The Unique Games Conjecture and optimal approximation thresholds

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. (Lean)

Manuscripts (5)

The Unique Games Theorem

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:The-Unique-Games-Theorem-September-23-2026}

We prove the Unique Games Conjecture. For every fixed \varepsilon,\delta\in(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-\varepsilon and soundness at most δ.

A Direct Proof of Optimal Max-Cut Hardness

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026}

We prove that approximating Max-Cut on simple unweighted graphs within any fixed factor greater than the Goemans–Williamson constant is NP-hard.

The Factor-Two Hardness Threshold for Vertex Cover

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026}

We prove that minimum Vertex Cover is NP-hard to approximate within every fixed factor below two, even on simple unweighted graphs.

Constant-factor hardness of Min-UnCut

September 23, 2026 · source · cite as \cite{OAI:Constant-factor-hardness-of-Min-UnCut-September-23-2026}

For every fixed C > 1, approximating Min-UnCut within factor C is NP-hard, even on simple undirected unweighted graphs.

Constant-factor hardness of directed feedback vertex set

September 23, 2026 · source · cite as \cite{OAI:Constant-factor-hardness-of-directed-feedback-vertex-set-September-23-2026}

Approximating minimum directed feedback vertex set within any fixed constant factor is NP-hard, even on unweighted digraphs.

Result 102 · Theoretical computer science. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Agora members.