{"paper":{"title":"Approximating Nash Equilibria and Dense Subgraphs via an Approximate Version of Carath\\'{e}odory's Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.GT","authors_text":"Siddharth Barman","submitted_at":"2014-06-09T19:39:50Z","abstract_excerpt":"We present algorithmic applications of an approximate version of Carath\\'{e}odory's theorem. The theorem states that given a set of vectors $X$ in $\\mathbb{R}^d$, for every vector in the convex hull of $X$ there exists an $\\varepsilon$-close (under the $p$-norm distance, for $2\\leq p < \\infty$) vector that can be expressed as a convex combination of at most $b$ vectors of $X$, where the bound $b$ depends on $\\varepsilon$ and the norm $p$ and is independent of the dimension $d$. This theorem can be derived by instantiating Maurey's lemma, early references to which can be found in the work of Pi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.2296","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"}