Maximum-cardinality matching in general graphs in almost-linear time

Theorem 1.1: a single uniform randomized algorithm finds a maximum-cardinality matching of any simple graph in $(n+m)^{1+o(1)}$ word operations on every path, with success probability at least $2/3$ and explicit output. Corollary: the same guarantees for deciding and constructing $f$ -factors (prescribed-degree spanning subgraphs). It is randomized (no deterministic version), unweighted (maximum-weight general matching is not addressed), and the $o(1)$ is not made explicit.

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Maximum-cardinality matching in general graphs in almost-linear time · Almost-linear general matching

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VibeMathed record: Maximum-cardinality matching in general graphs in almost-linear time

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No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against the general-graph matching question. Theorem 1.1 states one uniform randomized word-RAM algorithm that halts within $Cp^{1+\eta(p)}$ instructions on every computation path ( $p=n+m$ , $\eta\to0$ ) and outputs an explicit maximum matching with probability at least $2/3$ . It is Monte Carlo: correctness of the output is not certified. The algorithm uses the deterministic almost-linear flow theorem (van den Brand et al.) as a black box. Not refereed here. No Lean formalization.

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Maximum-cardinality matching in general graphs in almost-linear time

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