37 research outputs found
On The Hereditary Discrepancy of Homogeneous Arithmetic Progressions
We show that the hereditary discrepancy of homogeneous arithmetic
progressions is lower bounded by . This bound is tight up
to the constant in the exponent. Our lower bound goes via proving an
exponential lower bound on the discrepancy of set systems of subcubes of the
boolean cube .Comment: To appear in the Proceedings of the American Mathematical Societ
On largest volume simplices and sub-determinants
We show that the problem of finding the simplex of largest volume in the
convex hull of points in can be approximated with a factor
of in polynomial time. This improves upon the previously best
known approximation guarantee of by Khachiyan. On the other hand,
we show that there exists a constant such that this problem cannot be
approximated with a factor of , unless . % This improves over the
inapproximability that was previously known. Our hardness result holds
even if , in which case there exists a \bar c\,^{d}-approximation
algorithm that relies on recent sampling techniques, where is again a
constant. We show that similar results hold for the problem of finding the
largest absolute value of a subdeterminant of a matrix
An Algorithm for Koml\'os Conjecture Matching Banaszczyk's bound
We consider the problem of finding a low discrepancy coloring for sparse set
systems where each element lies in at most t sets. We give an efficient
algorithm that finds a coloring with discrepancy O((t log n)^{1/2}), matching
the best known non-constructive bound for the problem due to Banaszczyk. The
previous algorithms only achieved an O(t^{1/2} log n) bound. The result also
extends to the more general Koml\'{o}s setting and gives an algorithmic
O(log^{1/2} n) bound
Nearly Optimal Private Convolution
We study computing the convolution of a private input with a public input
, while satisfying the guarantees of -differential
privacy. Convolution is a fundamental operation, intimately related to Fourier
Transforms. In our setting, the private input may represent a time series of
sensitive events or a histogram of a database of confidential personal
information. Convolution then captures important primitives including linear
filtering, which is an essential tool in time series analysis, and aggregation
queries on projections of the data.
We give a nearly optimal algorithm for computing convolutions while
satisfying -differential privacy. Surprisingly, we follow
the simple strategy of adding independent Laplacian noise to each Fourier
coefficient and bounding the privacy loss using the composition theorem of
Dwork, Rothblum, and Vadhan. We derive a closed form expression for the optimal
noise to add to each Fourier coefficient using convex programming duality. Our
algorithm is very efficient -- it is essentially no more computationally
expensive than a Fast Fourier Transform.
To prove near optimality, we use the recent discrepancy lowerbounds of
Muthukrishnan and Nikolov and derive a spectral lower bound using a
characterization of discrepancy in terms of determinants
On Integer Programming, Discrepancy, and Convolution
Integer programs with a constant number of constraints are solvable in
pseudo-polynomial time. We give a new algorithm with a better pseudo-polynomial
running time than previous results. Moreover, we establish a strong connection
to the problem (min, +)-convolution. (min, +)-convolution has a trivial
quadratic time algorithm and it has been conjectured that this cannot be
improved significantly. We show that further improvements to our
pseudo-polynomial algorithm for any fixed number of constraints are equivalent
to improvements for (min, +)-convolution. This is a strong evidence that our
algorithm's running time is the best possible. We also present a faster
specialized algorithm for testing feasibility of an integer program with few
constraints and for this we also give a tight lower bound, which is based on
the SETH.Comment: A preliminary version appeared in the proceedings of ITCS 201
On a generalization of iterated and randomized rounding
We give a general method for rounding linear programs that combines the
commonly used iterated rounding and randomized rounding techniques. In
particular, we show that whenever iterated rounding can be applied to a problem
with some slack, there is a randomized procedure that returns an integral
solution that satisfies the guarantees of iterated rounding and also has
concentration properties. We use this to give new results for several classic
problems where iterated rounding has been useful