More Applications of the Polynomial Method to Algorithm Design

  title={More Applications of the Polynomial Method to Algorithm Design},
  author={Amir Abboud and Richard Ryan Williams and Huacheng Yu},
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 n3/2Ω([EQUATION]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… 
An Average-Case Depth Hierarchy Theorem for Boolean Circuits
The average-case depth hierarchy theorem implies that the polynomial hierarchy is infinite relative to a random oracle with probability 1, confirming a conjecture of Hastad [Has86a], Cai [Cai86], and Babai [Bab87].
Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates
The class FORMULA[s]∘G consists of Boolean functions computable by size-s De Morgan formulas whose leaves are any Boolean functions from a class G, for classes G of functions with low communication complexity, and lower bounds are given and algorithms are given.
The Polynomial Method in Circuit Complexity Applied to Algorithm Design (Invited Talk)
Old theorems proved by this method have recently found interesting applications to the design of algorithms for basic problems in the theory of computing, and a few new ones are given.
Probabilistic Polynomials and Hamming Nearest Neighbors
  • Josh AlmanRyan Williams
  • Computer Science, Mathematics
    2015 IEEE 56th Annual Symposium on Foundations of Computer Science
  • 2015
This paper shows how to compute any symmetric Boolean function on n variables over any field (as well as the integers) with a probabilistic polynomial of degree O( √nlog(1/ε) and error at most ε, and gives the first subquadratic time algorithm for computing a (worst-case) batch of Hamming distances in superlogarithmic dimensions.
Kronecker products, low-depth circuits, and matrix rigidity
A new rigidity upper bound is shown, showing that the following classes of matrices are not rigid enough to prove circuit lower bounds using Valiant’s approach, which generalizes recent results on non-rigidity.
Bounded Depth Circuits with Weighted Symmetric Gates: Satisfiability, Lower Bounds and Compression
This paper presents algorithms for the circuit satisfiability problem of bounded depth circuits with AND, OR, NOT gates and a limited number of weighted symmetric gates that run in time super-polynomially faster than 2^n even when the number of gates is super- polynomial and the maximum weight of asymmetric gates is nearly exponential.
Classical Algorithms from Quantum and Arthur-Merlin Communication Protocols
This paper studies another way of systematically constructing low-rank decompositions of matrices which could be used by algorithms, and applies its second connection to shed some light on long-standing open problems in communication complexity.
A Structural Investigation of the Approximability of Polynomial-Time Problems
It is shown how to rule out the existence of approximation schemes for a large class of problems admitting constant-factor approximations, under a hypothesis for exact Sparse Max- 3 -SAT algorithms posed by (Alman, Vassilevska Williams’20).
Polynomial Representations of Threshold Functions and Algorithmic Applications
New polynomials for representing threshold functions in three different regimes are designed: probabilistic polynomic constructions of low degree, which need far less randomness than previous constructions, and polynomial threshold functions (PTFs) with "nice" threshold behavior and degree almost as low as the probabilists.
Deterministic APSP, Orthogonal Vectors, and More: Quickly Derandomizing Razborov-Smolensky
These techniques can be used to deterministically count k-SAT assignments on n variable formulas in 2n--n/O(k) time, roughly matching the best known running times for detecting satisfiability and resolving an open problem of Santhanam (2013).


A Satisfiability Algorithm for Sparse Depth Two Threshold Circuits
This work gives a nontrivial algorithm for the satisfiability problem for threshold circuits of depth two with a linear number of wires which improves over exhaustive search by an exponential factor and is the first to achieve constant savings even for the special case of Integer Linear Programming.
The Complexity of Satisfiability of Small Depth Circuits
An improved randomized algorithm for the satisfiability problem for circuits of constant depth d and a linear number of gates cn is shown: for each d and c, the running time is 2(1 ? ?)n where the improvement $\delta\geq 1/O(c^{2-2-1}\lg^{3\cdot 2^{d-2}- 2}c)$, and the constant in the big-Oh depends only on d.
Fast approximation algorithms for the diameter and radius of sparse graphs
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Sublinear Space Algorithms for the Longest Common Substring Problem
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  • P. Indyk
  • Computer Science, Mathematics
    Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
  • 1998
This paper gives a randomized O(nlogn)-time algorithm for the string matching with don't cares problem, which improves the Fischer-Paterson bound from 1974 and answers the open problem posed by Weiner and Galil.
All-Pairs Shortest Paths with Real Weights in O(n3/log n) Time
An O(n3/log n)-time algorithm for the all-pairs-shortest-paths problem for a real-weighted directed graph with n vertices is described, surprisingly simple and different from previous ones.
On the complexity of K -SAT
The k-SAT problem is to determine if a given k-CNF has a satisfying assignment. It is a celebrated open question as to whether it requires exponential time to solve k-SAT for k?3. Here exponential
Higher Lower Bounds for Near-Neighbor and Further Rich Problems
  • M. PatrascuM. Thorup
  • Computer Science
    2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)
  • 2006
In the most important case of d = Theta (lg n), the first superconstant lower bound is obtained, which is the highest known for any static data-structure problem, significantly improving on previous records.
New Algorithms for Subset Query, Partial Match, Orthogonal Range Searching, and Related Problems
We consider the subset query problem, defined as follows: given a set P of N subsets of a universe U, |U| = m, build a data structure, which for any query set Q ? U detects if there is any P ? P such
A Geometric Approach to Lower Bounds for Approximate Near-Neighbor Search and Partial Match
This work investigates a geometric approach to proving cell probe lower bounds for data structure problems, and shows that any (randomized) data structure for the problem that answers c-approximate nearest neighbor search queries using t probes must use space at least $n^{1+\Omega(1/ct)}$.