{"paper":{"title":"Convex Minimization with Integer Minima in $\\widetilde O(n^4)$ Time","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["cs.DM","math.OC"],"primary_cat":"cs.DS","authors_text":"Haotian Jiang, Lichen Zhang, Yin Tat Lee, Zhao Song","submitted_at":"2023-04-07T00:44:51Z","abstract_excerpt":"Given a convex function $f$ on $\\mathbb{R}^n$ with an integer minimizer, we show how to find an exact minimizer of $f$ using $O(n^2 \\log n)$ calls to a separation oracle and $O(n^4 \\log n)$ time. The previous best polynomial time algorithm for this problem given in [Jiang, SODA 2021, JACM 2022] achieves $O(n^2\\log\\log n/\\log n)$ oracle complexity. However, the overall runtime of Jiang's algorithm is at least $\\widetilde{\\Omega}(n^8)$, due to expensive sub-routines such as the Lenstra-Lenstra-Lov\\'asz (LLL) algorithm [Lenstra, Lenstra, Lov\\'asz, Math. Ann. 1982] and random walk based cutting pl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2304.03426","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2304.03426/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}