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CS-151 · computer science · mathematics · Algorithms and data structures · Graph theory and combinatorics

Entropy bounds at perfect-matching marginals

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let G be a loopless multigraph on 2m vertices and x in its perfect-matching polytope. Is the maximum entropy H(x) of a distribution of perfect matchings having edge marginals x between $F(x)-(2-2/m)B(x)$ and F(x), where $F(x)=-\sum _e$ $x_e$ $\log (x_e)$ and B is the sum of $-(1-x_e)\log (1-x_e)$? Natural logarithms are used and 0 log 0 is zero.

Short English statement

Let G be a loopless multigraph on 2m vertices and x in its perfect-matching polytope. Is the maximum entropy H(x) of a distribution of perfect matchings having edge marginals x between $F(x)-(2-2/m)B(x)$ and F(x), where $F(x)=-\sum _e$ $x_e$ $\log (x_e)$ and B is the sum of $-(1-x_e)\log (1-x_e)$? Natural logarithms are used and 0 log 0 is zero.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-151-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-151

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).

Entropy bounds at perfect-matching marginals

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 feasible perfect-matching marginals satisfy the sharp entropy comparison $F-(2-2/m)B\le H\le F$, together with the face bound $|\operatorname{supp}x|-\dim F_x\le3m-2$.

Mathematical statement of the claimed result

Manuscript 1.
For any finite loopless multigraph $G$ on $2m\ge2$ vertices with a perfect matching, let $P(G)$ be the convex hull of matching indicators. For $x\in P(G)$ define \[ H(x)=\max_{\sum_Mp_M\mathbf1_M=x}(-\sum_Mp_M\log p_M),\quad F(x)=-\sum_ex_e\log x_e,\quad B(x)=-\sum_e(1-x_e)\log(1-x_e), \] where $p$ ranges over probability laws on perfect matchings and $0\log0=0$. The claimed pointwise estimate is \[ F(x)-(2-2/m)B(x)\le H(x)\le F(x). \] Parallel edges are separate choices and coordinates; connectedness and bipartiteness are not assumed.

Outline of the argument

For manuscript 1 (family 113; 2026-09-23), tight odd cuts are uncrossed and contracted to bound the tangent-space codimension. A finite exponential family on the minimal face then turns covariance trace and second-variation estimates into the entropy bound, including boundary marginals.

Formal verification

For manuscript 1 (family 113; 2026-09-23), the documented Lean scope is the sharp matching entropy comparison, boundary cases, and a triangle-expansion face identity; the general sharp face-dimension bound is not among the selected declarations.

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

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