We consider the parallel complexity of submodular function minimization (SFM). We provide a pair of methods which obtain two new query versus depth trade-offs a submodular function defined on subsets of $n$ elements that has integer values between $-M$ and $M$. The first method has depth $2$ and query complexity $n^{O(M)}$ and the second method has depth $\widetilde{O}(n^{1/3} M^{2/3})$ and query complexity $O(\mathrm{poly}(n, M))$. Despite a line of work on improved parallel lower bounds for SFM, prior to our work the only known algorithms for parallel SFM either followed from more general methods for sequential SFM or highly-parallel minimization of convex $\ell_2$-Lipschitz functions. Interestingly, to obtain our second result we provide the first highly-parallel algorithm for minimizing $\ell_\infty$-Lipschitz function over the hypercube which obtains near-optimal depth for obtaining constant accuracy.

翻译：本文研究子模函数最小化（SFM）的并行复杂度问题。我们提出两种方法，针对定义在$n$个元素子集上、取值范围为$-M$至$M$的整数值子模函数，实现了查询次数与并行深度之间的两种新权衡。第一种方法具有深度$2$和查询复杂度$n^{O(M)}$，第二种方法具有深度$\widetilde{O}(n^{1/3} M^{2/3})$和查询复杂度$O(\mathrm{poly}(n, M))$。尽管已有大量关于SFM并行下界改进的研究工作，但在我们的工作之前，已知的并行SFM算法要么源自更通用的顺序SFM方法，要么源自高度并行的凸$\ell_2$-Lipschitz函数最小化方法。值得注意的是，为获得第二个结果，我们首次提出了在超立方体上最小化$\ell_\infty$-Lipschitz函数的高度并行算法，该算法在实现恒定精度时获得了近乎最优的并行深度。