In this paper we study the problem of minimizing a submodular function $f : 2^V \rightarrow \mathbb{R}$ that is guaranteed to have a $k$-sparse minimizer. We give a deterministic algorithm that computes an additive $\epsilon$-approximate minimizer of such $f$ in $\widetilde{O}(\mathsf{poly}(k) \log(|f|/\epsilon))$ parallel depth using a polynomial number of queries to an evaluation oracle of $f$, where $|f| = \max_{S \subseteq V} |f(S)|$. Further, we give a randomized algorithm that computes an exact minimizer of $f$ with high probability using $\widetilde{O}(|V| \cdot \mathsf{poly}(k))$ queries and polynomial time. When $k = \widetilde{O}(1)$, our algorithms use either nearly-constant parallel depth or a nearly-linear number of evaluation oracle queries. All previous algorithms for this problem either use $\Omega(|V|)$ parallel depth or $\Omega(|V|^2)$ queries. In contrast to state-of-the-art weakly-polynomial and strongly-polynomial time algorithms for SFM, our algorithms use first-order optimization methods, e.g., mirror descent and follow the regularized leader. We introduce what we call {\em sparse dual certificates}, which encode information on the structure of sparse minimizers, and both our parallel and sequential algorithms provide new algorithmic tools for allowing first-order optimization methods to efficiently compute them. Correspondingly, our algorithm does not invoke fast matrix multiplication or general linear system solvers and in this sense is more combinatorial than previous state-of-the-art methods.

翻译：本文研究具有$k$-稀疏最小化子的子模函数$f : 2^V \rightarrow \mathbb{R}$的最小化问题。我们提出一个确定性算法，该算法在多项式次查询$f$的评估预言机条件下，以$\widetilde{O}(\mathsf{poly}(k) \log(|f|/\epsilon))$的并行深度计算$f$的加性$\epsilon$-近似最小化子，其中$|f| = \max_{S \subseteq V} |f(S)|$。进一步，我们给出一个随机化算法，以高概率使用$\widetilde{O}(|V| \cdot \mathsf{poly}(k))$次查询和多项式时间计算$f$的精确最小化子。当$k = \widetilde{O}(1)$时，我们的算法要么使用近乎常数的并行深度，要么使用近乎线性的评估预言机查询次数。此前该问题的所有算法要么需要使用$\Omega(|V|)$并行深度，要么需要$\Omega(|V|^2)$次查询。与SFM领域最先进的弱多项式与强多项式时间算法不同，我们的算法采用一阶优化方法，例如镜像下降和跟随正则化领导者。我们引入了称为"稀疏对偶证书"的概念，该概念编码了稀疏最小化子结构信息，并且我们的并行算法和顺序算法都提供了新的算法工具，使得一阶优化方法能够高效计算这些证书。相应地，我们的算法不涉及快速矩阵乘法或通用线性系统求解器，在此意义上比此前最先进方法更具组合性。