Related papers: On Tractable Exponential Sums
Let f be an arbitrary positive integer valued function. The goal of this note is to show that one can construct a finitely generated group in which the discrete log problem is polynomially equivalent to computing the function f. In…
We reveal a complexity chasm, separating the trinomial and tetranomial cases, for solving univariate sparse polynomial equations over certain local fields. First, for any fixed field $K\in\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$,…
We give new polynomial-time algorithms for testing isomorphism of a class of groups given by multiplication tables (GpI). Two results (Cannon & Holt, J. Symb. Comput. 2003; Babai, Codenotti & Qiao, ICALP 2012) imply that GpI reduces to the…
We systematically study the computational complexity of a broad class of computational problems in phylogenetic reconstruction. The class contains for example the rooted triple consistency problem, forbidden subtree problems, the quartet…
We give a method of constructing polynomials of arbitrarily large degree irreducible over a global field F but reducible modulo every prime of F. The method consists of finding quadratic f in F[x] whose iterates have the desired property,…
In this article, we consider the weighted partition function $p_f(n)$ given by the generating series $\sum_{n=1}^{\infty} p_f(n)z^n = \prod_{n\in\mathbb{N}^{*}}(1-z^n)^{-f(n)}$, where we restrict the class of weight functions to strongly…
We study the $generalized~model~counting~problem$, defined as follows: given a database, and a set of deterministic tuples, count the number of subsets of the database that include all deterministic tuples and satisfy the query. This…
We study optimization problems that are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. Specifically, we focus on Maximum Independent…
We study the problem of computing the isolated regular solutions of a system \((f_1,\ldots,f_n)\) of \(n\) polynomial equations in \(n\) variables \((X_1, \dots, X_n)\) over a field of characteristic zero \(k\). We focus on systems with a…
The study of the complexity of the equation satisfiability problem in finite groups had been initiated by Goldmann and Russell (2002) where they showed that this problem is in polynomial time for nilpotent groups while it is NP-complete for…
We consider the problem of learning high dimensional polynomial transformations of Gaussians. Given samples of the form $p(x)$, where $x\sim N(0, \mathrm{Id}_r)$ is hidden and $p: \mathbb{R}^r \to \mathbb{R}^d$ is a function where every…
Finding an irreducible factor, of a polynomial $f(x)$ modulo a prime $p$, is not known to be in deterministic polynomial time. Though there is such a classical algorithm that {\em counts} the number of irreducible factors of $f\bmod p$. We…
The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such…
We study the problems of counting copies and induced copies of a small pattern graph $H$ in a large host graph $G$. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns $H$. In…
The input of the popular roommates problem consists of a graph $G = (V, E)$ and for each vertex $v\in V$, strict preferences over the neighbors of $v$. Matching $M$ is more popular than $M'$ if the number of vertices preferring $M$ to $M'$…
We present algorithms to evaluate two types of multiple sums, which appear in higher-order loop computations. We consider expansions of a generalized hypergeometric-type sums, $\sum_{n_1,...,n_N} [Gamma(a1.n+c1) Gamma(a2.n}+c2) ...…
A temporal (constraint) language is a relational structure with a first-order definition in the rational numbers with the order. We study here the complexity of the Quantified Constraint Satisfaction Problem (QCSP) for temporal constraint…
The CSP of a first-order theory $T$ is the problem of deciding for a given finite set $S$ of atomic formulas whether $T \cup S$ is satisfiable. Let $T_1$ and $T_2$ be two theories with countably infinite models and disjoint signatures.…
We obtain general estimates for exponential integrals of the form \[ E_f(y)=\int_{\mathbb{Z}_{p}^{n}}\psi(\sum_{j=1}^r y_j f_j(x))|dx|, \] where the $f_j$ are restricted power series over $\mathbb{Q}_p$, $y_j\in\mathbb{Q}_p$, and $\psi$ a…
A popular method in combinatorial optimization is to express polytopes P, which may potentially have exponentially many facets, as solutions of linear programs that use few extra variables to reduce the number of constraints down to a…