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arXiv · 1811.04607

Streaming Hardness of Unique Games

Venkatesan Guruswami·Runzhou Tao
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Abstract

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 $(1+ε)$ accuracy. We prove that any single-pass streaming algorithm that achieves a $(p-ε)$-approximation requires $Ω_ε(\sqrt{n})$ 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 down\emph{stream} hardness results for streaming approximability for other CSP-like problems.

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Venkatesan Guruswami, Runzhou Tao. 2018-11-12. Streaming Hardness of Unique Games. https://doi.org/10.4230/lipics.approx-random.2019.5

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