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Reduction from Non-Unique Games to Boolean Unique Games

Authors: Ronen Eldan and Dana Moshkovitz

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)


Abstract
We reduce the problem of proving a "Boolean Unique Games Conjecture" (with gap 1-δ vs. 1-Cδ, for any C > 1, and sufficiently small δ > 0) to the problem of proving a PCP Theorem for a certain non-unique game. In a previous work, Khot and Moshkovitz suggested an inefficient candidate reduction (i.e., without a proof of soundness). The current work is the first to provide an efficient reduction along with a proof of soundness. The non-unique game we reduce from is similar to non-unique games for which PCP theorems are known. Our proof relies on a new concentration theorem for functions in Gaussian space that are restricted to a random hyperplane. We bound the typical Euclidean distance between the low degree part of the restriction of the function to the hyperplane and the restriction to the hyperplane of the low degree part of the function.

Cite as

Ronen Eldan and Dana Moshkovitz. Reduction from Non-Unique Games to Boolean Unique Games. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 64:1-64:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{eldan_et_al:LIPIcs.ITCS.2022.64,
  author =	{Eldan, Ronen and Moshkovitz, Dana},
  title =	{{Reduction from Non-Unique Games to Boolean Unique Games}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{64:1--64:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.64},
  URN =		{urn:nbn:de:0030-drops-156605},
  doi =		{10.4230/LIPIcs.ITCS.2022.64},
  annote =	{Keywords: Unique Games Conjecture, hyperplane encoding, concentration of measure, low degree testing}
}
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