Streaming Hardness of Unique Games.

We study the problem of approximating the value of a Unique Game instance in the streaming model. A simple count of the number of constraints divided by $p$, the alphabet size of the Unique Game, gives a trivial $p$-approximation that can be computed in $O(log n)$ space. Meanwhile, with high probability, a sample of $tilde{O}(n)$ constraints suffices to estimate the optimal value to $(1+epsilon)$ accuracy. We prove that any single-pass streaming algorithm that achieves a $(p-epsilon)$-approximation requires $Omega_epsilon(sqrt{n})$ space. Our proof is via a reduction from lower bounds for a communication problem that is a $p$-ary variant of the Boolean Hidden Matching problem studied in the literature. Given the utility of Unique Games as a starting point for reduction to other optimization problems, our strong hardness for approximating Unique Games could lead to downemph{stream} hardness results for streaming approximability for other CSP-like problems.