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Making the long code shorter
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0002-5379-345X
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2015 (English)In: SIAM journal on computing (Print), ISSN 0097-5397, E-ISSN 1095-7111, Vol. 44, no 5, 1287-1324 p.Article in journal (Refereed) PublishedText
Abstract [en]

The long code is a central tool in hardness of approximation especially in questions related to the Unique Games Conjecture. We construct a new code that is exponentially more efficient but can still be used in many of these applications. Using the new code we obtain exponential improvements over several known results including the following: (1) For any ε > 0, we show the existence of an n-vertex graph G where every set of o(n) vertices has expansion 1-ε but G's adjacency matrix has more than exp(logδ n) eigenvalues larger than 1 - ε, where δ depends only on ε. This answers an open question of Arora, Barak, and Steurer [Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, 2010, pp. 563-572] who asked whether one can improve over the noise graph on the Boolean hypercube that has poly(log n) such eigenvalues. (2) A gadget that reduces Unique Games instances with linear constraints modulo K into instances with alphabet k with a blowup of kpolylog(K) , improving over the previously known gadget with blowup of kω(K). (3) An n-variable integrality gap for Unique Games that survives exp(poly(log log n)) rounds of the semidefinite programming version of the Sherali-Adams hierarchy, improving on the previously known bound of poly(log log n). We show a connection between the local testability of linear codes and Small-Set Expansion in certain related Cayley graphs and use this connection to derandomize the noise graph on the Boolean hypercube.

Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics, 2015. Vol. 44, no 5, 1287-1324 p.
Keyword [en]
Cayley graphs, Expanders, Hardness of approximation, Locally testable codes, Codes (symbols), Eigenvalues and eigenfunctions, Hardness, Semi-definite programming, Sherali-adams hierarchies, Small-set expansions, Unique games conjecture, Graph theory
National Category
Discrete Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-181506DOI: 10.1137/130929394ISI: 000364454500006ScopusID: 2-s2.0-84945903744OAI: oai:DiVA.org:kth-181506DiVA: diva2:900131
Conference
53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012; New Brunswick, NJ; 20 October 2012 through 23 October 2012
Note

QC 20160203

Available from: 2016-02-03 Created: 2016-02-02 Last updated: 2016-02-03Bibliographically approved

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Håstad, Johan
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Theoretical Computer Science, TCS
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