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：本文研究在保证存在k-稀疏极小化子的条件下，最小化子模函数$f : 2^V \rightarrow \mathbb{R}$的问题。我们提出一种确定性算法，该算法通过多项式次对$f$的求值预言机查询，以$\widetilde{O}(\mathsf{poly}(k) \log(|f|/\epsilon))$的并行深度计算出$f$的加性$\epsilon$近似极小化子，其中$|f| = \max_{S \subseteq V} |f(S)|$。此外，我们提出一种随机算法，以高概率使用$\widetilde{O}(|V| \cdot \mathsf{poly}(k))$次查询和多项式时间计算出$f$的精确极小化子。当$k = \widetilde{O}(1)$时，我们的算法要么使用近似常数的并行深度，要么使用近似线性的求值预言机查询次数。此前所有针对该问题的算法均需使用$\Omega(|V|)$并行深度或$\Omega(|V|^2)$次查询。与当前最先进的弱多项式时间和强多项式时间子模函数最小化算法相比，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”的新概念，其编码了稀疏极小化子的结构信息，我们的并行与串行算法均为此提供了新的算法工具，使得一阶优化方法能够高效计算此类证书。相应地，我们的算法不调用快速矩阵乘法或通用线性系统求解器，在此意义上比现有最先进方法更具组合特性。