Abstract
This work investigates the hardness of computing sparse solutions to systems of linear equations over F_2. Consider the kEventSet problem: given a homogeneous system of linear equations over $\F_2$ on $n$ variables, decide if there exists a nonzero solution of Hamming weight at most k (i.e. a ksparse solution). While there is a simple O(n^{k/2})time algorithm for it, establishing fixed parameter intractability for kEventSet has been a notorious open problem. Towards this goal, we show that unless \kclq can be solved in n^{o(k)} time, kEventSet has no polynomial time algorithm when k = omega(log^2(n)).
Our work also shows that the nonhomogeneous generalization of the problem  which we call kVectorSum  is W[1]hard on instances where the number of equations is O(k*log(n)), improving on previous reductions which produced Omega(n) equations. We use the hardness of kVectorSum as a starting point to prove the result for kEventSet, and additionally strengthen the former to show the hardness of approximately learning kjuntas. In particular, we prove that given a system of O(exp(O(k))*log(n)) linear equations, it is W[1]hard to decide if there is a ksparse linear form satisfying all the equations or any function on at most kvariables (a kjunta) satisfies at most (1/2 + epsilon)fraction of the equations, for any constant epsilon > 0. In the setting of computational learning, this shows hardness of approximate nonproper learning of kparities.
In a similar vein, we use the hardness of kEventSet to show that that for any constant d, unless kClique can be solved in n^{o(k)} time, there is no poly(m,n)*2^{o(sqrt{k})} time algorithm to decide whether a given set of $m$ points in F_2^n satisfies: (i) there exists a nontrivial ksparse homogeneous linear form evaluating to 0 on all the points, or (ii) any nontrivial degree d polynomial P supported on at most k variables evaluates to zero on approx Pr_{F_2^n}[P({z}) = 0] fraction of the points i.e., P is fooled by the set of points.
Lastly, we study the approximation in the sparsity of the solution. Let the GapkVectorSum problem be: given an instance of kVectorSum of size n, decide if there exist a ksparse solution, or every solution is of sparsity at least k' = (1+delta_0)k. Assuming the Exponential Time Hypothesis, we show that for some constants c_0, delta_0 > 0 there is no poly(n) time algorithm for GapkVectorSum when k = omega((log(log( n)))^{c_0}).
BibTeX  Entry
@InProceedings{bhattacharyya_et_al:LIPIcs:2016:6362,
author = {Arnab Bhattacharyya and Ameet Gadekar and Suprovat Ghoshal and Rishi Saket},
title = {{On the Hardness of Learning Sparse Parities}},
booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)},
pages = {11:111:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770156},
ISSN = {18688969},
year = {2016},
volume = {57},
editor = {Piotr Sankowski and Christos Zaroliagis},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6362},
URN = {urn:nbn:de:0030drops63628},
doi = {10.4230/LIPIcs.ESA.2016.11},
annote = {Keywords: Fixed Parameter Tractable, Juntas, Minimum Distance of Code, Psuedorandom Generators}
}
Keywords: 

Fixed Parameter Tractable, Juntas, Minimum Distance of Code, Psuedorandom Generators 
Collection: 

24th Annual European Symposium on Algorithms (ESA 2016) 
Issue Date: 

2016 
Date of publication: 

18.08.2016 