Related papers: On Approximating Univariate NP-Hard Integrals
We demonstrate a new approach to the computation of ratios of elliptic integrals. It turns out that almost closed polygons interscribed between two conics retain some of the properties of such closed polygons. We apply these retained…
Inference algorithms based on evolving interactions between replicated solutions are introduced and analyzed on a prototypical NP-hard problem - the capacity of the binary Ising perceptron. The efficiency of the algorithm is examined…
Given any polynomial $p$ in $C[X]$, we show that the set of irreducible matrices satisfying $p(A)=0$ is finite. In the specific case $p(X)=X^2-nX$, we count the number of irreducible matrices in this set and analyze the arising sequences…
Approximation algorithms for classical constraint satisfaction problems are one of the main research areas in theoretical computer science. Here we define a natural approximation version of the QMA-complete local Hamiltonian problem and…
In this paper we consider inverse problems that are mathematically ill-posed. That is, given some (noisy) data, there is more than one solution that approximately fits the data. In recent years, deep neural techniques that find the most…
The Inverse Problem for the estimation of a point-wise approximation error occurring at the discretization and solving of the system of partial differential equations is addressed. The set of the differences between the numerical solutions…
A new sampling methodology based on incomplete cosine expansion series is presented as an alternative to the traditional sinc function approach. Numerical integration shows that this methodology is efficient and practical. Applying the…
Provided a special function of one variable and some of its derivatives can be accurately computed over a finite range, a method is presented to build a series of polynomial approximations of the function with a defined relative error over…
We study a class of functional problems reducible to computing $f^{(n)}(x)$ for inputs $n$ and $x$, where $f$ is a polynomial-time bijection. As we prove, the definition is robust against variations in the type of reduction used in its…
To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…
We study the complexity of computing the sign of the Tutte polynomial of a graph. As there are only three possible outcomes (positive, negative, and zero), this seems at first sight more like a decision problem than a counting problem.…
We explore an algorithm for approximating roots of integers, discuss its motivation and derivation, and analyze its convergence rates with varying parameters and inputs. We also perform comparisons with established methods for approximating…
In this paper, we numerically examine the precision challenges that emerge in automatic differentiation and numerical integration in various tasks now tackled by physics-informed neural networks (PINNs). Specifically, we illustrate how…
In order to formulate mathematical conjectures likely to be true, a number of base cases must be determined. However, many combinatorial problems are NP-hard and the computational complexity makes this research approach difficult using a…
Iterative rounding has enjoyed tremendous success in elegantly resolving open questions regarding the approximability of problems dominated by covering constraints. Although iterative rounding methods have been applied to packing problems,…
We present new algorithms for computing the low $n$ bits or the high $n$ bits of the product of two $n$-bit integers. We show that these problems may be solved in asymptotically 75% of the time required to compute the full $2n$-bit product,…
We establish the average-case hardness of the algorithmic problem of exact computation of the partition function associated with the Sherrington-Kirkpatrick model of spin glasses with Gaussian couplings and random external field. In…
We study robust convex quadratic programs where the uncertain problem parameters can contain both continuous and integer components. Under the natural boundedness assumption on the uncertainty set, we show that the generic problems are…
Classical algorithms in numerical analysis for numerical integration (quadrature/cubature) follow the principle of approximate and integrate: the integrand is approximated by a simple function (e.g. a polynomial), which is then integrated…
We give a high precision polynomial-time approximation scheme for the supremum of any honest n-variate (n+2)-nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and…