/pith-number/XIIYB7YV73ZB7CWLDSHBEHKEZW/graph.json","fetch_events":"https://pith.science/api/pith-number/XIIYB7YV73ZB7CWLDSHBEHKEZW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XIIYB7YV73ZB7CWLDSHBEHKEZW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XIIYB7YV73ZB7CWLDSHBEHKEZW/action/storage_attestation","attest_author":"https://pith.science/pith/XIIYB7YV73ZB7CWLDSHBEHKEZW/action/author_attestation","sign_citation":"https://pith.science/pith/XIIYB7YV73ZB7CWLDSHBEHKEZW/action/citation_signature","submit_replication":"https://pith.science/pith/XIIYB7YV73ZB7CWLDSHBEHKEZW/action/replication_record"}},"created_at":"2026-05-18T03:11:09.106642+00:00","updated_at":"2026-05-18T03:11:09.106642+00:00"}