Deterministic approximate counting of depth-2 circuits

The authors describe deterministic algorithms which for a given depth-2 circuit F approximate the probability that on a random input F outputs a specific value alpha . The approach gives an algorithm which for a given GF(2) multivariate polynomial p and given in >0 approximates the number of zeros (or ones) of p within a multiplicative factor 1+ in . The algorithm runs in time exp(exp(O( square root log(n/ in )))), where n is the size of the circuit. They also obtain an algorithm which given a DNF formula F and in >0 approximates the number of satisfying assignments of F within a factor of 1+ in and runs in time exp(O((log(n/ in ))/sup 4/)).<<ETX>>

[1]  Noam Nisan,et al.  Multiparty Protocols, Pseudorandom Generators for Logspace, and Time-Space Trade-Offs , 1992, J. Comput. Syst. Sci..

[2]  Avi Wigderson,et al.  Deterministic simulation of probabilistic constant depth circuits , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[3]  Noam Nisan,et al.  Hardness vs. randomness , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[4]  PolynomialMarek Karpinski.  Approximating the Number of Solutions of a GF ( 2 ) , 1993 .

[5]  Johan Håstad,et al.  On the power of small-depth threshold circuits , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[6]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[7]  Roman Smolensky,et al.  Algebraic methods in the theory of lower bounds for Boolean circuit complexity , 1987, STOC.

[8]  Richard J. Lipton,et al.  Multi-party protocols , 1983, STOC.

[9]  Richard M. Karp,et al.  Monte-Carlo Approximation Algorithms for Enumeration Problems , 1989, J. Algorithms.

[10]  Richard M. Karp,et al.  Monte-Carlo algorithms for enumeration and reliability problems , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[11]  Michael Luby,et al.  A simple parallel algorithm for the maximal independent set problem , 1985, STOC '85.

[12]  Marek Karpinski,et al.  Approximating the number of zeroes of a GF[2] polynomial , 1991, SODA '91.

[13]  J. Håstad.  Computational limitations of small-depth circuits , 1987 .

[14]  Noam Nisan,et al.  Pseudorandom bits for constant depth circuits , 1991, Comb..

[15]  Noam Nisan,et al.  Multiparty protocols and logspace-hard pseudorandom sequences , 1989, STOC '89.