Result 113, Theoretical computer science

Approximate counting and entropy of perfect matchings

Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts. Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector in a loopless labelled multigraph, including boundary points.

Lean formalization Proof

The bigger picture

Why it matters

A perfect matching pairs every vertex of a graph with exactly one other vertex along an edge. These manuscripts claim ways to approximate how many such pairings exist and bound how much randomness distributions over them can contain.

What changes?

The counting manuscript reports a randomized approximation scheme for arbitrary finite simple undirected graphs, which have neither loops nor repeated edges. It detects a zero count with certainty. Otherwise, relative error is at most epsilon with failure probability at most delta, in worst-case bit time polynomial in input length, 1/epsilon and log(1/delta). The entropy manuscript treats feasible edge-use probabilities in loopless labelled multigraphs, where repeated edges are allowed, on 2m vertices, with m at least one.

What does that help mathematicians do?

For fixed edge-use probabilities x, let H be the greatest entropy, or distributional uncertainty, among perfect-matching distributions realizing x. Define F as the sum over edges of -x log x, and B as the sum of -(1-x) log(1-x), with 0 log 0 interpreted as zero. The claimed bound is F - (2 - 2/m)B ≤ H ≤ F, including boundary points. Researchers can thus bound global uncertainty from individual edge probabilities, with an explicit limit on the gap.

Are there practical applications?

The counting guarantee offers a theoretical tool for estimating the number of complete pairings, not merely testing whether one exists. Its runtime guarantee specifies how computational cost scales with accuracy and confidence, but the abstracts provide no implementation benchmarks. The entropy result's immediate value is foundational: it constrains how much randomness can coexist with prescribed edge-use probabilities.

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.

2 manuscripts

A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs

September 23, 2026 45 pages

We give a fully polynomial randomized approximation scheme (FPRAS) for counting perfect matchings in arbitrary finite simple undirected graphs, resolving the general-graph perfect-matching approximation problem. The algorithm returns zero with certainty when no perfect matching exists. Otherwise, it achieves relative error ε with failure probability at most δ in worst-case bit time polynomial in the input length, ε−1\varepsilon ^{-1}, and log⁡δ−1\log\delta^{-1}.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026,
  author = {{OpenAI}},
  title = {{A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf}{OAI:A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026}},
  year = {2026}
}

Entropy and Face Dimension of the Perfect-Matching Polytope

September 23, 2026 45 pages Main result formalized in Lean

We prove the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant. For every feasible vector x of perfect-matching edge marginals in a loopless labelled multigraph on 2m≥22m\ge2 vertices, the maximum entropy H(x)H(x) of a matching law with marginals x satisfies

F(x)−(2−2/m)B(x)≤H(x)≤F(x),\displaystyle F(x)-(2-2/m)B(x)\le H(x)\le F(x),

where F(x)=−∑exelog⁡xeF(x)=-\sum_e x_e\log x_e and B(x)=−∑e(1−xe)log⁡(1−xe)B(x)=-\sum_e(1-x_e)\log(1-x_e). This bound holds throughout the polytope, including its boundary. We also prove the sharp bound ∣suppx∣−dim⁡Fx≤3m−2|\mathop{\mathrm{supp}}\nolimits x|-\dim F_x\le3m-2, where Fx is the minimal face of the perfect-matching polytope containing x.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026,
  author = {{OpenAI}},
  title = {{Entropy and Face Dimension of the Perfect-Matching Polytope}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026/main.pdf}{OAI:Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/113.md.

Approximate counting and entropy of perfect matchings

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization gives a fully polynomial randomized approximation scheme for counting perfect matchings in every finite simple undirected graph. For rational 0<ε<10<\varepsilon<1 and 0<δ<1/20<\delta<1/2, it returns a nonnegative rational estimate with relative error at most ε\varepsilon with probability at least 1−δ1-\delta. If the graph has no perfect matching, every execution returns zero.

The algorithm is a fixed finite-alphabet randomized machine. Its worst-case running time is polynomial in the binary input length, ε−1\varepsilon^{-1}, and log⁡(δ−1)\log(\delta^{-1}), including on unsuccessful random tapes.

For feasible edge marginals xx of perfect matchings in a loopless multigraph on 2m2m vertices, let F(x)=−∑exelog⁡xeF(x)=-\sum_e x_e\log x_e, B(x)=−∑e(1−xe)log⁡(1−xe)B(x)=-\sum_e(1-x_e)\log(1-x_e), and let H(x)H(x) be the maximum entropy of a matching law with those marginals. The formalization proves F(x)−(2−2/m)B(x)≤H(x)≤F(x)F(x)-(2-2/m)B(x)\le H(x)\le F(x) for m≥1m\ge1, including boundary points of the polytope. It retains the earlier coefficient-eight entropy bound and the deterministic counting approximations with factors 512n512^n for loopless graphs and 218n2^{18n} with singleton loops.

A further statement identifies the exact face obtained by expanding a degree-three vertex into a triangle and shows that minimal faces are carried to minimal faces by the expansion. The paper's sharp global face-dimension bound is not part of that selected statement.

Comparator links

Result Comparator statement
Randomized approximation of the perfect-matching count MatchingFPRAS.lean
Perfect-matching entropy bound MatchingEntropy.lean
Deterministic approximate matching count BinaryMatching.lean
Approximate counting with singleton loops SingletonLoopMatching.lean
Refined pointwise perfect-matching entropy bounds MatchingEntropyBounds.lean
Face correspondence under triangle expansion TriangleFace.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 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.