Result 120, Theoretical computer science

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

Gives a randomized algorithm finding an exact maximum-cardinality matching in any simple undirected graph in (n+m)1+o(1)(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.

Algorithm or complexity result

The bigger picture

Why it matters

Choosing as many nonconflicting pairs as possible is a basic graph problem. The manuscript claims an exact algorithm whose running time is almost linear in the graph's size, rather than settling for an approximation.

What changes?

The unreviewed manuscript reports one uniform randomized algorithm for any simple undirected graph: links have no direction, loops or duplicates. With n vertices and m edges, it returns an explicit largest set of edges sharing no endpoints with probability at least 2/3, in (n+m)^(1+o(1)) time using logarithmic-size machine words. This bound holds on every computation path. The same guarantees cover finding a subgraph retaining all vertices with prescribed valid degrees, meaning incident-edge counts, or deciding none exists.

What does that help mathematicians do?

Exactness lets a researcher determine whether a graph supports a required number of disjoint pairs, not merely obtain a near-best collection. The prescribed-degree extension addresses a broader feasibility question: can the available edges meet every vertex's target simultaneously? The exponent in the reported time bound approaches one as input size grows, while the runtime guarantee is not merely an expectation over random choices.

Are there practical applications?

The immediate relevance is algorithmic: matching models allocating compatible pairs without using an item twice, while prescribed degrees model exact connection quotas. The reported result provides an asymptotic foundation for these computations on general graphs. It does not by itself establish practical speedups; implementation costs and performance on real inputs are not supplied.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

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

September 24, 2026 86 pages

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)(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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026,
  author = {{OpenAI}},
  title = {{Almost-Linear-Time Maximum-Cardinality Matching in General Graphs}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026/main.pdf}{OAI:Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.