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CS-113 · computer science · mathematics · Computational complexity

Unique Games Approximation-Threshold Conjecture

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Unique Games Conjecture: for every finite CSP and every $\varepsilon>0$, a polynomial-time algorithm achieves within $\varepsilon$ of the approximation threshold beyond which approximation is UGC-hard, and this threshold factor can be efficiently approximated.

Formulation references: [S1], [S2].

Short English statement

Unique Games Conjecture: for every finite CSP and every $\varepsilon>0$, a polynomial-time algorithm achieves within $\varepsilon$ of the approximation threshold beyond which approximation is UGC-hard, and this threshold factor can be efficiently approximated.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-113-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-113

The Unique Games result supplies hardness and the finite-CSP SDP characterization. Computing or approximating the threshold itself is an additional algorithmic task not established by the hardness statement.

Unique Games Approximation-Threshold Conjecture

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 102; 2026-09-23), the asserted result is that for any fixed positive completeness and soundness errors, a deterministic polynomial reduction from 3SAT produces fixed-alphabet Unique Games with the requested gap.

Mathematical statement of the claimed result

Manuscript 1.
For every fixed $\varepsilon,\delta\in(0,1/2)$, the claim gives $s\ge1$ and a deterministic polynomial-time $3$SAT reduction to explicit unweighted Unique Games over $K=\mathbb F_2^s$, on a simple bipartite graph with translation constraints $a(v_e)=a(u_e)+c_e$, satisfying \[ \varphi\text{ satisfiable}\Rightarrow\operatorname{val}(G_{\varphi})\ge1-\varepsilon, \qquad\varphi\text{ unsatisfiable}\Rightarrow\operatorname{val}(G_{\varphi})\le\delta. \] Alphabet size and polynomial constants depend only on the two fixed errors. For any fixed constraint-language instance $J$ with $A=\operatorname{OPT}(J)<B=\operatorname{SDP}_{R}(J)$, the associated consequence claims NP-hardness of distinguishing value at least $B-\eta$ from value at most $A+\eta$ for any fixed $0<\eta<(B-A)/2$.

Outline of the argument

For manuscript 1 (family 102; 2026-09-23), a latent quadratic alphabet and matrix-slice tests supply an outer permutation game. High acceptance decodes agreeing advice strategies, whereas clean-coordinate estimates from the unsatisfiable parity instance force their agreement to decay, giving the soundness contradiction.

Formal verification

For manuscript 1 (family 102; 2026-09-23), OpenAI lists Lean statements for this assertion: for any fixed positive completeness and soundness errors, a deterministic polynomial reduction from 3SAT produces fixed-alphabet Unique Games with the requested gap.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are DirectedFeedback.lean, MinUncut.lean, OptimalMaxCut.lean, UniqueGamesTheorem.lean, VertexCover.lean.

Their named declarations include OAI.DirectedFeedback.main, OAI.MinUncut.main, OAI.OptimalMaxCut.main, OAI.UniqueGamesTheorem.theorem11, OAI.VertexCover.every_fixed_factor_below_two, OAI.VertexCover.explicit_cover_gap.

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

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