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Optimal Parallel Basis Finding in Graphic and Related Matroids

Authors Sanjeev Khanna , Aaron Putterman , Junkai Song

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LIPIcs.ICALP.2026.125.pdf
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ACM Subject Classification
  • Theory of computation → Parallel algorithms
Keywords
  • parallel algorithms
  • matroids

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Abstract

We study the parallel complexity of finding a basis of a graphic matroid under independence-oracle access. Karp, Upfal, and Wigderson (FOCS 1985, JCSS 1988) initiated the study of this problem and established two algorithms for finding a spanning forest: one running in O(log m) rounds with m^{Θ(log m)} queries, and another, for any d ∈ ℤ^+, running in O(m^{2/d}) rounds with Θ(m^d) queries. A key open question they posed was whether one could simultaneously achieve polylogarithmic rounds and polynomially many queries. 
We give a deterministic algorithm that uses O(log m) adaptive rounds and poly(m) non-adaptive queries per round to return a spanning forest on m edges, and complement this result with a matching Ω(log m) lower bound for any (even randomized) algorithm with poly(m) queries per round. Thus, the adaptive round complexity for graphic matroids is characterized exactly, settling this long-standing problem.
Beyond graphs, we show that our framework also yields an O(log m)-round, poly(m)-query algorithm for any binary matroid satisfying a smooth circuit counting property, implying, among others, an optimal O(log m)-round parallel algorithms for finding bases of cographic matroids. Finally, we conjecture a natural strengthening of known circuit-counting bounds for the much broader class of regular matroids and even an extension to so-called max-flow min-cut matroids; assuming it, our algorithm achieves the same O(log m) rounds and poly(m) queries for all such matroids - which includes graphic and cographic matroids as special cases.

Cite As Get BibTex

Sanjeev Khanna, Aaron Putterman, and Junkai Song. Optimal Parallel Basis Finding in Graphic and Related Matroids. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 125:1-125:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.125

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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