{"paper":{"title":"A Dichotomy Theorem for First-Fit Chain Partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kevin G. Milans, Michael C. Wigal","submitted_at":"2018-10-09T04:25:07Z","abstract_excerpt":"First-Fit is a greedy algorithm for partitioning the elements of a poset into chains. Let $\\textrm{FF}(w,Q)$ be the maximum number of chains that First-Fit uses on a $Q$-free poset of width $w$. A result due to Bosek, Krawczyk, and Matecki states that $\\textrm{FF}(w,Q)$ is finite when $Q$ has width at most $2$. We describe a family of posets $\\mathcal{Q}$ and show that the following dichotomy holds: if $Q\\in\\mathcal{Q}$, then $\\textrm{FF}(w,Q) \\le 2^{c(\\log w)^2}$ for some constant $c$ depending only on $Q$, and if $Q\\not\\in\\mathcal{Q}$, then $\\textrm{FF}(w,Q) \\ge 2^w - 1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.03807","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"}