Minimizing Convex Functions with Integral Minimizers

Given a separation oracle $\mathsf{SO}$ for a convex function $f$ that has an integral minimizer inside a box with radius $R$, we show how to find an exact minimizer of $f$ using at most (a) $O(n (n + \log(R)))$ calls to $\mathsf{SO}$ and $\mathsf{poly}(n, \log(R))$ arithmetic operations, or (b) $O(n \log(nR))$ calls to $\mathsf{SO}$ and $\exp(n) \cdot \mathsf{poly}(\log(R))$ arithmetic operations. When the set of minimizers of $f$ has integral extreme points, our algorithm outputs an integral minimizer of $f$. This improves upon the previously best oracle complexity of $O(n^2 (n + \log(R)))$ for polynomial time algorithms obtained by [Gr\"otschel, Lov\'asz and Schrijver, Prog. Comb. Opt. 1984, Springer 1988] over thirty years ago. For the Submodular Function Minimization problem, our result immediately implies a strongly polynomial algorithm that makes at most $O(n^3)$ calls to an evaluation oracle, and an exponential time algorithm that makes at most $O(n^2 \log(n))$ calls to an evaluation oracle. These improve upon the previously best $O(n^3 \log^2(n))$ oracle complexity for strongly polynomial algorithms given in [Lee, Sidford and Wong, FOCS 2015] and [Dadush, V\'egh and Zambelli, SODA 2018], and an exponential time algorithm with oracle complexity $O(n^3 \log(n))$ given in the former work. Our result is achieved via a reduction to the Shortest Vector Problem in lattices. We show how an approximately shortest vector of certain lattice can be used to effectively reduce the dimension of the problem. Our analysis of the oracle complexity is based on a potential function that captures simultaneously the size of the search set and the density of the lattice, which we analyze via technical tools from convex geometry.