On Monotonicity Testing and the 2-to-2 Games Conjecture
eBook - PDF

On Monotonicity Testing and the 2-to-2 Games Conjecture

  1. 233 pages
  2. English
  3. PDF
  4. Available on iOS & Android
eBook - PDF

On Monotonicity Testing and the 2-to-2 Games Conjecture

About this book

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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Yes, you can access On Monotonicity Testing and the 2-to-2 Games Conjecture by Dor Minzer in PDF and/or ePUB format, as well as other popular books in Computer Science & Computer Science General. We have over one million books available in our catalogue for you to explore.

Table of contents

  1. On Monotonicity Testing and the 2-to-2 Games Conjecture
  2. Contents
  3. Acknowledgements
  4. 1 Introduction
  5. 2 Monotonicity Testing and Directed Isoperimetric Inequalities
  6. 3 NP-hardness of 2-to-2 Games
  7. 4 Pseudo-random Sets in the Grassmann Graph Have Near-Perfect Expansion
  8. 5 The Biased Long Code and Hardness of Vertex Cover
  9. 6 Conclusions and Open Problems
  10. Bibliography
  11. Author's Biography
  12. Index