{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:PAMN7CEN4L77QHYLKYDXKY7EJA","short_pith_number":"pith:PAMN7CEN","schema_version":"1.0","canonical_sha256":"7818df888de2fff81f0b56077563e44803a03f94f442e280d4371793bd0acb38","source":{"kind":"arxiv","id":"2605.14428","version":1},"attestation_state":"computed","paper":{"title":"Branch-width of represented matroids in matrix multiplication time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A matroid given by an n by n matrix over a finite field has a branch-decomposition of width at most k found in O(n²) time plus one matrix multiplication, or the algorithm reports that branch-width exceeds k.","cross_cats":["math.CO"],"primary_cat":"cs.DS","authors_text":"Mujin Choi, Sang-il Oum, Tuukka Korhonen","submitted_at":"2026-05-14T06:19:53Z","abstract_excerpt":"For an $n$-element matroid $M$ given by an $n \\times n$ matrix representation over a finite field $\\mathbb F$ and an integer $k$, we present an $(O_{k,\\mathbb F}(n^2)+O(n^\\omega))$-time algorithm that either finds a branch-decomposition of $M$ of width at most $k$, or confirms that the branch-width of $M$ is more than $k$, where $\\omega < 2.3714$ is the matrix multiplication exponent, and the $O_{k,\\mathbb F}(\\cdot)$-notation hides factors that depend on $k$ and $\\mathbb F$ in a computable manner. All previous algorithms including Hlin\\v{e}n\\'y and Oum [SIAM J. Comput. (2008)] and Jeong, Kim, "},"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":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.14428","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2026-05-14T06:19:53Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"743d7b221a935709ce8f8cfb67302c1eda5b538bf470f57d87828c0003839287","abstract_canon_sha256":"b7f4e11a10bba13fc6b6bb3f43f7fd9c6fa38975a53f388d654bdfed7f7af182"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:07.168625Z","signature_b64":"53DNU9Hrcn3c8OBeWD0icv0jOQe7MIr9EK/OzSbpc+6+vMOCtKrmKshbZnfhmMyC1O9dWmtld2F2U+p7VT/vCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7818df888de2fff81f0b56077563e44803a03f94f442e280d4371793bd0acb38","last_reissued_at":"2026-05-17T23:39:07.167883Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:07.167883Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Branch-width of represented matroids in matrix multiplication time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A matroid given by an n by n matrix over a finite field has a branch-decomposition of width at most k found in O(n²) time plus one matrix multiplication, or the algorithm reports that branch-width exceeds k.","cross_cats":["math.CO"],"primary_cat":"cs.DS","authors_text":"Mujin Choi, Sang-il Oum, Tuukka Korhonen","submitted_at":"2026-05-14T06:19:53Z","abstract_excerpt":"For an $n$-element matroid $M$ given by an $n \\times n$ matrix representation over a finite field $\\mathbb F$ and an integer $k$, we present an $(O_{k,\\mathbb F}(n^2)+O(n^\\omega))$-time algorithm that either finds a branch-decomposition of $M$ of width at most $k$, or confirms that the branch-width of $M$ is more than $k$, where $\\omega < 2.3714$ is the matrix multiplication exponent, and the $O_{k,\\mathbb F}(\\cdot)$-notation hides factors that depend on $k$ and $\\mathbb F$ in a computable manner. All previous algorithms including Hlin\\v{e}n\\'y and Oum [SIAM J. Comput. (2008)] and Jeong, Kim, "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For an n-element matroid M given by an n × n matrix representation over a finite field F and an integer k, we present an (O_{k,F}(n²)+O(n^ω))-time algorithm that either finds a branch-decomposition of M of width at most k, or confirms that the branch-width of M is more than k.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The input matroid is supplied as a matrix representation over a finite field F; the algorithm's hidden factors that depend on k and F are computable, and the overhead of converting to standard form is accounted for when the matrix is not already in that form.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"O(n² + n^ω)-time algorithm decides if branch-width of a matrix-represented matroid over a finite field is at most k.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A matroid given by an n by n matrix over a finite field has a branch-decomposition of width at most k found in O(n²) time plus one matrix multiplication, or the algorithm reports that branch-width exceeds k.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"82238c72c1a35c93b378baf7d72b579a7e77bcff8ffba9be5bc5988b70a410be"},"source":{"id":"2605.14428","kind":"arxiv","version":1},"verdict":{"id":"e7344d5a-9e34-4e59-b693-e8c0aeb037f3","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:10:41.184580Z","strongest_claim":"For an n-element matroid M given by an n × n matrix representation over a finite field F and an integer k, we present an (O_{k,F}(n²)+O(n^ω))-time algorithm that either finds a branch-decomposition of M of width at most k, or confirms that the branch-width of M is more than k.","one_line_summary":"O(n² + n^ω)-time algorithm decides if branch-width of a matrix-represented matroid over a finite field is at most k.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The input matroid is supplied as a matrix representation over a finite field F; the algorithm's hidden factors that depend on k and F are computable, and the overhead of converting to standard form is accounted for when the matrix is not already in that form.","pith_extraction_headline":"A matroid given by an n by n matrix over a finite field has a branch-decomposition of width at most k found in O(n²) time plus one matrix multiplication, or the algorithm reports that branch-width exceeds k."},"references":{"count":24,"sample":[{"doi":"","year":2025,"title":"Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou, More asymmetry yields faster matrix multiplication , Proceedings of the 2025 Annual ACM-SIAM Symposium on D","work_id":"18e4ac5d-932c-4976-b76a-0a51dd7daadc","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1996,"title":"Hans L. Bodlaender and Ton Kloks, Efficient and constructive algorithms for the pathwidth and treewidth of graphs, J. Algorithms 21 (1996), no. 2, 358–402. MR 98g:68122 27","work_id":"ba12edc2-b80f-4ea0-817a-3181d9be3418","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1974,"title":"Bunch and John E","work_id":"99e4b3aa-36c7-47ca-8096-47203d78bf5a","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1990,"title":"Bruno Courcelle, The monadic second-order logic of graphs I: Recognizable sets of ﬁnite graphs , Inform. and Comput. 85 (1990), no. 1, 12–75. MR 91g:05107","work_id":"8c7cd507-89ff-4a11-bb8d-950d73f93adf","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"138, Cambridge University Press, Cam- bridge, 2012","work_id":"0e78156c-f856-44b0-b617-5ee5ac7f89fa","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":24,"snapshot_sha256":"5e82240e5d05514ddf012d38e448e8cc2281a60dd018705b37e95ca9c5406a0b","internal_anchors":0},"formal_canon":{"evidence_count":3,"snapshot_sha256":"38d655cb42d60aaa3bef112e2594cb916e93aecf3d2dff7f6fbbafb4052609d6"},"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":"2605.14428","created_at":"2026-05-17T23:39:07.168005+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.14428v1","created_at":"2026-05-17T23:39:07.168005+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14428","created_at":"2026-05-17T23:39:07.168005+00:00"},{"alias_kind":"pith_short_12","alias_value":"PAMN7CEN4L77","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_16","alias_value":"PAMN7CEN4L77QHYL","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_8","alias_value":"PAMN7CEN","created_at":"2026-05-18T12:33:37.589309+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":3,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PAMN7CEN4L77QHYLKYDXKY7EJA","json":"https://pith.science/pith/PAMN7CEN4L77QHYLKYDXKY7EJA.json","graph_json":"https://pith.science/api/pith-number/PAMN7CEN4L77QHYLKYDXKY7EJA/graph.json","events_json":"https://pith.science/api/pith-number/PAMN7CEN4L77QHYLKYDXKY7EJA/events.json","paper":"https://pith.science/paper/PAMN7CEN"},"agent_actions":{"view_html":"https://pith.science/pith/PAMN7CEN4L77QHYLKYDXKY7EJA","download_json":"https://pith.science/pith/PAMN7CEN4L77QHYLKYDXKY7EJA.json","view_paper":"https://pith.science/paper/PAMN7CEN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.14428&json=true","fetch_graph":"https://pith.science/api/pith-number/PAMN7CEN4L77QHYLKYDXKY7EJA/graph.json","fetch_events":"https://pith.science/api/pith-number/PAMN7CEN4L77QHYLKYDXKY7EJA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PAMN7CEN4L77QHYLKYDXKY7EJA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PAMN7CEN4L77QHYLKYDXKY7EJA/action/storage_attestation","attest_author":"https://pith.science/pith/PAMN7CEN4L77QHYLKYDXKY7EJA/action/author_attestation","sign_citation":"https://pith.science/pith/PAMN7CEN4L77QHYLKYDXKY7EJA/action/citation_signature","submit_replication":"https://pith.science/pith/PAMN7CEN4L77QHYLKYDXKY7EJA/action/replication_record"}},"created_at":"2026-05-17T23:39:07.168005+00:00","updated_at":"2026-05-17T23:39:07.168005+00:00"}