Result 117, Theoretical computer science

Uniform sparsest cut: hardness and semidefinite gaps

Proves that approximating Uniform Sparsest Cut within any fixed constant factor is NP-hard, even with nonnegative rational capacities and unit demands. The Goemans–Linial semidefinite relaxation also has integrality gaps of order at least $\sqrt{\log n}/(\log\log n)^3$ , approaching the square-root-logarithmic upper bound.

Lean formalization New or sharp bound

The bigger picture

Why it matters

Uniform Sparsest Cut seeks a division of a network minimizing the total capacity of crossing edges per separated vertex pair. The unreviewed manuscripts report strong barriers to approximating this deceptively simple partitioning problem.

What changes?

For every fixed approximation factor C greater than one, the first manuscript claims NP-hardness with nonnegative rational edge capacities and unit demand between every distinct vertex pair. The second constructs instances with nonnegative real capacities and the same demands where the Goemans-Linial semidefinite relaxation, an optimization surrogate, has integrality gap at least c times the square root of log n divided by (log log n) cubed, for a positive constant c along a sequence of vertex counts n tending to infinity.

What does that help mathematicians do?

The hardness claim would rule out polynomial-time constant-factor approximation unless P equals NP. The gap measures how much smaller the relaxation's answer can be than the best actual cut value. These examples would show that this particular surrogate can be substantially overoptimistic even with uniform demands. They approach the Arora-Rao-Vazirani upper bound up to a power of log log n, sharply limiting the guarantees obtainable from this relaxation.

Are there practical applications?

The immediate value is foundational for network partitioning: it separates what efficient algorithms might achieve from guarantees that would contradict the hardness claim. For researchers using semidefinite optimization, the examples identify a specific limitation of the relaxation, rather than providing a faster partitioning method or evidence about performance on practical networks.

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.

2 manuscripts

Constant-factor hardness of uniform sparsest cut

September 24, 2026 48 pages

We prove that, for every fixed C > 1, approximating Uniform Sparsest Cut within factor C is NP-hard. The output graphs have nonnegative rational capacities and unit demand between every pair of distinct vertices.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026,
  author = {{OpenAI}},
  title = {{Constant-factor hardness of uniform sparsest cut}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026/Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026.pdf}{OAI:Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026}},
  year = {2026}
}

Near-square-root logarithmic integrality gaps for uniform sparsest cut

September 24, 2026 22 pages Main result formalized in Lean

We construct uniform sparsest-cut instances whose Goemans–Linial semidefinite integrality gap is at least $c\sqrt{\log n}/(\log\log n)^3$ along a sequence $n\to\infty$ . The demand is one between every pair of distinct vertices, and the capacities are nonnegative real numbers. This matches the Arora–Rao–Vazirani upper bound up to a power of $\log\log n$ .

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026,
  author = {{OpenAI}},
  title = {{Near-square-root logarithmic integrality gaps for uniform sparsest cut}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026.pdf}{OAI:Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/117.md.

Uniform sparsest cut: hardness and semidefinite gaps

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

Near-square-root logarithmic integrality gaps for uniform sparsest cut

Scope

The formalized result gives integrality gaps for the Goemans–Linial relaxation of uniform sparsest cut. Along a sequence of instance sizes $n\to\infty$ , the ratio of the integral optimum to the positive relaxation optimum is at least $c\sqrt{\log n}/(\log\log n)^3$ for one absolute $c>0$ . Every unordered pair has unit demand, and the relaxation uses squared Euclidean distances satisfying the triangle inequalities. The result is an existential sequence, not a bound for every size or a computational hardness theorem.

Comparator links

Result	Comparator statement
Uniform sparsest-cut integrality gaps	UniformSparsestCut.lean