Polynomial Representations of Threshold Functions and Algorithmic Applications

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 ddimensional 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( √ c log 2/3 c) or deterministic time n 2−1/O(c log 2 c) .These improve on a randomized n 2−1/O(c log 2 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 ddimensional ℓ 1 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−Ω(ε 1/3 / log(1/ε)) . This improves on an algorithm by Valiant (FOCS'12) with randomized time near dn + n 2−Ω(

Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

“…Recent results (e.g. [ACW16]) obtained 2 n−o(n) time improvements, but there is still no O((2 − ε) n ) time algorithm. Generalizing the reduction from [Wil07] (see Section 9), one can reduce Max-k-SAT to -hyperclique in a k-uniform hypergraph for any > k, so that if the latter problem can be solved in O(n −ε ) time for n node graphs and ε > 0, then Max-k-SAT can be solved in O(2 (1−δ )n ) time for formulas on n variables.…”

“…m for number of clauses and n for number of variables). The current fastest known algorithms for Max-k-SAT come from Alman, Chan and Williams [ACW16]. LEMMA 9.1.…”

An Equivalence Class for Orthogonal Vectors

“…The state-of-the-art algorithm for (1+ε) approximation to Bichrom.-ℓ p -Closest-Pair runs in n 2− O(ε 1/3 ) time, and for Max-IP n,c log n , the best running time n 2− O(1/ √ c) . Both algorithms are presented in [ACW16], and relied on probabilistic threshold functions.…”

“…It was recently shown in [Che18] that Apx-Max-IP can be solved in n 2−1/O(log c) time, while the best known algorithm for solving Apx-Min-IP just applies the n 2−1/ O( √ c) time algorithm for Min-IP [ACW16].…”

LSH Forest: Practical Algorithms Made Theoretical

“…This decomposition is not particularly efficient, and generously contributes to the n o(1) factors in the bounds for space and query time. Making such reductions for finite subsets of R d more efficient would be of independent interest-in particular, for problems where we know algorithms with better performance for the pseudo-random datasets than the worst-case datasets (e.g., [Val15,KKK16,KKKÓ16,ACW16]). …”