{"paper":{"title":"Number Balancing is as hard as Minkowski's Theorem and Shortest Vector","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.CG","cs.DS"],"primary_cat":"cs.DM","authors_text":"Harishchandra Ramadas, Rebecca Hoberg, Thomas Rothvoss, Xin Yang","submitted_at":"2016-11-26T22:59:07Z","abstract_excerpt":"The number balancing (NBP) problem is the following: given real numbers $a_1,\\ldots,a_n \\in [0,1]$, find two disjoint subsets $I_1,I_2 \\subseteq [n]$ so that the difference $|\\sum_{i \\in I_1}a_i - \\sum_{i \\in I_2}a_i|$ of their sums is minimized. An application of the pigeonhole principle shows that there is always a solution where the difference is at most $O(\\frac{\\sqrt{n}}{2^n})$. Finding the minimum, however, is NP-hard. In polynomial time,the differencing algorithm by Karmarkar and Karp from 1982 can produce a solution with difference at most $n^{-\\Theta(\\log n)}$, but no further improvem"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.08757","kind":"arxiv","version":2},"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"}