Almost-linear-time exact matching and prescribed-degree factors in general graphs

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Gives a randomized algorithm finding an exact maximum-cardinality matching in any simple undirected graph in \((n+m)^{1+o(1)}\) word time, with success probability at least 2/3. The time bound holds on every computation path. The same guarantees apply to finding a spanning subgraph with prescribed admissible vertex degrees, or deciding that none exists.

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Theorem 1.1 (Exact matching in almost-linear input-size time)

In Almost-Linear-Time Maximum-Cardinality Matching in General Graphs, 24 September 2026

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There are a single uniform randomized word-RAM algorithm, fixed constants c_0\ge2 and C\ge1, and a nonincreasing function \eta:\mathbb N_{\ge1}\to\mathbb R_{\ge0} with \eta(q)\to0 as q\to\infty such that the following holds. For every input graph G=([n],E) with m=|E| and p=n+m, the algorithm halts on every computation path in at most

C p^{1+\eta(p)}

instructions. With probability at least 2/3 it outputs an explicit list of edges of a matching of size \nu(G).

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Almost-Linear-Time Maximum-Cardinality Matching in General Graphs

24 September 2026, current edition.

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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.

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