Polynomial Representations of Threshold Functions and Algorithmic Applications

TL;DR

This paper introduces new polynomial representations for threshold functions, enabling faster algorithms for nearest neighbor search, SAT problems, and circuit lower bounds, with significant improvements in efficiency and applicability.

Contribution

The paper develops novel polynomial constructions for threshold functions, leading to improved algorithms for nearest neighbor problems, SAT solving, and circuit lower bounds.

Findings

Faster algorithms for Hamming and Euclidean nearest neighbors.

Efficient SAT algorithms for specific threshold circuit classes.

New circuit lower bounds derived from polynomial representations.

Abstract

We design new polynomials for representing threshold functions in three different regimes: probabilistic polynomials of low degree, which need far less randomness than previous constructions, polynomial threshold functions (PTFs) with "nice" threshold behavior and degree almost as low as the probabilistic polynomials, and a new notion of probabilistic PTFs where we combine the above techniques to achieve even lower degree with similar "nice" threshold behavior. Utilizing these polynomial constructions, we design faster algorithms for a variety of problems: $\bullet$ Offline Hamming Nearest (and Furthest) Neighbors: Given $n$ red and $n$ blue points in $d$-dimensional Hamming space for $d=c\log n$, we can find an (exact) nearest (or furthest) blue neighbor for every red point in randomized time $n^{2-1/O(\sqrt{c}\log^{2/3}c)}$ or deterministic time $n^{2-1/O(c\log^2c)}$. These also lead to faster MAX-SAT algorithms for sparse CNFs. $\bullet$ Offline Approximate Nearest (and Furthest) Neighbors: Given $n$ red and $n$ blue points in $d$-dimensional $\ell_1$ or Euclidean space, we can find a $(1+\epsilon)$-approximate nearest (or furthest) blue neighbor for each red point in randomized time near $dn+n^{2-\Omega(\epsilon^{1/3}/\log(1/\epsilon))}$. $\bullet$ SAT Algorithms and Lower Bounds for Circuits With Linear Threshold Functions: We give a satisfiability algorithm for $AC^0[m]\circ LTF\circ LTF$ circuits with a subquadratic number of linear threshold gates on the bottom layer, and a subexponential number of gates on the other layers, that runs in deterministic $2^{n-n^\epsilon}$ time. This also implies new circuit lower bounds for threshold circuits. We also give a randomized $2^{n-n^\epsilon}$-time SAT algorithm for subexponential-size $MAJ\circ AC^0\circ LTF\circ AC^0\circ LTF$ circuits, where the top $MAJ$ gate and middle $LTF$ gates have $O(n^{6/5-\delta})$ fan-in.