1 research outputs found
New algorithms and lower bounds for circuits with linear threshold gates
Let be the class of constant-depth circuits comprised of AND,
OR, and MOD gates (for some constant ), with a bottom layer of gates
computing arbitrary linear threshold functions. This class of circuits can be
seen as a "midpoint" between (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
circuits of size , on all possible
inputs, in time. Several consequences are derived:
The number of satisfying assignments to an circuit
of subexponential size can be computed in time (where
depends on the depth and modulus of the circuit).
does not have quasi-polynomial size
circuits, nor does have quasi-polynomial size circuits.
Nontrivial size lower bounds were not known even for
circuits.
Every 0-1 integer linear program with Boolean variables and
linear constraints is solvable in time with high probability, where upper bounds the bit
complexity of the coefficients. (For example, 0-1 integer programs with weights
in and constraints can be solved in
time.)
We also present an algorithm for evaluating depth-two linear threshold
circuits (a.k.a., ) with exponential weights and size
on all input assignments, running in time. This is
evidence that non-uniform lower bounds for are within reach