{"paper":{"title":"Clustering with Locally Bounded Ignorance","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Correlation Clustering admits polynomial kernels when the fuzzy edge graph has bounded degeneracy or closure, for parameters combining solution cost with that bound.","cross_cats":["cs.CC"],"primary_cat":"cs.DS","authors_text":"Christian Komusiewicz, Jaroslav Garvardt","submitted_at":"2026-05-13T11:03:37Z","abstract_excerpt":"In Correlation Clustering, the input is a graph $G=(V,E)$ with weight function $\\omega: {V \\choose 2}\\to Z$\n  and the task is to partition the vertex set into clusters such that\n  the total weight of edges between clusters and missing edges\n  inside clusters is minimized. Due to close connections\n  between Correlation Clustering and Edge Multicut,\n  deciding whether there is a partition with total cost at most $k$ is\n  FPT with respect to $k$ but a polynomial kernel is presumably\n  impossible. We study the influence of the structure of the fuzzy\n  edge graph, that is, the graph induced by the "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show in particular that Correlation Clustering admits a polynomial problem kernel when parameterized by k+d, where d is the degeneracy of the fuzzy edge graph, and when parameterized by k+c, where c is the closure of the fuzzy edge graph.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The fuzzy edge graph (induced by weight-0 edges) has bounded degeneracy or closure; the kernelization algorithms rely on this structural restriction to produce a polynomial-size reduced instance.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Correlation Clustering admits polynomial kernels parameterized by k plus degeneracy or closure of the fuzzy edge graph.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Correlation Clustering admits polynomial kernels when the fuzzy edge graph has bounded degeneracy or closure, for parameters combining solution cost with that bound.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3bb81925b1de7068be2942550b4acc980f4eeb997a399a8c4bd1576e163cdb6c"},"source":{"id":"2605.13917","kind":"arxiv","version":1},"verdict":{"id":"25b7406d-7772-485c-9da7-2bce856bc9b1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T03:01:26.258429Z","strongest_claim":"We show in particular that Correlation Clustering admits a polynomial problem kernel when parameterized by k+d, where d is the degeneracy of the fuzzy edge graph, and when parameterized by k+c, where c is the closure of the fuzzy edge graph.","one_line_summary":"Correlation Clustering admits polynomial kernels parameterized by k plus degeneracy or closure of the fuzzy edge graph.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The fuzzy edge graph (induced by weight-0 edges) has bounded degeneracy or closure; the kernelization algorithms rely on this structural restriction to produce a polynomial-size reduced instance.","pith_extraction_headline":"Correlation Clustering admits polynomial kernels when the fuzzy edge graph has bounded degeneracy or closure, for parameters combining solution cost with that bound."},"references":{"count":15,"sample":[{"doi":"10.1023/b:mach.0000033116.57574.95","year":2004,"title":"Correlation clustering.Mach","work_id":"20fefd66-76ed-43aa-861e-739ef45decd7","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/j.jda.2012.04.005","year":2012,"title":"1016/J.JDA.2012.04.005","work_id":"52ef55f1-9219-4b0b-a583-bc27b39de8da","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/j.tcs.2009.05.006","year":2009,"title":"URL:https://doi.org/10.1016/j.tcs.2009.05.006, doi:10.1016/J.TCS.2009. 05.006. 5 Hans L. Bodlaender, Michael R. Fellows, Pinar Heggernes, Federico Mancini, Charis Pa- padopoulos, and Frances A. Rosamo","work_id":"c876a106-36b5-4b90-9567-adbebcf8cf01","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2011,"title":"Multicut is FPT","work_id":"b6bfe120-bc4c-4642-8114-407565fc5899","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/s00453-011-9595-1","year":1994,"title":"8 Alain Cournier and Michel Habib","work_id":"b0d15abf-c8e7-4eec-9792-980ca501ab8f","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":15,"snapshot_sha256":"0b8bf5783f5c18405ccc364510cea23d83d5f69201e56e01284deabb5b41b4fe","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"}