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DOI: 10.4230/LIPIcs.ITCS.2026.3

Linear Matroid Intersection Is in Catalytic Logspace

Authors: Aryan Agarwala, Yaroslav Alekseev, and Antoine Vinciguerra

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

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.

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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)

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@InProceedings{agarwala_et_al:LIPIcs.ITCS.2026.3,
  author =	{Agarwala, Aryan and Alekseev, Yaroslav and Vinciguerra, Antoine},
  title =	{{Linear Matroid Intersection Is in Catalytic Logspace}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{3:1--3:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.3},
  URN =		{urn:nbn:de:0030-drops-252908},
  doi =		{10.4230/LIPIcs.ITCS.2026.3},
  annote =	{Keywords: Catalytic Computing, Computational Complexity, Matroid Theory, Algorithms}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2025.6

Catalytic Computing and Register Programs Beyond Log-Depth

Authors: Yaroslav Alekseev, Yuval Filmus, Ian Mertz, Alexander Smal, and Antoine Vinciguerra

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract

In a seminal work, Buhrman et al. (STOC 2014) defined the class CSPACE(s,c) of problems solvable in space s with an additional catalytic tape of size c, which is a tape whose initial content must be restored at the end of the computation. They showed that uniform TC¹ circuits are computable in catalytic logspace, i.e., CL = CSPACE(O(log{n}), 2^{O(log{n})}), thus giving strong evidence that catalytic space gives L strict additional power. Their study focuses on an arithmetic model called register programs, which has been a focal point in development since then. Understanding CL remains a major open problem, as TC¹ remains the most powerful containment to date. In this work, we study the power of catalytic space and register programs to compute circuits of larger depth. Using register programs, we show that for every ε > 0, SAC² ⊆ CSPACE (O((log²n)/(log log n)), 2^{O(log^{1+ε} n)}) . On the other hand, we know that SAC² ⊆ TC² ⊆ CSPACE(O(log²{n}) , 2^{O(log{n})}). Our result thus shows an O(log log n) factor improvement on the free space needed to compute SAC², at the expense of a nearly-polynomial-sized catalytic tape. We also exhibit non-trivial register programs for matrix powering, which is a further step towards showing NC² ⊆ CL.

Cite as

Yaroslav Alekseev, Yuval Filmus, Ian Mertz, Alexander Smal, and Antoine Vinciguerra. Catalytic Computing and Register Programs Beyond Log-Depth. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{alekseev_et_al:LIPIcs.MFCS.2025.6,
  author =	{Alekseev, Yaroslav and Filmus, Yuval and Mertz, Ian and Smal, Alexander and Vinciguerra, Antoine},
  title =	{{Catalytic Computing and Register Programs Beyond Log-Depth}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.6},
  URN =		{urn:nbn:de:0030-drops-241136},
  doi =		{10.4230/LIPIcs.MFCS.2025.6},
  annote =	{Keywords: catalytic computing, circuit classes, polynomial method}
}

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Linear Matroid Intersection Is in Catalytic Logspace

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Catalytic Computing and Register Programs Beyond Log-Depth

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