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 $\widetilde{O}(n^2)$ 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 plane method [Bertsimas, Vempala, JACM 2004]. Our significant speedup is obtained by a nontrivial combination of a faster version of the LLL algorithm due to [Neumaier, Stehl\'e, ISSAC 2016] that gives similar guarantees, the volumetric center cutting plane method (CPM) by [Vaidya, FOCS 1989] and its fast implementation given in [Jiang, Lee, Song, Wong, STOC 2020]. For the special case of submodular function minimization (SFM), our result implies a strongly polynomial time algorithm for this problem using $O(n^3 \log n)$ calls to an evaluation oracle and $O(n^4 \log n)$ additional arithmetic operations. Both the oracle complexity and the number of arithmetic operations of our more general algorithm are better than the previous best-known runtime algorithms for this specific problem given in [Lee, Sidford, Wong, FOCS 2015] and [Dadush, V\'egh, Zambelli, SODA 2018, MOR 2021].

翻译：给定$\mathbb{R}^n$上一个具有整数极小值点的凸函数$f$，我们展示了如何通过$O(n^2 \log n)$次分离预言机调用和$O(n^4 \log n)$时间找到$f$的一个精确极小值点。此前该问题的最佳多项式时间算法由[Jiang, SODA 2021, JACM 2022]给出，实现了$\widetilde{O}(n^2)$次预言机复杂度。然而，由于使用了Lenstra-Lenstra-Lovász (LLL)算法[Lenstra, Lenstra, Lovász, Math. Ann. 1982]和基于随机游走的割平面法[Bertsimas, Vempala, JACM 2004]等高耗时子程序，Jiang算法的总运行时间至少为$\widetilde{\Omega}(n^8)$。我们的大幅加速得益于以下方法的非平凡组合：由[Neumaier, Stehlé, ISSAC 2016]提出的保证相似性能的LLL加速版本、[Vaidya, FOCS 1989]的体积中心割平面法(CPM)及其在[Jiang, Lee, Song, Wong, STOC 2020]中给出的快速实现。对于子模函数极小化(SFM)这一特例，我们的结果蕴含一个强多项式时间算法，该算法仅需$O(n^3 \log n)$次评估预言机调用和$O(n^4 \log n)$次额外算术运算。我们通用算法的预言机复杂度与算术运算次数均优于[Lee, Sidford, Wong, FOCS 2015]和[Dadush, Végh, Zambelli, SODA 2018, MOR 2021]针对该特例给出的当前最佳已知运行时算法。