New algorithms and lower bounds for circuits with linear threshold gates

Let ACC o THR be the class of constant-depth circuits comprised of AND, OR, and MODm gates (for some constant m > 1), with a bottom layer of gates computing arbitrary linear threshold functions. This class of circuits can be seen as a \"midpoint\" between ACC (where we know nontrivial lower bounds) and depth-two linear threshold circuits (where nontrivial lower bounds remain open). We give an algorithm for evaluating an arbitrary symmetric function of 2no(1) ACC o THR circuits of size 2no(1), on all possible inputs, in 2n · poly(n) time. Several consequences are derived: • The number of satisfying assignments to an ACC o THR circuit of 2no(1) size is computable in 2n-ne time (where ε > 0 depends on the depth and modulus of the circuit). • NEXP does not have quasi-polynomial size ACC o THR circuits, and NEXP does not have quasi-polynomial size ACC o SYM circuits. Nontrivial size lower bounds were not known even for AND o OR o THR circuits. • Every 0-1 integer linear program with n Boolean variables and s linear constraints is solvable in 2n-Ω(n/log M)(log s)5) poly(s, n,M) time with high probability, where M upper bounds the bit complexity of the coefficients. (For example, 0-1 integer programs with weights in [--2poly(n), 2poly(n)] and poly(n) constraints can be solved in 2n-Ω(n/log6n) time.) Impagliazzo, Paturi, and Schneider [IPS13] recently gave an algorithm for Õ (n) constraints; ours is the first asymptotic improvement over exhaustive search for up to subexponentially many constraints. We also present an algorithm for evaluating depth-two linear threshold circuits (a.k.a., THR o THR) with exponential weights and 2n/24 size on all 2n input assignments, running in 2n · poly(n) time. This is evidence that non-uniform lower bounds for THR o THR are within reach.