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Certification with an NP Oracle

Authors Guy Blanc, Caleb Koch, Jane Lange, Carmen Strassle, Li-Yang Tan

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ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Certificate complexity
  • Boolean functions
  • polynomial hierarchy
  • hardness of approximation

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Abstract

In the certification problem, the algorithm is given a function f with certificate complexity k and an input x^⋆, and the goal is to find a certificate of size ≤ poly(k) for f’s value at x^⋆. This problem is in NP^NP, and assuming 𝖯 ≠ NP, is not in 𝖯. Prior works, dating back to Valiant in 1984, have therefore sought to design efficient algorithms by imposing assumptions on f such as monotonicity. 
Our first result is a BPP^NP algorithm for the general problem. The key ingredient is a new notion of the balanced influence of variables, a natural variant of influence that corrects for the bias of the function. Balanced influences can be accurately estimated via uniform generation, and classic BPP^NP algorithms are known for the latter task. 
We then consider certification with stricter instance-wise guarantees: for each x^⋆, find a certificate whose size scales with that of the smallest certificate for x^⋆. In sharp contrast with our first result, we show that this problem is NP^NP-hard even to approximate. We obtain an optimal inapproximability ratio, adding to a small handful of problems in the higher levels of the polynomial hierarchy for which optimal inapproximability is known. Our proof involves the novel use of bit-fixing dispersers for gap amplification.

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Guy Blanc, Caleb Koch, Jane Lange, Carmen Strassle, and Li-Yang Tan. Certification with an NP Oracle. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 18:1-18:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ITCS.2023.18

Author Details

Guy Blanc
  • Department of Computer Science, Stanford University, CA, USA
Caleb Koch
  • Department of Computer Science, Stanford University, CA, USA
Jane Lange
  • Department of Electrical Engineering and Computer Science, MIT, Cambridge, MA, USA
Carmen Strassle
  • Department of Computer Science, Stanford University, CA, USA
Li-Yang Tan
  • Department of Computer Science, Stanford University, CA, USA

Funding

Guy, Caleb, Carmen, and Li-Yang are supported by NSF awards 1942123, 2211237, and 2224246. Jane is supported by NSF Award 2006664. Caleb is also supported by an NDSEG fellowship.

Acknowledgements

We thank the ITCS reviewers for their useful comments and feedback.

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