{"paper":{"title":"Boundary optimization for rough sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Konrad Engel, Tran Dan Thu","submitted_at":"2017-09-15T08:52:34Z","abstract_excerpt":"Let $n > m\\ge 2$ be integers and let $\\mathcal{A}=\\{A_1,\\dots,A_m\\}$ be a partition of $[n]=\\{1,\\dots,n\\}$. For $X \\subseteq [n]$, its $\\mathcal{A}$-boundary region $\\mathcal{A}(X)$ is defined to be the union of those blocks $A_i$ of $\\mathcal{A}$ for which $A_i\\cap X\\neq \\emptyset$ and $A_i\\cap ([n] \\setminus X)\\neq \\emptyset$. For three different probability distributions on the power set of $[n]$, partitions $\\mathcal{A}$ of $[n]$ are determined such that the expected cardinality of the $\\mathcal{A}$-boundary region of a randomly chosen subset of $[n]$ is minimal and maximal, respectively. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.05109","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"}