Using nondeterminism to amplify hardness

  • Authors:

  • Alexander Healy;Salil Vadhan;Emanuele Viola

  • Affiliations:

  • Harvard University, Cambridge, MA;Harvard University, Cambridge, MA;Harvard University, Cambridge, MA

  • Venue:
  • STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
  • Year:
  • 2004

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Abstract

We revisit the problem of hardness amplification in NP, as recently studied by O'Donnell (STOC '02). We prove that if NP has a balanced function f such that any circuit of size s(n) fails to compute f on a 1/poly(n) fraction of inputs, then NP has a function f′ such that any circuit of size s′(n)=s(√n)Ω(1) fails to compute f′ on a 1/2 - 1/s′(n) fraction of inputs. In particular, 1. If s(n)=nω(1), we amplify to hardness 1/2-1/nω(1). 2. If s(n)=2nω(1), we amplify to hardness 1/2-1/2nΩ(1). 3. If s(n)=2(n), we amplify to hardness 1/2-1/2 Ω(sqrtn).These improve the results of O'Donnell, which only amplified to 1/2-1/√n. O'Donnell also proved that no construction of a certain general form could amplify beyond 1/2-1/n. We bypass this barrier by using both derandomization and nondeterminism in the construction of f′.We also prove impossibility results demonstrating that both our use of nondeterminism and the hypothesis that f is balanced are necessary for "black-box" hardness amplification procedures (such as ours).