OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
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}/(loglog n)^3, approaching the square-root-logarithmic upper bound.
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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.
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We construct uniform sparsest-cut instances whose Goemans–Linial semidefinite integrality gap is at least c\sqrt{log n}/(loglog n)^3 along a sequence n→∞. 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…
We construct uniform sparsest-cut instances whose Goemans–Linial semidefinite integrality gap is at least c\sqrt{log n}/(loglog n)^3 along a sequence n→∞. 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 loglog n.
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