Understanding the query complexity for testing linear-invariant properties has been a central open problem in the study of algebraic property testing. Triangle-freeness 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 upper-bounded only by a tower of twos whose height is polynomial in 1 / epsislon, where epsislon is the distance between the tested function triple and triangle-freeness, 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 one-sided 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.
@InProceedings{fu_et_al:LIPIcs.APPROX-RANDOM.2014.669, author = {Fu, Hu and Kleinberg, Robert}, title = {{Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {669--676}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.669}, URN = {urn:nbn:de:0030-drops-47304}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.669}, annote = {Keywords: Property testing, linear invariance, fast matrix multiplication, uniquely solvable puzzles} }
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