Polynomial Representations of Threshold Functions and Algorithmic Applications

Josh Alman, Timothy M. Chan, Ryan Williams

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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: 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 n2-1/O(√c log2/3 c) or deterministic time n2-1/O(c log2 c). These improve on a randomized n2-1/O(c log2 c) bound by Alman and Williams (FOCS'15), and also lead to faster MAX-SAT algorithms for sparse CNFs. 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+n2-Ω(ϵ1/3/log(1/ϵ)). This improves on an algorithm by Valiant (FOCS'12) with randomized time near dn+n2-Ω(√ϵ), which in turn improves previous methods based on locality-sensitive hashing. SAT Algorithms and Lower Bounds for Circuits With Linear Threshold Functions: We give a satisfiability algorithm for AC0[m] ○ LTF LTF circuits with a subquadratic number of LTF gates on the bottom layer, and a subexponential number of gates on the other layers, that runs in deterministic 2n-nϵ time. This strictly generalizes a SAT algorithm for ACC0 ○ LTF circuits of subexponential size by Williams (STOC'14) and also implies new circuit lower bounds for threshold circuits, improving a recent gate lower bound of Kane and Williams (STOC'16). We also give a randomized 2n-nϵ-time SAT algorithm for subexponential-size MAJ ○ AC0 ○ LTF ○ AC0 ○ LTF circuits, where the top MAJ gate and middle LTF gates have O(n6/5-δ) fan-in.

Original languageEnglish (US)
Title of host publicationProceedings - 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016
PublisherIEEE Computer Society
Pages467-476
Number of pages10
ISBN (Electronic)9781509039333
DOIs
StatePublished - Dec 14 2016
Externally publishedYes
Event57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016 - New Brunswick, United States
Duration: Oct 9 2016Oct 11 2016

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2016-December
ISSN (Print)0272-5428

Other

Other57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016
CountryUnited States
CityNew Brunswick
Period10/9/1610/11/16

ASJC Scopus subject areas

  • Computer Science(all)

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