Volume 14 (2018)
Article 17 pp. 1-25

New Algorithms and Lower Bounds for Circuits With Linear Threshold Gates

Received: June 28, 2015

Revised: June 4, 2018

Published: December 10, 2018

**Keywords:** circuit complexity, satisfiability algorithms, linear threshold functions

**ACM Classification: **F.1.1, F.2.3, G.1.6

**AMS Classification: **68Q15, 68Q17, 68Q15

**Abstract:**
[Plain Text Version]

$
\newcommand \poly {{\operatorname{poly}}}
\newcommand \SYM {{\sf SYM}}
\newcommand \THR {{\sf THR}}
\newcommand \ACC {{\sf ACC}}
\newcommand \NEXP {{\sf NEXP}}
\newcommand \eps {{\varepsilon}}
$

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:

The number of satisfying assignments to an $\ACC\circ\THR$ circuit of $2^{n^{o(1)}}$ size is computable in $2^{n-n^{\eps}}$ time (where $\eps > 0$ depends on the depth and modulus of the circuit).

$\NEXP$ does not have quasi-polynomial size $\ACC \circ \THR$ circuits, and $\NEXP$ does not have quasi-polynomial size $\ACC \circ \SYM$ circuits. Nontrivial size lower bounds were not known even for ${\sf AND} \circ {\sf OR} \circ \THR$ circuits.

Every 0-1 integer linear program with $n$ Boolean variables and $s$ linear constraints is solvable in $2^{n-\Omega(n/\log^4(sM(\log n)))}\cdot \poly(s,n,M)$ time with high probability, where $M \leq 2^{n^{o(1)}}$
is an upper bound on
the bit complexity of the coefficients. (For example, 0-1 integer programs with weights in $[-2^{n^{o(1)}},2^{n^{o(1)}}]$ and $\poly(n)$ constraints can be solved in $2^{n-\Omega(n/\log^4 n)}$ time.)
We also present an algorithm for evaluating depth-two linear threshold circuits (also known as $\THR \circ \THR$) 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.