License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2019.9
URN: urn:nbn:de:0030-drops-101022
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/10102/
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Barak, Boaz ; Kothari, Pravesh K. ; Steurer, David

Small-Set Expansion in Shortcode Graph and the 2-to-2 Conjecture

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Abstract

Dinur, Khot, Kindler, Minzer and Safra (2016) recently showed that the (imperfect completeness variant of) Khot's 2 to 2 games conjecture follows from a combinatorial hypothesis about the soundness of a certain "Grassmanian agreement tester". In this work, we show that soundness of Grassmannian agreement tester follows from a conjecture we call the "Shortcode Expansion Hypothesis" characterizing the non-expanding sets of the degree-two Short code graph. We also show the latter conjecture is equivalent to a characterization of the non-expanding sets in the Grassman graph, as hypothesized by a follow-up paper of Dinur et al. (2017).
Following our work, Khot, Minzer and Safra (2018) proved the "Shortcode Expansion Hypothesis". Combining their proof with our result and the reduction of Dinur et al. (2016), completes the proof of the 2 to 2 conjecture with imperfect completeness. We believe that the Shortcode graph provides a useful view of both the hypothesis and the reduction, and might be suitable for obtaining new hardness reductions.

BibTeX - Entry

@InProceedings{barak_et_al:LIPIcs:2018:10102,
  author =	{Boaz Barak and Pravesh K. Kothari and David Steurer},
  title =	{{Small-Set Expansion in Shortcode Graph and the 2-to-2 Conjecture}},
  booktitle =	{10th Innovations in Theoretical Computer Science  Conference (ITCS 2019)},
  pages =	{9:1--9:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{124},
  editor =	{Avrim Blum},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/10102},
  URN =		{urn:nbn:de:0030-drops-101022},
  doi =		{10.4230/LIPIcs.ITCS.2019.9},
  annote =	{Keywords: Unique Games Conjecture, Small-Set Expansion, Grassmann Graph, Shortcode}
}

Keywords: Unique Games Conjecture, Small-Set Expansion, Grassmann Graph, Shortcode
Collection: 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)
Issue Date: 2018
Date of publication: 08.01.2019

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