we-solve (beta)

CS-135 · computer science · mathematics · Algorithms and data structures · Graph theory and combinatorics

FPRAS for perfect matchings in general graphs

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Can the number of perfect matchings of an arbitrary explicitly given finite general graph be estimated to relative error $\varepsilon $ with failure probability $\delta $ in worst-case bit time polynomial in input length, $\varepsilon ^{-1}$ and $\log (\delta ^{-1})$, with zero returned certainly when there are no perfect matchings?

Short English statement

Can the number of perfect matchings of an arbitrary explicitly given finite general graph be estimated to relative error $\varepsilon $ with failure probability $\delta $ in worst-case bit time polynomial in input length, $\varepsilon ^{-1}$ and $\log (\delta ^{-1})$, with zero returned certainly when there are no perfect matchings?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-135-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-135

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 113; 2026-09-23).

FPRAS for perfect matchings in general graphs

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 113; 2026-09-23), the asserted result is that perfect matchings in arbitrary simple undirected graphs admit an FPRAS with polynomial work on every tape and certain zero output when no matching exists.

Mathematical statement of the claimed result

Manuscript 1.
For every explicitly encoded finite general graph $G$, let $Z(G)$ count its perfect matchings. Given rational $0<\varepsilon<1$ and $0<\delta<1/2$, the claimed uniform classical randomized algorithm returns $\widehat Z\in\mathbb Q_{\ge0}$ with \[ \Pr[(1-\varepsilon)Z(G)\le\widehat Z\le(1+\varepsilon)Z(G)]\ge1-\delta. \] It outputs zero surely if $Z(G)=0$, and its worst-case bit cost is polynomial in the complete input length, $\varepsilon^{-1}$ and $\log\delta^{-1}$; bipartiteness is not required.

Outline of the argument

For manuscript 1 (family 113; 2026-09-23), weighted matching laws are enlarged to expose local partition ratios as marked edges. Subdivision controls unmatched-vertex costs, and alternating overlays plus a two-coordinate energy inequality bound the chain variance needed for annealing and counting.

Formal verification

For manuscript 1 (family 113; 2026-09-23), OpenAI lists Lean statements for this assertion: perfect matchings in arbitrary simple undirected graphs admit an FPRAS with polynomial work on every tape and certain zero output when no matching exists.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are BinaryMatching.lean, MatchingEntropy.lean, MatchingEntropyBounds.lean, MatchingFPRAS.lean, SingletonLoopMatching.lean, TriangleFace.lean.

Their named declarations include OAI.BinaryMatching.deterministic_approximate_counting, OAI.LoopMatching.fullEndpoint, OAI.MatchingEntropy.entropy_main, OAI.MatchingEntropyBounds.Refined.pointwise_entropy, OAI.MatchingFPRAS.thm_main, OAI.TriangleFace198.triangle_expansion_face.

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

Go to contribution →