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.FSTTCS.2018.11
URN: urn:nbn:de:0030-drops-99105
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9910/
Goldenberg, Elazar ;
C. S., Karthik
Towards a General Direct Product Testing Theorem
Abstract
The Direct Product encoding of a string a in {0,1}^n on an underlying domain V subseteq ([n] choose k), is a function DP_V(a) which gets as input a set S in V and outputs a restricted to S. In the Direct Product Testing Problem, we are given a function F:V -> {0,1}^k, and our goal is to test whether F is close to a direct product encoding, i.e., whether there exists some a in {0,1}^n such that on most sets S, we have F(S)=DP_V(a)(S). A natural test is as follows: select a pair (S,S')in V according to some underlying distribution over V x V, query F on this pair, and check for consistency on their intersection. Note that the above distribution may be viewed as a weighted graph over the vertex set V and is referred to as a test graph.
The testability of direct products was studied over various domains and test graphs: Dinur and Steurer (CCC '14) analyzed it when V equals the k-th slice of the Boolean hypercube and the test graph is a member of the Johnson graph family. Dinur and Kaufman (FOCS '17) analyzed it for the case where V is the set of faces of a Ramanujan complex, where in this case V=O_k(n). In this paper, we study the testability of direct products in a general setting, addressing the question: what properties of the domain and the test graph allow one to prove a direct product testing theorem?
Towards this goal we introduce the notion of coordinate expansion of a test graph. Roughly speaking a test graph is a coordinate expander if it has global and local expansion, and has certain nice intersection properties on sampling. We show that whenever the test graph has coordinate expansion then it admits a direct product testing theorem. Additionally, for every k and n we provide a direct product domain V subseteq (n choose k) of size n, called the Sliding Window domain for which we prove direct product testability.
BibTeX - Entry
@InProceedings{goldenberg_et_al:LIPIcs:2018:9910,
author = {Elazar Goldenberg and Karthik C. S.},
title = {{Towards a General Direct Product Testing Theorem}},
booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)},
pages = {11:1--11:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-093-4},
ISSN = {1868-8969},
year = {2018},
volume = {122},
editor = {Sumit Ganguly and Paritosh Pandya},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9910},
URN = {urn:nbn:de:0030-drops-99105},
doi = {10.4230/LIPIcs.FSTTCS.2018.11},
annote = {Keywords: Property Testing, Direct Product, PCP, Johnson graph, Ramanujan Complex, Derandomization}
}
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Keywords: |
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Property Testing, Direct Product, PCP, Johnson graph, Ramanujan Complex, Derandomization |
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Collection: |
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38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018) |
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Issue Date: |
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2018 |
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Date of publication: |
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05.12.2018 |
