Maximum Matching and Related Problems in Catalytic Logspace

2026-04-27

AI Summary

This work investigates the efficient solvability of general graph maximum matching and related algebraic combinatorial optimization problems within catalytic logspace (CL/CLP). Building upon Geelen’s linear algebraic matching algorithm and integrating tools from matroid theory, rank approximation, and PTAS techniques, the study achieves—for the first time—the complete constructive solution to maximum matching in CLP. It further extends the scope of tractable problems in this model to include maximum rank matrix completion and linear matroid intersection. Additionally, the paper presents a (1−ε)-approximation algorithm in CLP for Edmonds’ problem. These results substantially broaden the frontier of both exact and approximate problems solvable in catalytic logspace.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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 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 in the \emph{linear matroid matching} problem.

Problem

• maximum matching
• catalytic logspace
• maximum rank completion
• linear matroid intersection
• Edmond's problem

Innovation

• catalytic logspace
• maximum matching
• general graphs
• linear matroid intersection
• Edmonds' problem