{"paper":{"title":"Fast and Compact Graph Cuts for the Boykov-Kolmogorov Algorithm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The fcBK algorithm computes minimum s-t cuts in O(m|C|) time using a compact graph representation for graphs with billions of vertices.","cross_cats":["cs.DS"],"primary_cat":"cs.CV","authors_text":"Anders Bjorholm Dahl, Christian M{\\o}ller Mikkelstrup, Inge Li G{\\o}rtz, Philip Bille, Vedrana Andersen Dahl","submitted_at":"2026-05-13T11:57:35Z","abstract_excerpt":"Computing a minimum $s$-$t$ cut in a graph is a solution to a wide range of computer vision problems, and is often done using the Boykov-Kolmogorov (BK) algorithm. In this paper, we revisit the BK algorithm from both a theoretical and practical point of view. We improve the analysis of the time complexity of the BK algorithm to $O(mn|C|)$ and propose a new algorithm, the fast and compact BK (fcBK) algorithm, with a time complexity of $O(m|C|)$, where $m$, $n$, and $|C|$ are the number of edges, number of vertices, and the capacity of the cut, respectively. We additionally propose a compact gra"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We propose a new algorithm, the fast and compact BK (fcBK) algorithm, with a time complexity of O(m|C|), where m and |C| are the number of edges and the capacity of the cut, respectively.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The compact graph representation preserves the correctness of the minimum-cut computation and incurs no hidden asymptotic overhead on the claimed time bound.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The fcBK algorithm computes minimum s-t cuts in O(m|C|) time and supports graphs with up to 10^9 vertices using a compact representation.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The fcBK algorithm computes minimum s-t cuts in O(m|C|) time using a compact graph representation for graphs with billions of vertices.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"902307208967bd106259c29c8594bfc6e98c55a205e343876e59e93462d52da6"},"source":{"id":"2605.13402","kind":"arxiv","version":1},"verdict":{"id":"0e891b51-fb6d-4859-b8a9-2b81dfc3b37f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:41:44.333762Z","strongest_claim":"We propose a new algorithm, the fast and compact BK (fcBK) algorithm, with a time complexity of O(m|C|), where m and |C| are the number of edges and the capacity of the cut, respectively.","one_line_summary":"The fcBK algorithm computes minimum s-t cuts in O(m|C|) time and supports graphs with up to 10^9 vertices using a compact representation.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The compact graph representation preserves the correctness of the minimum-cut computation and incurs no hidden asymptotic overhead on the claimed time bound.","pith_extraction_headline":"The fcBK algorithm computes minimum s-t cuts in O(m|C|) time using a compact graph representation for graphs with billions of vertices."},"references":{"count":60,"sample":[{"doi":"","year":2004,"title":"An experimental comparison of min- cut/max-flow algorithms for energy minimization in vision,","work_id":"be597a52-54bc-4572-93f7-9387eafaaae2","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2001,"title":"Interactive graph cuts for optimal boundary & region segmentation of objects in n-d images,","work_id":"05a58ed5-bdbb-4b66-bc56-b8ce7eed2032","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2005,"title":"Efficiently solving dynamic markov random fields using graph cuts,","work_id":"d5bea19f-b0ca-4015-a73f-f10fb2f202c8","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"Graph cuts and efficient n-d image segmentation,","work_id":"f04c6a05-9e51-46b9-8969-7da48b8f6071","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"Fast approximate energy minimization with label costs,","work_id":"e02cf547-53b2-46c9-8319-2618c33633a7","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":60,"snapshot_sha256":"2ae27956d09394e44fcd163371f258dafed2b2154f86733ccfe7eb74ad81645d","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"}