A fundamental question in parallel computation, posed by Karp, Upfal, and Wigderson (FOCS 1985, JCSS 1988), asks: \emph{given only independence-oracle access to a matroid on $n$ elements, how many rounds are required to find a basis using only polynomially many queries?} This question generalizes, among others, the complexity of finding bases of linear spaces, partition matroids, and spanning forests in graphs. In their work, they established an upper bound of $O(\sqrt{n})$ rounds and a lower bound of $\widetilde{\Omega}(n^{1/3})$ rounds for this problem, and these bounds have remained unimproved since then. In this work, we make the first progress in narrowing this gap by designing a parallel algorithm that finds a basis of an arbitrary matroid in $\tilde{O}(n^{7/15})$ rounds (using polynomially many independence queries per round) with high probability, surpassing the long-standing $O(\sqrt{n})$ barrier. Our approach introduces a novel matroid decomposition technique and other structural insights that not only yield this general result but also lead to a much improved new algorithm for the class of \emph{partition matroids} (which underlies the $\widetilde\Omega(n^{1/3})$ lower bound of Karp, Upfal, and Wigderson). Specifically, we develop an $\tilde{O}(n^{1/3})$-round algorithm, thereby settling the round complexity of finding a basis in partition matroids.

翻译：并行计算中的一个基本问题由Karp、Upfal和Wigderson（FOCS 1985，JCSS 1988）提出：\emph{在仅通过独立性预言机访问一个定义在$n$个元素上的拟阵时，若每轮仅使用多项式次查询，需要多少轮才能找到一个基？}该问题推广了包括寻找线性空间基、划分拟阵基以及图中生成森林在内的多个问题的复杂性。在他们的工作中，他们为该问题建立了$O(\sqrt{n})$轮的上界和$\widetilde{\Omega}(n^{1/3})$轮的下界，且这些界自提出以来一直未被改进。在本工作中，我们首次通过设计一个并行算法来缩小这一差距，该算法以高概率在$\tilde{O}(n^{7/15})$轮内（每轮使用多项式次独立性查询）找到任意拟阵的一个基，从而突破了长期存在的$O(\sqrt{n})$障碍。我们的方法引入了一种新颖的拟阵分解技术及其他结构性洞见，不仅得到了这一普适性结果，还为\emph{划分拟阵}类（该类构成了Karp、Upfal和Wigderson所提$\widetilde\Omega(n^{1/3})$下界的基础）带来了显著改进的新算法。具体而言，我们开发了一个$\tilde{O}(n^{1/3})$轮的算法，从而确定了在划分拟阵中寻找基的轮数复杂性。