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 Bläser, 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 size in the \emph{linear matroid matching} problem.

翻译：理解带催化空间的空间有界计算能力是近年来复杂性理论中的一个重要主题。该领域的关键算法成果之一是二分图最大匹配可在催化对数空间中以多项式时间界计算（Agarwala 和 Mertz, 2025）。在本文中，我们证明可以在 CL 中构建一般图上的 \emph{最大匹配}，实际上更是在 CLP 中实现。首先，我们证明一般图上 \emph{最大匹配} 的大小可在 CL 中确定。我们的算法基于 Geelen (2000) 提出的最大匹配线性代数算法。随后，我们展示该算法结合一些新思想可用于在一般图上 \emph{寻找} 最大匹配。利用 Geelen (1999) 的类似算法，我们还在 CLP 中解决了 \emph{最大秩完成问题}，该问题先前已知可在确定性多项式时间内求解（Geelen）。此问题等价于 \emph{线性拟阵交} 问题（由 Murota, 1995 指出），而 Agarwala、Alekseev 和 Vinciguerra (2026) 已证明该问题属于 CLP。最后，利用 Bläser、Jindal 和 Pandey (2018) 提出的用于近似 Edmond 问题中秩的多项式时间近似方案（PTAS），我们推导出一个 CLP 算法，该算法能够对任意 \emph{Edmond 问题} 实例的秩进行近似，近似因子可达 $(1-\eps)$，其中 $\eps\in(0,1)$。这一结果的一个应用是为 \emph{线性拟阵匹配} 问题中最大独立匹配大小的近似提供了 CLP 界。