eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-11-28
39:1
39:16
10.4230/LIPIcs.ISAAC.2023.39
article
Regularization of Low Error PCPs and an Application to MCSP
Hirahara, Shuichi
1
Moshkovitz, Dana
2
National Institute of Informatics, Tokyo, Japan
Department of Computer Science, University of Texas at Austin, TX, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol283-isaac2023/LIPIcs.ISAAC.2023.39/LIPIcs.ISAAC.2023.39.pdf
PCP theorem
regularization
Minimum Circuit Size Problem