Fu, Hu ;
Kleinberg, Robert
Improved Lower Bounds for Testing Trianglefreeness in Boolean Functions via Fast Matrix Multiplication
Abstract
Understanding the query complexity for testing linearinvariant properties has been a central open problem in the study of algebraic property testing. Trianglefreeness in Boolean functions is a simple property whose testing complexity is unknown. Three Boolean functions f1, f2 and f3, mapping {0,1}^k to {0,1}, are said to be triangle free if there is no x, y in {0,1}^k such that f1(x) = f2(y) = f3(x + y) = 1. This property is known to be strongly testable (Green 2005), but the number of queries needed is upperbounded only by a tower of twos whose height is polynomial in 1 / epsislon, where epsislon is the distance between the tested function triple and trianglefreeness, i.e., the minimum fraction of function values that need to be modified to make the triple triangle free. A lower bound of (1 / epsilon)^2.423 for any onesided tester was given by Bhattacharyya and Xie (2010). In this work we improve this bound to (1 / epsilon)^6.619. Interestingly, we prove this by way of a combinatorial construction called uniquely solvable puzzles that was at the heart of Coppersmith and Winograd's renowned matrix multiplication algorithm.
BibTeX  Entry
@InProceedings{fu_et_al:LIPIcs:2014:4730,
author = {Hu Fu and Robert Kleinberg},
title = {{Improved Lower Bounds for Testing Trianglefreeness in Boolean Functions via Fast Matrix Multiplication}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
pages = {669676},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897743},
ISSN = {18688969},
year = {2014},
volume = {28},
editor = {Klaus Jansen and Jos{\'e} D. P. Rolim and Nikhil R. Devanur and Cristopher Moore},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2014/4730},
URN = {urn:nbn:de:0030drops47304},
doi = {10.4230/LIPIcs.APPROXRANDOM.2014.669},
annote = {Keywords: Property testing, linear invariance, fast matrix multiplication, uniquely solvable puzzles}
}
04.09.2014
Keywords: 

Property testing, linear invariance, fast matrix multiplication, uniquely solvable puzzles 
Seminar: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Issue date: 

2014 
Date of publication: 

04.09.2014 