3 research outputs found
Probabilistic existence of rigid combinatorial structures
We show the existence of rigid combinatorial objects which previously were
not known to exist. Specifically, for a wide range of the underlying
parameters, we show the existence of non-trivial orthogonal arrays,
-designs, and -wise permutations. In all cases, the sizes of the objects
are optimal up to polynomial overhead. The proof of existence is probabilistic.
We show that a randomly chosen such object has the required properties with
positive yet tiny probability. The main technical ingredient is a special local
central limit theorem for suitable lattice random walks with finitely many
steps.Comment: Extended abstract for STOC 201
A Fourier-Analytic Approach for the Discrepancy of Random Set Systems
One of the prominent open problems in combinatorics is the discrepancy of set
systems where each element lies in at most sets. The Beck-Fiala conjecture
suggests that the right bound is , but for three decades the only
known bound not depending on the size of the set system has been .
Arguably we currently lack techniques for breaking that barrier.
In this paper we introduce discrepancy bounds based on Fourier analysis. We
demonstrate our method on random set systems. Suppose one has elements and
sets containing each element independently with probability . We prove
that in the regime of , the discrepancy is at most
with high probability. Previously, a result of Ezra and Lovett gave a bound
of under the stricter assumption that .Comment: Added acknowledgment of independent wor
The Discrepancy of Random Rectangular Matrices
A recent approach to the Beck-Fiala conjecture, a fundamental problem in
combinatorics, has been to understand when random integer matrices have
constant discrepancy. We give a complete answer to this question for two
natural models: matrices with Bernoulli or Poisson entries. For Poisson
matrices, we further characterize the discrepancy for any rectangular aspect
ratio. These results give sharp answers to questions of Hoberg and Rothvoss
(SODA 2019) and Franks and Saks (Random Structures Algorithms 2020). Our main
tool is a conditional second moment method combined with Stein's method of
exchangeable pairs. While previous approaches are limited to dense matrices,
our techniques allow us to work with matrices of all densities. This may be of
independent interest for other sparse random constraint satisfaction problems.Comment: corrected typo