Almost-linear maximum-cardinality matching
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
Can one uniform randomized logarithmic-word RAM algorithm find an explicit maximum-cardinality matching in every finite general graph with probability $\geq 2/3$, halting on every path within $(n+m)^{1+o(1)}$ instructions?Short English statement
Can one uniform randomized logarithmic-word RAM algorithm find an explicit maximum-cardinality matching in every finite general graph with probability $\geq 2/3$, halting on every path within $(n+m)^{1+o(1)}$ instructions?Sources
- S1 Original question source or mathematical context. https://arxiv.org/pdf/1704.01929.
- S2 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for CS-157
The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 120; 2026-09-24).
Almost-linear maximum-cardinality matching
OpenAI's claim
OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 120; 2026-09-24), the asserted result is that general simple graphs admit a uniform randomized algorithm producing a maximum matching in $(n+m)^{1+o(1)}$ time on every path with success probability at least $2/3$, also yielding prescribed-degree subgraphs.Mathematical statement of the claimed result
Manuscript 1.
For each finite simple general graph supplied as adjacency lists (a boundary array and two neighbor occurrences per undirected edge) $G=([n],E)$, put $m=|E|$, $p=n+m$ and let $\nu(G)$ be its maximum matching size. One uniform randomized word-RAM algorithm is claimed with absolute constants $C\ge1,c_0\ge2$ and a nonincreasing $\eta(p)\ge0$ tending to zero, using words of $\lceil c_0\log_2(n+2)\rceil$ bits, such that \[ T(G)\le Cp^{1+\eta(p)}\quad\hbox{on every random path},\qquad \Pr[\text{output is a matching of size }\nu(G)]\ge2/3. \] The matching is explicitly output as an edge list, and the graph need not be bipartite.Outline of the argument
For manuscript 1 (family 120; 2026-09-24), factor-critical contraction and Hall reassignment maintain explicit matching states. Multiscale allocation ledgers control debt and resets; a cut-matching routing helper with conductance estimates keeps the required searches within the almost-linear budget.Formal verification
No Lean formalization is listed at this snapshot for manuscript 1 (family 120; 2026-09-24).This site has not executed the Lean code. Described statement coverage is not an independent proof check.
Status caveat
The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.Sources
- OpenAI, Almost-Linear-Time Maximum-Cardinality Matching in General Graphs, 2026-09-24; citation key
OAI:Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.