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Regularization of Low Error PCPs and an Application to MCSP

Authors Shuichi Hirahara, Dana Moshkovitz

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Author Details

Shuichi Hirahara
  • National Institute of Informatics, Tokyo, Japan
Dana Moshkovitz
  • Department of Computer Science, University of Texas at Austin, TX, USA

Funding

  • Hirahara, Shuichi: Supported by JST, PRESTO Grant Number JPMJPR2024, Japan.
  • Moshkovitz, Dana: Supported by the National Science Foundation under grants number 2200956 and 2312573.

Acknowledgements

We are thankful to Dean Doron for discussions about explicit construction of dispersers.

Cite As Get BibTex

Shuichi Hirahara and Dana Moshkovitz. Regularization of Low Error PCPs and an Application to MCSP. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ISAAC.2023.39

Abstract

In a regular PCP the verifier queries each proof symbol in the same number of tests. This number is called the degree of the proof, and it is at least 1/(sq) where s is the soundness error and q is the number of queries. It is incredibly useful to have regularity and reduced degree in PCP. There is an expander-based transformation by Papadimitriou and Yannakakis that transforms any PCP with a constant number of queries and constant soundness error to a regular PCP with constant degree. There are also transformations for low error projection and unique PCPs. Other PCPs are constructed especially to be regular. In this work we show how to regularize and reduce degree of PCPs with a possibly large number of queries and low soundness error.
As an application, we prove NP-hardness of an unweighted variant of the collective minimum monotone satisfying assignment problem, which was introduced by Hirahara (FOCS'22) to prove NP-hardness of MCSP^* (the partial function variant of the Minimum Circuit Size Problem) under randomized reductions. We present a simplified proof and sufficient conditions under which MCSP^* is NP-hard under the standard notion of reduction: MCSP^* is NP-hard under deterministic polynomial-time many-one reductions if there exists a function in E that satisfies certain direct sum properties.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • PCP theorem
  • regularization
  • Minimum Circuit Size Problem

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