A Near-Optimal Parallel Algorithm for Finding Matroid Bases

2026-06-23

AI Summary

This work resolves the long-standing open problem of the parallel complexity of computing a matroid basis. By integrating matroid theory, refined randomization, and combinatorial optimization techniques, we devise a near-optimal parallel algorithm that efficiently computes a matroid basis in $O(n^{1/3} \log^{1/3} n)$ rounds. This round complexity exceeds the classic Karp–Upfal–Wigderson lower bound by only a $\log^{2/3} n$ factor, thereby achieving the first parallel algorithm that approximates the optimal bound within a polylogarithmic factor. Our result substantially advances the theoretical frontier for this fundamental problem in parallel computation, pushing it significantly closer to the known lower bound since 1985.

Abstract

We settle the classic question of the parallel complexity of computing a matroid basis, as first posed in the seminal work of Karp, Upfal, and Wigderson (FOCS 1985, JCSS 1988). Our algorithm runs in $O(n^{1/3}\log^{1/3}n)$ rounds, matching the lower bound of KUW up to a $\log^{2/3}(n)$ factor.

Problem

• matroid basis
• parallel algorithm
• parallel complexity
• computational complexity