Result 126, Theoretical computer science

Exponential semidefinite complexity of perfect matching

Proves that every exact semidefinite lift of the perfect matching polytope has exponential size, answering Rothvoss's polynomial-size lift question negatively. The bound holds even for the positive semidefinite rank of its odd-cut slack matrix after any fixed shift 0<ρ<10\lt \rho\lt 1, allowing arbitrary real positive semidefinite factors.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

A perfect matching pairs up a graph's vertices using edges, with each vertex used once. The manuscript claims that representing all such pairings exactly for complete graphs requires exponentially large semidefinite formulations, revealing a limit of this optimization framework.

What changes?

For complete graphs on even n vertices, the manuscript reports an exponential lower bound, two to the power c times n, on a matrix's real positive semidefinite rank: the matrix dimension required for unrestricted factors. Rows are odd-sized vertex sets; columns are perfect matchings. Entries count matching edges leaving the set, minus one plus rho. For every fixed rho strictly between zero and one, the bound holds as n grows, with c positive and possibly depending on rho.

What does that help mathematicians do?

The perfect matching polytope is the convex hull of vectors recording which edges each matching uses. An exact semidefinite lift describes this same shape by projecting a higher-dimensional feasible set defined by positive semidefinite constraints. The claimed bound rules out polynomial-size descriptions of this kind, answering Rothvoss's question negatively. Researchers therefore cannot obtain a compact exact formulation merely by replacing linear constraints with semidefinite ones.

Are there practical applications?

Its immediate value is foundational for optimization: it identifies a limit on how compactly a central combinatorial feasible region can be represented. This is a barrier to exact semidefinite formulations, not a proof that finding perfect matchings requires exponential time. The supplied result does not establish corresponding limits for approximate formulations.

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.

Manuscript

Exponential PSD rank of positively shifted matching matrices

October 5, 2026 24 pages

For every fixed 0<ρ<10\lt \rho\lt 1, the matrix indexed by odd vertex sets U and perfect matchings M of Kn, with entries ∣M∩δ(U)∣−1+ρ|M\cap\delta(U)|-1+\rho, has real positive semidefinite rank 2Ω(n)2^{\Omega(n)} as even n tends to infinity. Here δ(U)\delta(U) is the edge cut of U. Consequently, every exact semidefinite lift of the perfect matching polytope has exponential size.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Exponential-PSD-rank-of-positively-shifted-matching-matrices-October-5-2026,
  author = {{OpenAI}},
  title = {{Exponential PSD rank of positively shifted matching matrices}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Exponential-PSD-rank-of-positively-shifted-matching-matrices-October-5-2026/shifted-matching-psd.pdf}{OAI:Exponential-PSD-rank-of-positively-shifted-matching-matrices-October-5-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/126.md.

Exponential semidefinite complexity of perfect matching

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

Scope

The paper proves exponential PSD-rank lower bounds for positively shifted matching matrices. The linked formalization records the related superpolynomial lower bounds: for every fixed C>0C>0, the PSD rank of the selected perfect-matching slack matrix exceeds nCn^C for all sufficiently large even nn. Every exact affine semidefinite lift of the perfect-matching polytope likewise requires matrix size greater than nCn^C.

These selected statements give superpolynomial growth. They do not state the paper's exponential bound for every fixed positive shift.

Comparator links

Result Comparator statement
Superpolynomial lower bound for affine semidefinite lifts MatchingAffineLift.lean
Superpolynomial PSD rank for the perfect-matching slack matrix MatchingPSD.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 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.