{"paper":{"title":"Inverting sets and the packing problem","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Emanuel Knill, Mark Goldberg, Thomas Spencer, Vance Faber","submitted_at":"1994-09-16T00:00:00Z","abstract_excerpt":"Given a set $V$, a subset $S$, and a permutation $\\pi$ of $V$, we say that $\\pi$ permutes $S$ if $\\pi (S) \\cap S = \\emptyset$. Given a collection $\\cS = \\{V; S_1,\\ldots , S_m\\}$, where $S_i \\subseteq V ~~(i=1,\\ldots ,m)$, we say that $\\cS$ is invertible if there is a permutation $\\pi$ of $V$ such that $\\pi (S_i) \\subseteq V-S_i$. In this paper, we present necessary and sufficient conditions for the invertibility of a collection and construct a polynomial algorithm which determines whether a given collection is invertible. For an arbitrary collection, we give a lower bound for the maximum numbe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9409213","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"}