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CS-139 · computer science · mathematics · Algorithms and data structures · Computational complexity · Graph theory and combinatorics · Optimization

Exponential semidefinite complexity for perfect matching

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every fixed shift $\rho \in (0,1)$, do the matrices indexed by odd vertex sets U and perfect matchings M, with entry $|M\cap \delta (U)|-1+\rho $, where $\delta (U)$ is the set of edges crossing U, of complete graphs on even n vertices require real positive-semidefinite factorization rank at least $2^{cn}$, for one $c>0$ independent of $\rho $ and all sufficiently large n depending on $\rho $? This also asks for the resulting exponential exact semidefinite-lift lower bound.

Short English statement

For every fixed shift $\rho \in (0,1)$, do the matrices indexed by odd vertex sets U and perfect matchings M, with entry $|M\cap \delta (U)|-1+\rho $, where $\delta (U)$ is the set of edges crossing U, of complete graphs on even n vertices require real positive-semidefinite factorization rank at least $2^{cn}$, for one $c>0$ independent of $\rho $ and all sufficiently large n depending on $\rho $? This also asks for the resulting exponential exact semidefinite-lift lower bound.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-139-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-139

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 126; 2026-10-05).

Exponential semidefinite complexity for perfect 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 126; 2026-10-05), the asserted result is that each fixed positive shift below one in the matching slack matrix has exponential real PSD rank, implying exponential size for exact semidefinite lifts of the matching polytope.

Mathematical statement of the claimed result

Manuscript 1.
For even $n\ge4$, index rows by odd subsets $U\subseteq[n]$ and columns by perfect matchings $M$ of $K_n$, and define \[ A_n(\rho)_{U,M}=|M\cap\delta(U)|-1+\rho. \] Real PSD rank is the smallest $r$ permitting exact factorizations $A_{U,M}=\operatorname{tr}(F_UG_M)$ with real symmetric positive semidefinite $r\times r$ matrices. The asserted bound is \[ \exists c>0\ \forall\rho\in(0,1)\ \exists n_0(\rho)\ \forall\text{ even }n\ge n_0(\rho):\quad \operatorname{rank}_{\rm psd}A_n(\rho)\ge2^{cn}. \] The exponential constant is uniform in $\rho$, but the threshold may depend on it. No symmetry or precision restrictions are imposed on the factors.

Outline of the argument

For manuscript 1 (family 126; 2026-10-05), coordinatewise product averages are expressed as sums of matrix squares. Orthogonal frequency splitting bounds a Gram potential through the averaging process; matching smoothing and an expanding parity system force a rank large enough to support the remaining Fourier mass.

Formal verification

For manuscript 1 (family 126; 2026-10-05), the documented Lean scope is superpolynomial PSD rank for the selected matching slack matrix and superpolynomial size of exact lifts; the claimed exponential bound for every fixed positive shift is outside the selected statement.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are MatchingAffineLift.lean, MatchingPSD.lean.

Their named declarations include OAI.PerfectMatchingPSD.affine_lift_lower_bound, OAI.PerfectMatchingPSD.main.

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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