More Applications of the Polynomial Method to Algorithm Design

Abstract

In low-depth circuit complexity, the polynomial method is a way to prove lower bounds by translating weak circuits into low-degree polynomials, then analyzing properties of these polynomials. Recently, this method found an application to algorithm design: Williams (STOC 2014) used it to compute all-pairs shortest paths in n 3 /2 Ω(√ log n) time on dense n-node graphs. In this paper, we extend this methodology to solve a number of problems in combinatorial pattern matching and Boolean algebra, considerably faster than previously known methods. First, we give an algorithm for BOOLEAN ORTHOGONAL DETECTION, which is to detect among two sets A, B ⊆ {0, 1} d of size n if there is an x ∈ A and y ∈ B such that x, y = 0. For vectors of dimension d = c(n) log n, we solve BOOLEAN ORTHOGONAL DETECTION in n 2−1/O(log c(n)) time by a Monte Carlo randomized algorithm. We apply this as a subroutine in several other new algorithms:

Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications

“…Surprisingly, both results are obtained by showing completeness of the Sparse Orthogonal Vectors problem for the class of first-order properties under fine-grained reductions. To obtain improved algorithms, we apply the fast Orthogonal Vectors algorithm of [3,16].…”

“…Surprisingly, both results are obtained by showing completeness of the Sparse Orthogonal Vectors problem for the class of first-order properties under fine-grained reductions. To obtain improved algorithms, we apply the fast Orthogonal Vectors algorithm of [3,16].While fine-grained reductions (reductions that closely preserve the conjectured complexities of problems) have been used to relate the hardness of disparate specific problems both within P and beyond, this is the first such completeness result for a standard complexity class. …”

Untitled

“…1 A polynomial is multilinear if the degree of each individual variable is at most 1 in the polynomial.…”

“…The latter paper uses them to prove lower bounds for AC 0 . More recently, in a remarkable result, Williams [24] (see also [25,1]) used polynomial approximations in Hamming metric to obtain the best known algorithms for all-pairs shortest path and other related algorithmic questions. Here, we study lower bounds for the existence of such approximations.…”

“…Consider The first result in this direction, due to Costello, Tao, and the third author, [8], is The exponent 2 −(d 2 +d)/2 tends very fast to zero with d, and it is desirable to improve this bound. For the case d = 2, Costello [7] obtained the optimal bound n −1/2+o (1) . In a more recent paper [21], Razborov and Viola proved Theorem 1.3.…”