{"paper":{"title":"Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.LG"],"primary_cat":"cs.DS","authors_text":"Jan Vondrak, Vitaly Feldman","submitted_at":"2013-07-12T00:41:01Z","abstract_excerpt":"We investigate the approximability of several classes of real-valued functions by functions of a small number of variables ({\\em juntas}). Our main results are tight bounds on the number of variables required to approximate a function $f:\\{0,1\\}^n \\rightarrow [0,1]$ within $\\ell_2$-error $\\epsilon$ over the uniform distribution: 1. If $f$ is submodular, then it is $\\epsilon$-close to a function of $O(\\frac{1}{\\epsilon^2} \\log \\frac{1}{\\epsilon})$ variables. This is an exponential improvement over previously known results. We note that $\\Omega(\\frac{1}{\\epsilon^2})$ variables are necessary even"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.3301","kind":"arxiv","version":3},"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"}