On the Parallel Complexity of Finding a Matroid Basis

TL;DR

This paper advances the understanding of parallel algorithms for matroid basis problems by introducing a new algorithm that reduces the number of rounds needed, improving over previous bounds and settling the complexity for partition matroids.

Contribution

It presents the first algorithm to find a matroid basis in fewer than O(√n) rounds, using novel decomposition techniques, and determines the exact complexity for partition matroids.

Findings

New algorithm finds matroid basis in rac{n^{7/15}}{} rounds.

Achieves rac{n^{1/3}}{} rounds for partition matroids.

Surpasses the longstanding O(rac{rac{n}{}}) barrier.

Abstract

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.