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 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点的凸函数$f$，我们展示了如何通过$O(n^2 \log n)$次分离神谕调用和$O(n^4 \log n)$时间找到$f$的精确极小点。此前该问题的最佳多项式时间算法由[Jiang, SODA 2021, JACM 2022]给出，其神谕复杂度为$O(n^2\log\log n/\log n)$。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]等高代价子程序，Jiang算法的总运行时间至少为$\widetilde{\Omega}(n^8)$。我们的显著加速源自对以下方法非平凡的组合：Neumaier和Stehlé[ISSAC 2016]提出的具有类似保证的LLL算法快速版本、Vaidya[FOCS 1989]的体积中心割平面法(CPM)及其由Jiang、Lee、Song和Wong[STOC 2020]给出的快速实现。对于子模函数极小化(SFM)这一特例，我们的结果意味着一个强多项式时间算法，该算法使用$O(n^3 \log n)$次求值神谕调用和$O(n^4 \log n)$次额外算术运算。我们更通用算法的神谕复杂度和算术运算次数均优于此前针对该特定问题的最佳已知运行时算法[Lee, Sidford, Wong, FOCS 2015]及[Dadush, Végh, Zambelli, SODA 2018, MOR 2021]。