{"paper":{"title":"Clustering with Local Restrictions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Daniel Lokshtanov, D\\'aniel Marx","submitted_at":"2017-11-10T15:41:00Z","abstract_excerpt":"We study a family of graph clustering problems where each cluster has to satisfy a certain local requirement. Formally, let $\\mu$ be a function on the subsets of vertices of a graph $G$. In the $(\\mu,p,q)$-PARTITION problem, the task is to find a partition of the vertices into clusters where each cluster $C$ satisfies the requirements that (1) at most $q$ edges leave $C$ and (2) $\\mu(C)\\le p$. Our first result shows that if $\\mu$ is an {\\em arbitrary} polynomial-time computable monotone function, then $(\\mu,p,q)$-PARTITION can be solved in time $n^{O(q)}$, i.e., it is polynomial-time solvable "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.03885","kind":"arxiv","version":1},"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"}