OpenAI research collection · 5 manuscripts · Source checked 2026-10-19
Some optimisation problems are so difficult that computers aim for a good approximation. This family claims proofs of the Unique Games Conjecture and several precise limits on how good efficient approximations can be.
Knowing which improvements are computationally hard can guide algorithm design. These are hardness results under specific definitions and assumptions, not a proof that all difficult computation is impossible.
Proves Khot's Unique Games Conjecture. Independent direct reductions also establish NP-hardness, on unweighted graphs, of approximation beyond the Goemans–Williamson ratio for Max-Cut, below factor two for Vertex Cover, and within any fixed constant factor for Min-UnCut and directed feedback vertex set. These direct proofs use established PCP and Label Cover hardness results.
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
We prove the Unique Games Conjecture. For every fixed ε,δ∈(0,1/2), we give a deterministic polynomial-time reduction from 3SAT to Unique Games over a fixed finite alphabet, with completeness at least 1-ε and soundness at most δ.
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We prove that approximating Max-Cut on simple unweighted graphs within any fixed factor greater than the Goemans–Williamson constant is NP-hard.
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We prove that minimum Vertex Cover is NP-hard to approximate within every fixed factor below two, even on simple unweighted graphs.
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For every fixed C > 1, approximating Min-UnCut within factor C is NP-hard, even on simple undirected unweighted graphs.
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Approximating minimum directed feedback vertex set within any fixed constant factor is NP-hard, even on unweighted digraphs.
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New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.
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