arxiv: 2604.24275 · v1 · submitted 2026-04-27 · 💻 cs.CC

Recognition: unknown

Maximum Matching and Related Problems in Catalytic Logspace

Srijan Chakraborty , Samir Datta , Aryan Kusre , Partha Mukhopadhyay , Amit Sinhababu

Authors on Pith no claims yet

Pith reviewed 2026-05-07 17:10 UTC · model grok-4.3

classification 💻 cs.CC

keywords catalytic logspacemaximum matchinggeneral graphslinear algebrarank computationEdmond's problemmatroid intersection

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The pith

The size of a maximum matching in general graphs can be determined in catalytic logspace, and the matching itself can be constructed in CLP.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that the size of a maximum matching in general graphs can be determined in catalytic logspace by adapting an existing linear-algebraic method. It then extends the technique to construct an actual maximum matching within the CLP model. Related tasks such as maximum rank completion and a constant-factor approximation for the rank in Edmond's problem are also placed in CLP. A reader cares because these results move a core combinatorial problem from polynomial time into a model that uses only logarithmic working space plus a reusable catalytic tape.

Core claim

The size of a maximum matching in general graphs can be determined in CL by simulating Geelen's linear-algebraic algorithm. This algorithm, together with new ideas, can be used to find a maximum matching in general graphs in CLP. Using a similar algorithm of Geelen, the maximum rank completion problem is solved in CLP. A PTAS for approximating the rank in Edmond's problem yields a CLP algorithm that approximates the rank to within a factor of (1-ε) for any ε in (0,1).

What carries the argument

Geelen's linear-algebraic algorithm for maximum matching, which reduces the problem to matrix rank computations over fields and is simulated inside catalytic logspace.

If this is right

The size of a maximum matching in general graphs is in CL.
A maximum matching in general graphs can be constructed in CLP.
The maximum rank completion problem is in CLP.
The rank in any instance of Edmond's problem can be approximated to within (1-ε) in CLP.
Approximating the maximum independent matching in the linear matroid matching problem is in CLP.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Catalytic logspace may be sufficient for other algebraic reductions that underlie graph algorithms.
Similar simulation techniques could place additional matroid problems in CLP.
The result suggests that polynomial-time algebraic algorithms for matching can often be made space-efficient when catalytic space is available.
If the approach generalizes, it may connect catalytic space to derandomized versions of randomized algebraic algorithms.

Load-bearing premise

Geelen's linear-algebraic algorithm and the PTAS for rank approximation can be implemented or simulated inside catalytic logspace without exceeding the space or catalytic restoration constraints.

What would settle it

An explicit family of graphs where the rank computations in Geelen's reduction require more than logarithmic catalytic space or fail to restore the catalyst exactly would falsify the claim.

read the original abstract

Understanding the power of space-bounded computation with access to catalytic space has been an important theme in complexity theory over the recent years. One of the key algorithmic results in this area is that bipartite maximum matching can be computed in catalytic logspace with a polynomial-time bound, Agarwala and Mertz (2025). In this paper, we show that we can construct a \emph{maximum matching} in \emph{general graphs} in CL, and, in fact, in CLP. We first show that the size of a \emph{maximum matching} in \emph{general graphs} can be determined in CL. Our algorithm is based on the linear-algebraic algorithm for maximum matching by Geelen (2000). We then show that this algorithm, along with some new ideas, can be used to \emph{find} a maximum matching in general graphs. Using a similar algorithm of Geelen (1999), we also solve the \emph{maximum rank completion problem} in CLP, which was previously known to be solvable in deterministic polynomial time, Geelen. This problem turns out to be equivalent to the \emph{linear matroid intersection} problem (shown by Murota, 1995) which has been shown to be in CLP by Agarwala, Alekseev, and Vinciguerra (2026). Finally, using a PTAS algorithm Bl\"{a}ser, Jindal and Pandey (2018), for approximating the rank in Edmond's problem, we derive a CLP algorithm that can approximate the rank given by any instance of the \emph{Edmond's problem} upto a factor of $(1-\eps)$ for any $\eps\in(0,1)$. An application of this is a CLP bound for approximating the maximum independent matching in the \emph{linear matroid matching} problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

General matching reaches CL and CLP by adapting Geelen's algebraic reduction, but the catalytic restoration steps need close checking.

read the letter

The main point is that maximum matching in general graphs is now claimed to be in CL for the size and in CLP for constructing the matching itself. The paper also puts maximum rank completion into CLP and gives a CLP approximation algorithm for rank in Edmond's problem up to (1-eps). This extends the earlier bipartite matching result in catalytic logspace and ties into matroid intersection via known equivalences. They rely on Geelen's 2000 linear-algebraic approach for the core reduction to matrix rank, plus some new ideas for the actual matching construction, and they simulate the Bläser et al. PTAS for the approximation part. The citation pattern is straightforward and builds cleanly on the cited prior work without obvious gaps. What stands out is the clean extension from bipartite to general graphs while staying inside the catalytic model. The matroid applications follow directly once rank completion is handled. The approach stays grounded in existing algorithms rather than requiring entirely new combinatorial machinery. The soft spot is the space accounting for the algebraic steps. Standard rank or determinant routines involve matrix operations that overwrite data or use non-reversible arithmetic. The abstract notes that new ideas handle the construction and restoration, but it is not obvious how these ideas ensure every catalytic cell returns exactly to its starting value on non-bipartite instances. The stress-test concern about reversible Gaussian elimination or field arithmetic is reasonable until the proofs are examined in detail. If the restoration arguments hold, the result is solid; if they require extra logspace or fail to recompute intermediates from the work tape alone, the claim would need revision. This paper is for complexity researchers who follow catalytic space and its algorithmic reach. Readers already familiar with the bipartite CL result and matroid problems in restricted models will get the most out of it. It deserves a serious referee because the claims are specific, the prior-art connections are clear, and the potential advance is meaningful if the details check out. I would send it to peer review, asking referees to focus on the catalytic restoration proofs and the exact space usage in the algebraic subroutines.

Referee Report

3 major / 2 minor

Summary. The manuscript claims to show that the size of a maximum matching in general graphs can be computed in catalytic logspace (CL) by adapting Geelen's (2000) linear-algebraic algorithm based on rank computation over a Tutte-like matrix. It further claims that a maximum matching itself can be constructed in CLP using this algorithm together with new ideas for the catalytic model. Additional results include solving the maximum rank completion problem in CLP (equivalent to linear matroid intersection) via Geelen (1999), and a CLP algorithm for (1-ε)-approximating the rank in Edmond's problem using the PTAS of Bläser et al. (2018), with an application to approximating maximum independent matching in linear matroid matching.

Significance. If the catalytic-space implementations hold, the results would meaningfully extend the known power of CL from bipartite matching (Agarwala-Mertz 2025) to general graphs and connect catalytic space to matroid problems. The technical contribution would lie in showing that linear-algebraic routines can be made to satisfy the exact catalytic-tape restoration invariant, which is a non-trivial constraint. The approximation result adds a quantitative angle with potential for further algorithmic applications.

major comments (3)

[Abstract and algorithm for determining matching size] The central claim that maximum-matching size lies in CL rests on implementing Geelen (2000)'s reduction to matrix rank in the catalytic model. Standard algebraic primitives (Gaussian elimination, field arithmetic, determinant evaluation) are not obviously reversible; any intermediate matrix or overwrite that cannot be recomputed solely from the O(log n) work tape would violate the catalytic restoration requirement. The abstract invokes 'some new ideas' for this adaptation on non-bipartite instances, but without an explicit description of the reversible implementation or a space-accounting argument showing that the catalytic tape is returned to its exact pre-computation state, the claim cannot be verified. This is load-bearing for the entire result.
[Section on constructing the maximum matching] The extension from computing the size to constructing an actual maximum matching in CLP requires additional mechanisms (e.g., recovering the matching edges or performing matrix modifications while preserving the catalytic invariant). The manuscript must supply a concrete space analysis or pseudocode showing how the new ideas handle output and search steps without irreversible overwrites on the catalytic tape.
[Section on approximation for Edmond's problem] The CLP algorithm for (1-ε)-approximating rank in Edmond's problem (via Bläser-Jindal-Pandey 2018) and its application to linear matroid matching must confirm that the PTAS can be simulated inside catalytic logspace while maintaining polynomial time and exact restoration. Any derandomization or space overhead introduced by the approximation procedure needs explicit accounting.

minor comments (2)

Clarify the precise relationship between the new CLP result for maximum rank completion and the prior CLP result of Agarwala-Alekseev-Vinciguerra (2026) for linear matroid intersection; state whether the Geelen (1999) route yields a genuinely different proof or merely an alternative presentation.
Ensure all citations include full bibliographic details (e.g., complete reference for Geelen 2000 and Bläser et al. 2018) and that any new technical lemmas introduced for catalytic restoration are clearly labeled.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight the need for greater explicitness in describing the catalytic-space implementations, which we will address in the revision.

read point-by-point responses

Referee: [Abstract and algorithm for determining matching size] The central claim that maximum-matching size lies in CL rests on implementing Geelen (2000)'s reduction to matrix rank in the catalytic model. Standard algebraic primitives (Gaussian elimination, field arithmetic, determinant evaluation) are not obviously reversible; any intermediate matrix or overwrite that cannot be recomputed solely from the O(log n) work tape would violate the catalytic restoration requirement. The abstract invokes 'some new ideas' for this adaptation on non-bipartite instances, but without an explicit description of the reversible implementation or a space-accounting argument showing that the catalytic tape is returned to its exact pre-computation state, the claim cannot be verified. This is load-bearing for the entire result.

Authors: We agree that an explicit reversible implementation and space accounting are essential for verifying the CL claim. In the revised manuscript we will add a dedicated subsection that spells out the reversible Gaussian-elimination procedure for the Tutte-like matrix, showing step-by-step how every overwrite on the catalytic tape is undone using only the O(log n) work tape and the original input. We will also supply a precise space-accounting argument confirming that the catalytic tape is restored to its exact pre-computation state after each rank-computation phase. revision: yes
Referee: [Section on constructing the maximum matching] The extension from computing the size to constructing an actual maximum matching in CLP requires additional mechanisms (e.g., recovering the matching edges or performing matrix modifications while preserving the catalytic invariant). The manuscript must supply a concrete space analysis or pseudocode showing how the new ideas handle output and search steps without irreversible overwrites on the catalytic tape.

Authors: We will include both pseudocode and a detailed space analysis for the matching-construction procedure. The new ideas allow us to recover a maximum matching by a sequence of matrix modifications that are performed while maintaining an auxiliary catalytic copy of the original matrix; each modification is reversed after the corresponding edge is recorded on the output tape. The revised section will explicitly verify that the catalytic tape returns to its initial state after every search step. revision: yes
Referee: [Section on approximation for Edmond's problem] The CLP algorithm for (1-ε)-approximating rank in Edmond's problem (via Bläser-Jindal-Pandey 2018) and its application to linear matroid matching must confirm that the PTAS can be simulated inside catalytic logspace while maintaining polynomial time and exact restoration. Any derandomization or space overhead introduced by the approximation procedure needs explicit accounting.

Authors: We will expand the approximation section with an explicit simulation of the PTAS inside the CLP model. The revised text will detail how the randomized steps of Bläser et al. are derandomized using the catalytic tape (via standard space-bounded derandomization techniques that fit within logarithmic work space) and will provide a concrete accounting of the resulting polynomial-time bound and the exact restoration invariant. No additional asymptotic space overhead is incurred beyond the logarithmic work tape. revision: yes

Circularity Check

0 steps flagged

No circularity; central claims adapt external algorithms to CL model

full rationale

The paper derives its CL and CLP bounds for maximum matching and related problems by adapting the linear-algebraic method of Geelen (2000) and the PTAS of Bläser et al. (2018), both independent prior results by non-overlapping authors. It also references Murota (1995) for equivalence to matroid intersection and prior CLP results by Agarwala et al. These are treated as black-box inputs whose implementation inside catalytic logspace (with restoration) is the novel contribution. No equation or claim reduces a derived quantity to a fitted parameter from the same data, no self-citation chain bears the load of a uniqueness theorem, and no ansatz is smuggled via the authors' own prior work. The derivation therefore remains non-circular and externally falsifiable against the cited algorithms.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background results in complexity theory (properties of catalytic space, linear algebra over fields, and known polynomial-time algorithms) rather than new free parameters or invented entities. No ad-hoc constants or new postulates are introduced in the abstract.

axioms (2)

domain assumption Catalytic logspace machines can simulate the linear-algebraic operations of Geelen's algorithm while restoring catalytic memory.
Invoked when claiming the size and construction of maximum matching are in CL/CLP.
domain assumption The PTAS of Bläser, Jindal and Pandey can be adapted to catalytic space.
Used for the approximation result for Edmond's problem.

pith-pipeline@v0.9.0 · 5658 in / 1424 out tokens · 56859 ms · 2026-05-07T17:10:54.076072+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references

[1]

Linear matroid intersection is in catalytic logspace

[AAV26] Aryan Agarwala, Yaroslav Alekseev, and Antoine Vinciguerra. Linear matroid intersection is in catalytic logspace. InProceedings of the 17th Innovations in Theoretical Computer Science Conference (ITCS 2026), volume 362 ofLeibniz International Proceedings in Informatics (LIPIcs), pages 3:1–3:23,

2026
[2]

Bipartite matching is in catalytic logspace

[AM25] Aryan Agarwala and Ian Mertz. Bipartite matching is in catalytic logspace. In66th IEEE Annual Symposium on Foundations of Com- puter Science, FOCS 2025, Sydney, Australia, December 14-17, 2025, pages 360–372. IEEE,

2025
[3]

Catalytic computation

[BCK+14] Harry Buhrman, Richard Cleve, Michal Kouck´ y, Bruno Loff, and Florian Speelman. Catalytic computation. InProceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC 2014), pages 101–110. ACM,

2014
[4]

Derandomizing logspace with a small shared hard drive

[Pyn24] Edward Pyne. Derandomizing logspace with a small shared hard drive. In Rahul Santhanam, editor,39th Computational Complex- ity Conference, CCC 2024, July 22-25, 2024, Ann Arbor, MI, USA, volume 300 ofLIPIcs, pages 4:1–4:20. Schloss Dagstuhl - Leibniz- Zentrum f¨ ur Informatik,

2024