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Streaming Hardness of Unique Games

Authors Venkatesan Guruswami , Runzhou Tao

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Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Streaming models
Keywords
  • Communication complexity
  • CSP
  • Fourier Analysis
  • Lower bounds
  • Streaming algorithms
  • Unique Games

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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 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 downstream hardness results for streaming approximability for other CSP-like problems.

Cite As Get BibTex

Venkatesan Guruswami and Runzhou Tao. Streaming Hardness of Unique Games. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 5:1-5:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.5

Author Details

Venkatesan Guruswami
  • Computer Science Department, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA, USA, 15213
Runzhou Tao
  • Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China 100084

Funding

  • Guruswami, Venkatesan: Research supported in part by NSF grants CCF-1422045 and CCF-1526092.
  • Tao, Runzhou: Most of this work was done during a visit by the author to Carnegie Mellon University.

References

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