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On Pseudodeterministic Approximation Algorithms

Authors Peter Dixon, A. Pavan, N. V. Vinodchandran

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Subject Classification

ACM Subject Classification
  • Theory of computation → Probabilistic computation
  • Theory of computation → Circuit complexity
Keywords
  • Approximation Algorithms
  • Circuit lower bounds
  • Pseudodeterminism

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Abstract

We investigate the notion of pseudodeterminstic approximation algorithms. A randomized approximation algorithm A for a function f is pseudodeterministic if for every input x there is a unique value v so that A(x) outputs v with high probability, and v is a good approximation of f(x). We show that designing a pseudodeterministic version of Stockmeyer's well known approximation algorithm for the NP-membership counting problem will yield a new circuit lower bound: if such an approximation algorithm exists, then for every k, there is a language in the complexity class ZPP^{NP}_{tt} that does not have n^k-size circuits. While we do not know how to design such an algorithm for the NP-membership counting problem, we show a general result that any randomized approximation algorithm for a counting problem can be transformed to an approximation algorithm that has a constant number of influential random bits. That is, for most settings of these influential bits, the approximation algorithm will be pseudodeterministic.

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Peter Dixon, A. Pavan, and N. V. Vinodchandran. On Pseudodeterministic Approximation Algorithms. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 61:1-61:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.MFCS.2018.61

Author Details

Peter Dixon
  • Iowa State University, Ames, USA
A. Pavan
  • Iowa State University, Ames, USA
N. V. Vinodchandran
  • University of Nebraska, Lincoln, USA

Funding

  • Dixon, Peter: Research Supported in part by NSF grant 1421163.
  • Pavan, A.: Research Supported in part by NSF grant 1421163.
  • Vinodchandran, N. V.: Research Supported in part by NSF grant 1422668.

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