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An Õ(n^{3/7}) Round Parallel Algorithm for Matroid Bases

Authors Sanjeev Khanna , Aaron Putterman , Junkai Song

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LIPIcs.ICALP.2026.124.pdf
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ACM Subject Classification
  • Theory of computation → Constraint and logic programming
Keywords
  • parallel algorithms
  • matroids

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Abstract

We study the parallel (adaptive) complexity of the classic problem of finding a basis in an n-element matroid, given access via an 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(√n) adaptive rounds suffice for any matroid, and that Ω̃(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 Õ(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 Õ(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. Specifically, our framework introduces three new ideas:  
1) A new subset-based decomposition that provides precise guarantees on how random circuits intersect groups of elements, yet remains computable in few adaptive rounds. 
2) A new method for identifying and removing redundant elements in bulk, based on short circuit witnesses that certify redundancy across large portions of the matroid. 
3) An adaptive early-stopping strategy that uses the evolving structure of the matroid to decide when to contract or delete, preventing wasted rounds. 
Each of these contributions, in isolation, already yields meaningful improvements over the round complexity achieved in KPS; their combination enables our main result of Õ(n^{3/7}) rounds.
As further consequences, incorporating our improved basis-finding algorithm into known reductions yields an Õ(n^{17/21})-round parallel algorithm for matroid intersection, as well as an Õ(n^{3/7})-round parallel algorithm for approximate monotone submodular maximization under a matroid constraint.

Cite As Get BibTex

Sanjeev Khanna, Aaron Putterman, and Junkai Song. An Õ(n^{3/7}) Round Parallel Algorithm for Matroid Bases. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 124:1-124:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.124

Author Details

Sanjeev Khanna
  • Courant Institute, Warren Weaver Hall, New York University, NY, USA
Aaron Putterman
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Junkai Song
  • Courant Institute, Warren Weaver Hall, New York University, NY, USA

Funding

  • Khanna, Sanjeev: Supported in part by NSF award CCF-2625203 and AFOSR award FA9550-25-1-0107.
  • Putterman, Aaron: Supported in part by the Simons Investigator Awards of Madhu Sudan and Salil Vadhan and AFOSR award FA9550-25-1-0112.
  • Song, Junkai: Supported in part by NSF award CCF-2625203.

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