Revisiting Alphabet Reduction in Dinur’s PCP

Authors Venkatesan Guruswami , Jakub Opršal , Sai Sandeep

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Author Details

Venkatesan Guruswami
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Jakub Opršal
  • Computer Science Department, Durham University, UK
Sai Sandeep
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA


We would like to thank Irit Dinur, Margalit Glasgow, Oded Goldreich, Prahladh Harsha, Johan Håstad, Jaikumar Radhakrishnan, and Madhu Sudan for useful comments and feedback. We also thank the anonymous reviewers for valuable comments on the presentation.

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Venkatesan Guruswami, Jakub Opršal, and Sai Sandeep. Revisiting Alphabet Reduction in Dinur’s PCP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Dinur’s celebrated proof of the PCP theorem alternates two main steps in several iterations: gap amplification to increase the soundness gap by a large constant factor (at the expense of much larger alphabet size), and a composition step that brings back the alphabet size to an absolute constant (at the expense of a fixed constant factor loss in the soundness gap). We note that the gap amplification can produce a Label Cover CSP. This allows us to reduce the alphabet size via a direct long-code based reduction from Label Cover to a Boolean CSP. Our composition step thus bypasses the concept of Assignment Testers from Dinur’s proof, and we believe it is more intuitive - it is just a gadget reduction. The analysis also uses only elementary facts (Parseval’s identity) about Fourier Transforms over the hypercube.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Approximation algorithms analysis
  • PCP theorem
  • CSP
  • discrete Fourier analysis
  • label cover
  • long code


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