Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH scholarly article en Guruswami, Venkatesan; Tao, Runzhou http://www.dagstuhl.de/lipics License
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URN: urn:nbn:de:0030-drops-112209
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Streaming Hardness of 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.

BibTeX - Entry

@InProceedings{guruswami_et_al:LIPIcs:2019:11220,
  author =	{Venkatesan Guruswami and Runzhou Tao},
  title =	{{Streaming Hardness of Unique Games}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{5:1--5:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Dimitris Achlioptas and L{\'a}szl{\'o} A. V{\'e}gh},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/11220},
  URN =		{urn:nbn:de:0030-drops-112209},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.5},
  annote =	{Keywords: Communication complexity, CSP, Fourier Analysis, Lower bounds, Streaming algorithms, Unique Games}
}

Keywords: Communication complexity, CSP, Fourier Analysis, Lower bounds, Streaming algorithms, Unique Games
Seminar: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)
Issue date: 2019
Date of publication: 2019


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