Maximum Matching and Related Problems in Catalytic Logspace

TL;DR

This paper advances the understanding of space-bounded computation with catalytic space, showing that maximum matching in general graphs and related problems can be solved within the class CL and CLP, extending previous results.

Contribution

It introduces algorithms for maximum matching in general graphs within CL and CLP, and applies these to related problems like maximum rank completion and Edmond's problem approximation.

Findings

Maximum matching in general graphs can be computed in CL.

Maximum rank completion problem is solvable in CLP.

Approximate solutions for Edmond's problem can be obtained in CLP.

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.