An $\widetilde{O} (n^{3/7})$ Round Parallel Algorithm for Matroid Bases

TL;DR

This paper presents a new parallel algorithm that finds a basis in an $n$-element matroid within approximately $O(n^{3/7})$ rounds, improving previous bounds and introducing a novel framework for analyzing dependencies.

Contribution

It introduces a new algorithm with $ ilde{O}(n^{3/7})$ rounds for matroid basis finding and develops a structural framework for analyzing random circuits.

Findings

Achieved $ ilde{O}(n^{3/7})$ rounds for matroid basis finding.

Developed a new framework analyzing dependencies across multiple elements.

Improved upon previous algorithms with better round complexity.

Abstract

We study the parallel (adaptive) complexity of the classic problem of finding a basis in an $n$-element matroid, given access via an \emph{independence oracle}. In this model, the algorithm may submit polynomially many independence queries in each round, and the central question is: how many rounds are necessary and sufficient to find a basis? Karp, Upfal, and Wigderson (FOCS~1985, JCSS~1988; hereafter KUW) initiated this study, showing that $O(\sqrt{n})$ adaptive rounds suffice for any matroid, and that $\widetilde\Omega(n^{1/3})$ rounds are necessary even for partition matroids. This left a substantial gap that persisted for nearly four decades, until Khanna, Putterman, and Song (FOCS~2025; hereafter KPS) achieved $\widetilde O(n^{7/15})$ rounds, the first improvement since~KUW. In this work, we make another conceptual advance beyond KPS, giving a new algorithm that finds a matroid basis in $\widetilde O(n^{3/7})$ rounds. We develop a structural and algorithmic framework that brings a new lens to the analysis of random circuits, moving from reasoning about individual elements to understanding how dependencies span multiple elements simultaneously.