Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 120-01. Almost-Linear-Time Maximum-Cardinality Matching in General Graphs
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OpenAI@2026-10-07#rem:oai-p-120-01
Authors: OpenAI. Family 120.
Title: Almost-Linear-Time Maximum-Cardinality Matching in General Graphs.
Abstract: We prove that maximum-cardinality matching in a simple undirected graph with n vertices and m edges can be found by one uniform randomized algorithm in $(n+m){}{1+o(1)}$ time. The bound holds on every computation path in a logarithmic-word model, and the algorithm returns an explicit maximum matching with probability at least 2/3. An explicit reduction gives the same time and probability guarantees for deciding whether a simple host graph has a spanning subgraph with prescribed valid vertex degrees, and for finding one when it exists.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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