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Linear Matroid Intersection Is in Catalytic Logspace

Authors Aryan Agarwala , Yaroslav Alekseev , Antoine Vinciguerra

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ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Submodular optimization and polymatroids
  • Theory of computation → Computational complexity and cryptography
  • Mathematics of computing → Graph algorithms
Keywords
  • Catalytic Computing
  • Computational Complexity
  • Matroid Theory
  • Algorithms

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Abstract

Linear matroid intersection is an important problem in combinatorial optimization. Given two linear matroids over the same ground set, the linear matroid intersection problem asks you to find a common independent set of maximum size. The deep interest in linear matroid intersection is due to the fact that it generalises many classical problems in theoretical computer science, such as bipartite matching, edge disjoint spanning trees, rainbow spanning tree, and many more.
We study this problem in the model of catalytic computation: space-bounded machines are granted access to catalytic space, which is additional working memory that is full with arbitrary data that must be preserved at the end of its computation. 
Although linear matroid intersection has had a polynomial time algorithm for over 50 years, it remains an important open problem to show that linear matroid intersection belongs to any well studied subclass of {P}. We address this problem for the class catalytic logspace (CL) with a polynomial time bound (CLP).
Recently, Agarwala and Mertz (2025) showed that bipartite maximum matching can be computed in the class CLP ⊆ {P}. This was the first subclass of {P} shown to contain bipartite matching, and additionally the first problem outside TC¹ shown to be contained in CL. We significantly improve the result of Agarwala and Mertz by showing that linear matroid intersection can be computed in CLP.

Cite As Get BibTex

Aryan Agarwala, Yaroslav Alekseev, and Antoine Vinciguerra. Linear Matroid Intersection Is in Catalytic Logspace. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 3:1-3:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.3

Author Details

Aryan Agarwala
  • Max-Planck-Institut für Informatik, Saarbrücken, Germany
Yaroslav Alekseev
  • Technion - Israel Institute of Technology, Haifa, Israel
Antoine Vinciguerra
  • Technion - Israel Institute of Technology, Haifa, Israel

Funding

  • Alekseev, Yaroslav: Supported by ISF grant 507/24.
  • Vinciguerra, Antoine: Supported by ISF grant 507/24.

Acknowledgements

All three authors thank Ian Mertz and Yuval Filmus for extensive discussions.

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