On the Parallel Complexity of Finding a Matroid Basis

2025-07-10

AI Summary

This paper studies the parallel query complexity of finding a basis in a matroid: given independence oracle access to an $n$-element matroid, minimize the number of parallel rounds—each allowing polynomially many queries—required to identify a basis. Since its introduction by Karp et al. in 1985, the problem has exhibited a long-standing gap between the $Omega(sqrt{n})$ lower bound and the $O(n)$ upper bound. We introduce a novel matroid decomposition framework and structural analysis techniques, breaking the $sqrt{n}$-round barrier for the first time. For general matroids, we design a randomized algorithm succeeding with high probability in $ ilde{O}(n^{7/15})$ rounds; for partition matroids, we achieve the tight asymptotically optimal bound of $ ilde{O}(n^{1/3})$ rounds. Our approach integrates deep matroid structural properties, probabilistic analysis, and parallel query scheduling, significantly advancing the theoretical frontier of parallel matroid computation.

Technology Category

• Search and Optimization: Other Foundations of Search & Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 $widetildeΩ(n^{1/3})$ 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 $ ilde{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 $widetildeΩ(n^{1/3})$ lower bound of Karp, Upfal, and Wigderson). Specifically, we develop an $ ilde{O}(n^{1/3})$-round algorithm, thereby settling the round complexity of finding a basis in partition matroids.

Problem

• Determine round complexity for finding matroid basis
• Improve bounds on parallel matroid basis computation
• Develop efficient algorithm for partition matroids

Innovation

• Novel matroid decomposition technique introduced
• Parallel algorithm achieves O(n^7/15) rounds
• Improved O(n^1/3) algorithm for partition matroids