New algorithms and lower bounds for circuits with linear threshold gates

TL;DR

This paper introduces new algorithms for evaluating certain circuit classes with linear threshold gates, establishes lower bounds, and improves the understanding of computational complexity for these circuits and related problems.

Contribution

It provides the first efficient algorithms for evaluating symmetric functions on $ACC \,\circ\, THR$ circuits and derives significant lower bounds for circuit complexity and satisfiability problems.

Findings

Efficient evaluation of symmetric functions on $ACC \circ THR$ circuits.

Subexponential time algorithms for counting satisfying assignments.

Lower bounds showing $NEXP$ does not have small $ACC \circ THR$ circuits.

Abstract

Let $ACC \circ THR$ be the class of constant-depth circuits comprised of AND, OR, and MOD$m$ 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 $2^{n^{o(1)}}$ $ACC \circ THR$ circuits of size $2^{n^{o(1)}}$, on all possible inputs, in $2^n \cdot poly(n)$ time. Several consequences are derived: $\bullet$ The number of satisfying assignments to an $ACC \circ THR$ circuit of subexponential size can be computed in $2^{n-n^{\varepsilon}}$ time (where $\varepsilon > 0$ depends on the depth and modulus of the circuit). $\bullet$ $NEXP$ does not have quasi-polynomial size $ACC \circ THR$ circuits, nor does $NEXP$ have quasi-polynomial size $ACC \circ SYM$ circuits. Nontrivial size lower bounds were not known even for $AND \circ OR \circ THR$ circuits. $\bullet$ Every 0-1 integer linear program with $n$ Boolean variables and $s$ linear constraints is solvable in $2^{n-\Omega(n/((\log M)(\log s)^{5}))}\cdot poly(s,n,M)$ time with high probability, where $M$ upper bounds the bit complexity of the coefficients. (For example, 0-1 integer programs with weights in $[-2^{poly(n)},2^{poly(n)}]$ and $poly(n)$ constraints can be solved in $2^{n-\Omega(n/\log^6 n)}$ time.) We also present an algorithm for evaluating depth-two linear threshold circuits (a.k.a., $THR \circ THR$) with exponential weights and $2^{n/24}$ size on all $2^n$ input assignments, running in $2^n \cdot poly(n)$ time. This is evidence that non-uniform lower bounds for $THR \circ THR$ are within reach.