arXiv:2304.03426 · v3
Convex Minimization with Integer Minima in $\widetilde O(n^4)$ Time
摘要
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.
作者备注：SODA 2024. This version fixes errors in the cost of the shortest vector computations and in the leverage score maintenance. The oracle complexity bound still holds, and for submodular function minimization the arithmetic bound becomes $O(n^4 log^3 n)$ instead of $O(n^4 log n)$