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This book discusses two questions in Complexity Theory: the
Monotonicity Testing problem and the 2-to-2 Games
Conjecture.Monotonicity testing is a problem from the field of
property testing, first considered by Goldreich et al. in 2000. The
input of the algorithm is a function, and the goal is to design a
tester that makes as few queries to the function as possible,
accepts monotone functions and rejects far-from monotone functions
with a probability close to 1. The first result of this book is an
essentially optimal algorithm for this problem. The analysis of the
algorithm heavily relies on a novel, directed, and robust analogue
of a Boolean isoperimetric inequality of Talagrand from 1993. The
probabilistically checkable proofs (PCP) theorem is one of the
cornerstones of modern theoretical computer science. One area in
which PCPs are essential is the area of hardness of approximation.
Therein, the goal is to prove that some optimization problems are
hard to solve, even approximately. Many hardness of approximation
results were proved using the PCP theorem; however, for some
problems optimal results were not obtained. This book touches on
some of these problems, and in particular the 2-to-2 games problem
and the vertex cover problem. The second result of this book is a
proof of the 2-to-2 games conjecture (with imperfect completeness),
which implies new hardness of approximation results for problems
such as vertex cover and independent set. It also serves as strong
evidence towards the unique games conjecture, a notorious related
open problem in theoretical computer science. At the core of the
proof is a characterization of small sets of vertices in Grassmann
graphs whose edge expansion is bounded away from 1.
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