Bounded depth circuits with weighted symmetric gates

A Boolean function f : { 0 , 1 } n → { 0 , 1 } is weighted symmetric if there exist a function g : Z → { 0 , 1 } and integers w 0 , w 1 , … , w n such that f ( x 1 , … , x n ) = g ( w 0 + ∑ i = 1 n w i x i ) holds. In this paper, we present algorithms for the circuit satisfiability problem of bounded depth circuits with AND, OR, NOT gates and a limited number of weighted symmetric gates. Our algorithms run in time super-polynomially faster than 2 n even when the number of gates is super-polynomial and the maximum weight of symmetric gates is nearly exponential. As a special case, we obtain an algorithm for the maximum satisfiability problem that runs in time poly ( n t ) ⋅ 2 n − n 1 / O ( t ) for instances with n variables and O ( n t ) clauses. Through the analysis of our algorithms, we show average-case lower bounds and compression algorithms for such circuits and worst-case lower bounds for majority votes of such circuits.