Polynomial Representations of Threshold Functions and Algorithmic Applications

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=clog n, we can find an (exact) nearest (or furthest) blue neighbor for every red point in randomized time n^2-1/O(√clog2/3c) or deterministic time n2-1/O(clog2c). 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 ℓ1 or Euclidean space, we can find a (1+ε)-approximate nearest (or furthest) blue neighbor for each red point in randomized time near dn+n^2-Omega(ε1/3/log(1/ε)). bullet SAT Algorithms and Lower Bounds for Circuits With Linear Threshold Functions: We give a satisfiability algorithm for AC0[m]circ LTFcirc 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 2n-nε time. This also implies new circuit lower bounds for threshold circuits. We also give a randomized 2n-nε-time SAT algorithm for subexponential-size MAJcirc AC0circ LTFcirc AC0circ LTF circuits, where the top MAJ gate and middle LTF gates have O(n6/5-δ) fan-in.