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Smooth and Strong PCPs

Author Orr Paradise

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Orr Paradise
  • University of California, Berkeley, CA, USA


This work was done during my time at the Weizmann Institute of Science. It originated in a question of Irit Dinur, and I am grateful to her. My heartfelt appreciation goes to Oded Goldreich for guiding me through all stages of this work, from communication of Irit’s question to me and up to this very write-up. Many thanks to Madhu Sudan and Oded for providing the vector-valued low-degree test (Appendix C). I also thank Elad Granot and Roei Tell for the helpful discussions, and Amey Bhangale, Tom Gur, and Eylon Yogev for suggesting improvements to the write-up. I wish to thank an anonymous reviewer for pointing out an issue with the smoothness of the construction of Section 5.3 as it appeared in a previous version.

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Orr Paradise. Smooth and Strong PCPs. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 2:1-2:41, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs: - A PCP is strong if it rejects an alleged proof of a correct claim with probability proportional to its distance from some correct proof of that claim. - A PCP is smooth if each location in a proof is queried with equal probability. We prove that all sets in NP have PCPs that are both smooth and strong, are of polynomial length, and can be verified based on a constant number of queries. This is achieved by following the proof of the PCP theorem of Arora, Lund, Motwani, Sudan and Szegedy (JACM, 1998), providing a stronger analysis of the Hadamard and Reed - Muller based PCPs and a refined PCP composition theorem. In fact, we show that any set in NP has a smooth strong canonical PCP of Proximity (PCPP), meaning that there is an efficiently computable bijection of NP witnesses to correct proofs. This improves on the recent construction of Dinur, Gur and Goldreich (ITCS, 2019) of PCPPs that are strong canonical but inherently non-smooth. Our result implies the hardness of approximating the satisfiability of "stable" 3CNF formulae with bounded variable occurrence, where stable means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric). This proves a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (SODA, 2019), suggesting a connection between the hardness of these instances and other stable optimization problems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Interactive proof systems
  • Interactive and probabilistic proof systems
  • Probabilistically checkable proofs
  • Hardness of approximation


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