5,369 research outputs found
On the hardness of learning sparse parities
This work investigates the hardness of computing sparse solutions to systems
of linear equations over F_2. Consider the k-EvenSet 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 k-sparse
solution). While there is a simple O(n^{k/2})-time algorithm for it,
establishing fixed parameter intractability for k-EvenSet has been a notorious
open problem. Towards this goal, we show that unless k-Clique can be solved in
n^{o(k)} time, k-EvenSet has no poly(n)2^{o(sqrt{k})} time algorithm and no
polynomial time algorithm when k = (log n)^{2+eta} for any eta > 0.
Our work also shows that the non-homogeneous generalization of the problem --
which we call k-VectorSum -- 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 also show that for any constant eps > 0, given a system
of O(exp(O(k))log n) linear equations, it is W[1]-hard to decide if there is a
k-sparse linear form satisfying all the equations or if every function on at
most k-variables (k-junta) satisfies at most (1/2 + eps)-fraction of the
equations. In the setting of computational learning, this shows hardness of
approximate non-proper learning of k-parities. In a similar vein, we use the
hardness of k-EvenSet to show that that for any constant d, unless k-Clique 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 non-trivial k-sparse homogeneous linear form evaluating to 0 on all the
points, or (ii) any non-trivial 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
Inapproximability of Combinatorial Optimization Problems
We survey results on the hardness of approximating combinatorial optimization
problems
New Approximability Results for the Robust k-Median Problem
We consider a robust variant of the classical -median problem, introduced
by Anthony et al. \cite{AnthonyGGN10}. In the \emph{Robust -Median problem},
we are given an -vertex metric space and client sets . The objective is to open a set of
facilities such that the worst case connection cost over all client sets is
minimized; in other words, minimize . Anthony
et al.\ showed an approximation algorithm for any metric and
APX-hardness even in the case of uniform metric. In this paper, we show that
their algorithm is nearly tight by providing
approximation hardness, unless . This hardness result holds even for uniform and line
metrics. To our knowledge, this is one of the rare cases in which a problem on
a line metric is hard to approximate to within logarithmic factor. We
complement the hardness result by an experimental evaluation of different
heuristics that shows that very simple heuristics achieve good approximations
for realistic classes of instances.Comment: 19 page
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