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We show 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( n log(1/ε)) and error at most ε. The degree dependence on n and ε is optimal, matching a lower bound of Razborov (1987) and Smolensky (1987) for the MAJORITY function. The proof is constructive: a low-degree polynomial can be efficiently sampled from the distribution.This polynomial construction is combined with other algebraic ideas to give the first subquadratic time algorithm for computing a (worst-case) batch of Hamming distances in superlogarithmic dimensions, exactly. To illustrate, let c(n) : N → N. Suppose we are given a database D of n vectors in {0, 1} c(n) logn and a collection of n query vectors Q in the same dimension. For all u ∈ Q, we wish to compute a v ∈ D with minimum Hamming distance from u. We solve this problem in n 2−1/O(c(n) log 2 c(n)) randomized time. Hence, the problem is in "truly subquadratic" time for O(log n) dimensions, and in subquadratic time for d = o((log 2 n)/(log log n) 2 ). We apply the algorithm to computing pairs with maximum inner product, closest pair in ℓ 1 for vectors with bounded integer entries, and pairs with maximum Jaccard coefficients.

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(

Fixed-parameter algorithms and kernelization are two powerful methods to solve NP-hard problems. Yet, so far those algorithms have been largely restricted to static inputs. In this paper we provide fixed-parameter algorithms and kernelizations for fundamental NPhard problems with dynamic inputs. We consider a variety of parameterized graph and hitting set problems which are known to have f (k)n 1+o(1) time algorithms on inputs of size n, and we consider the question of whether there is a data structure that supports small updates (such as edge/vertex/set/element insertions and deletions) with an update time of g(k)n o(1) ; such an update time would be essentially optimal. Update and query times independent of n are particularly desirable. Among many other results, we show that Feedback Vertex Set and k-Path admit dynamic algorithms with f (k) log O(1) n update and query times for some function f depending on the solution size k only. We complement our positive results by several conditional and unconditional lower bounds. For example, we show that unlike their undirected counterparts, Directed Feedback Vertex Set and Directed k-Path do not admit dynamic algorithms with n o(1) update and query times even for constant solution sizes k ≤ 3, assuming popular hardness hypotheses. We also show that unconditionally, in the cell probe model, Directed Feedback Vertex Set cannot be solved with update time that is purely a function of k.