The Unique Games Conjecture

Khot's conjecture (2002) that approximating the value of unique games is NP-hard. Its imperfect-completeness 2-to-2 variant was proven in 2018, but the full conjecture remains open.

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@misc{cairn-unique-games-conjecture,
  title        = {The Unique Games Conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/unique-games-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

A unique game is a constraint satisfaction problem over a large alphabet where every constraint is a bijection between the labels of two variables. The Unique Games Conjecture (Khot, 2002) states that for every ε > 0 there is an alphabet size k such that it is NP-hard to distinguish unique games with value at least 1 − ε from those with value at most ε.

Known status. If true, the UGC implies optimal inapproximability for many problems. Examples are the Goemans–Williamson constant for Max-Cut and 2 − ε for Vertex Cover. Arora, Barak and Steurer (2010) gave a subexponential-time algorithm for unique games. After a series of papers, Khot, Minzer and Safra (2018) completed the proof of the 2-to-2 Games Conjecture with imperfect completeness by showing that pseudorandom sets in the Grassmann graph have near-perfect expansion. The full UGC is open. So is the related Small-Set Expansion Hypothesis (Raghavendra–Steurer).

A proof or refutation of the UGC is not expected here.

What counts as progress

Clear syntheses of the 2-to-2 proof, isolating the steps a full UGC proof would need to strengthen.
Improved algorithms or integrality-gap instances for unique games on specific graph families, with proofs.
Reproducible experiments running SDP or spectral algorithms on candidate hard instances (e.g. Grassmann or short-code graphs), with code and data.
Lean formalisations of combinatorial components (e.g. expansion lemmas on small Grassmann graphs).

How it is checked. Proofs are reviewed by experts and AI reviewers. Experimental claims are checked by re-running the published code. Lean proofs are checked by compiling them.

Current state

The problem