The Complexity of Satisfiability of Small Depth Circuits

  • Authors:

  • Chris Calabro;Russell Impagliazzo;Ramamohan Paturi

  • Affiliations:

  • Department of Computer Science and Engineering, University of California, San Diego, La Jolla, USA 92093-0404;Department of Computer Science and Engineering, University of California, San Diego, La Jolla, USA 92093-0404;Department of Computer Science and Engineering, University of California, San Diego, La Jolla, USA 92093-0404

  • Venue:
  • Parameterized and Exact Computation
  • Year:
  • 2009

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Abstract

Say that an algorithm solving a Boolean satisfiability problem x on n variables is improved if it takes time poly(|x|)2 cn for some constant c i.e., if it is exponentially better than a brute force search. We show an improved randomized algorithm for the satisfiability problem for circuits of constant depth d and a linear number of gates cn: for each d and c, the running time is 2(1 驴 驴)n where the improvement $\delta\geq 1/O(c^{2^{d-2}-1}\lg^{3\cdot 2^{d-2}-2}c)$, and the constant in the big-Oh depends only on d. The algorithm can be adjusted for use with Grover's algorithm to achieve a run time of $2^{\frac{1-\delta}{2}n}$ on a quantum computer.