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CS-169-A1 · revision r1 · on CS-169

OpenAI claim summary for CS-169

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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 117; 2026-09-24).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Constant-factor hardness of uniform sparsest cut
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Constant-factor hardness of uniform sparsest cut

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 117; 2026-09-24), the asserted result is that Uniform Sparsest Cut is NP-hard to approximate by any fixed factor greater than one, for rational capacities and unit pair demands.

Mathematical statement of the claimed result

Manuscript 1.
For an undirected graph whose capacities are nonnegative rational numbers, with each pair of distinct vertices assigned demand one, set \[ \Phi(G)=\min_{\varnothing\ne S\subsetneq V} \frac{\sum_{u\in S,v\notin S}c_{uv}}{|S|\,|V\setminus S|}. \] For each fixed $C>1$, the claimed polynomial-time reduction from $3$CNF satisfiability outputs $G$ and positive rational $a$ with \[ \varphi\text{ satisfiable}\Rightarrow\Phi(G)\le a, \qquad\varphi\text{ unsatisfiable}\Rightarrow\Phi(G)>Ca. \] All output bit lengths are polynomial in formula size for fixed $C$; the demands remain exactly one.

Outline of the argument

For manuscript 1 (family 117; 2026-09-24), an additive tower produces a short explicit direction list with robust quadratic support. A mixing parity test and deterministic correction enforce almost-quadratic consistency; the resulting gap is converted to a graph while preserving uniform demands.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 117; 2026-09-24).

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