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Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems
Semidefinite programming is a powerful tool in the design and analysis of
approximation algorithms for combinatorial optimization problems. In
particular, the random hyperplane rounding method of Goemans and Williamson has
been extensively studied for more than two decades, resulting in various
extensions to the original technique and beautiful algorithms for a wide range
of applications. Despite the fact that this approach yields tight approximation
guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and
Max-DiCut, the tight approximation ratio is still unknown. One of the main
reasons for this is the fact that very few techniques for rounding semidefinite
relaxations are known.
In this work, we present a new general and simple method for rounding
semi-definite programs, based on Brownian motion. Our approach is inspired by
recent results in algorithmic discrepancy theory. We develop and present tools
for analyzing our new rounding algorithms, utilizing mathematical machinery
from the theory of Brownian motion, complex analysis, and partial differential
equations. Focusing on constraint satisfaction problems, we apply our method to
several classical problems, including Max-Cut, Max-2SAT, and MaxDiCut, and
derive new algorithms that are competitive with the best known results. To
illustrate the versatility and general applicability of our approach, we give
new approximation algorithms for the Max-Cut problem with side constraints that
crucially utilizes measure concentration results for the Sticky Brownian
Motion, a feature missing from hyperplane rounding and its generalization
Upper Tail Estimates with Combinatorial Proofs
We study generalisations of a simple, combinatorial proof of a Chernoff bound
similar to the one by Impagliazzo and Kabanets (RANDOM, 2010).
In particular, we prove a randomized version of the hitting property of
expander random walks and apply it to obtain a concentration bound for expander
random walks which is essentially optimal for small deviations and a large
number of steps. At the same time, we present a simpler proof that still yields
a "right" bound settling a question asked by Impagliazzo and Kabanets.
Next, we obtain a simple upper tail bound for polynomials with input
variables in which are not necessarily independent, but obey a certain
condition inspired by Impagliazzo and Kabanets. The resulting bound is used by
Holenstein and Sinha (FOCS, 2012) in the proof of a lower bound for the number
of calls in a black-box construction of a pseudorandom generator from a one-way
function.
We then show that the same technique yields the upper tail bound for the
number of copies of a fixed graph in an Erd\H{o}s-R\'enyi random graph,
matching the one given by Janson, Oleszkiewicz and Ruci\'nski (Israel J. Math,
2002).Comment: Full version of the paper from STACS 201
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