The Unique Games Conjecture

Theorem 1.1: for every fixed $\varepsilon,\delta\in(0,1/2)$ there are $s\ge1$ and a deterministic polynomial-time reduction from 3SAT to explicit unweighted, simple bipartite Unique Games over $K=\mathbb F_2^s$ whose constraints are translations $a(v)=a(u)+c_e$, with value at least $1-\varepsilon$ on satisfiable formulas and at most $\delta$ on unsatisfiable ones. This is the conjecture in its usual edge-value form. The route keeps the Barak-Kothari-Steurer degree-two matrix shortcode and its Khot-Minzer-Safra inverse theorem and changes the noise through a nonlinear map to push completeness to $1-\varepsilon$. Section 7 transfers known UG-hardness reductions (cut, covering, CSP, ordering, deletion, clustering) to NP-hardness; those reductions are credited to their authors. Companion manuscripts give direct NP-hardness proofs for Max-Cut beyond $\alpha_{GW}$, Vertex Cover below factor $2$, and every constant factor for Min-UnCut and directed feedback vertex set.

Theorem 1.1: for every fixed $\varepsilon,\delta\in(0,1/2)$ there are $s\ge1$ and a deterministic polynomial-time reduction from 3SAT to explicit unweighted, simple bipartite Unique Games over $K=\mathbb F_2^s$ whose constraints are translations $a(v)=a(u)+c_e$, with value at least $1-\varepsilon$ on satisfiable formulas and at most $\delta$ on unsatisfiable ones. This is the conjecture in its usual edge-value form. The route keeps the Barak-Kothari-Steurer degree-two matrix shortcode and its Khot-Minzer-Safra inverse theorem and changes the noise through a nonlinear map to push completeness to $1-\varepsilon$. Section 7 transfers known UG-hardness reductions (cut, covering, CSP, ordering, deletion, clustering) to NP-hardness; those reductions are credited to their authors. Companion manuscripts give direct NP-hardness proofs for Max-Cut beyond $\alpha_{GW}$, Vertex Cover below factor $2$, and every constant factor for Min-UnCut and directed feedback vertex set.