Given a convex function $f$ on $\mathbb{R}^n$ whose set of minimizers is nonempty and closed, lies in a box of radius $R$ with $\log R\le\mathrm{poly}(n)$, and has integral extreme points, we give a randomized algorithm that finds an integral minimizer of $f$ with high probability using $O(n^2\log n+n\log R)$ calls to a separation oracle, $O(n+\log R/\log n)$ calls to an approximate shortest vector routine such as the LLL algorithm, and $O(n^4\log(nR)\log\log n)$ other arithmetic operations. The previous best polynomial time algorithm [Jiang, SODA 2021, JACM 2022] achieves $O(n(n\log\log n/\log n+\log R))$ oracle complexity, but its overall runtime is at least $\widetildeΩ(n^8)$. We obtain our speedup by combining a variant of the volumetric center cutting plane method of [Vaidya, FOCS 1989], run on the hybrid barrier of [Lee, Sidford, Wong, FOCS 2015] with the fast implementation of [Jiang, Lee, Song, Wong, STOC 2020], and a variant of the LLL algorithm that reduces segments of consecutive basis vectors recursively. The method runs in blocks, so that shortest vectors are computed only $O(n+\log R/\log n)$ times, and each computation takes $O(n^3\log^3 n)$ arithmetic operations in exact arithmetic. For submodular function minimization, our result implies an algorithm using $O(n^3\log n)$ calls to an evaluation oracle and $O(n^4\log^3 n)$ arithmetic operations in total. Its oracle complexity is better than that of the algorithms of [Lee, Sidford, Wong, FOCS 2015] and [Dadush, Végh, Zambelli, SODA 2018, MOR 2021], which had the best running time before this work.

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