Related papers: A note on the quality of simultaneous Diophantine …
In this article algorithmic methods are presented that have essentially been introduced into computer algebra systems like Mathematica within the last decade. The main ideas are due to Stanley and Zeilberger. Some of them had already been…
We revisit the problem of integer factorization with number-theoretic oracles, including a well-known problem: can we factor an integer $N$ unconditionally, in deterministic polynomial time, given the value of the Euler totient $$\Phi$(N)$?…
We discuss a proposal for a continued fraction-like algorithm to determine simultaneous rational approximations to $d$ real numbers $\alpha_1,\ldots,\alpha_d$. It combines an algorithm of Hermite and Lagarias with ideas from LLL-reduction.…
The complexity of Philip Wolfe's method for the minimum Euclidean-norm point problem over a convex polytope has remained unknown since he proposed the method in 1974. The method is important because it is used as a subroutine for one of the…
We describe the first strongly subquadratic time algorithm with subexponential approximation ratio for approximately computing the Fr\'echet distance between two polygonal chains. Specifically, let $P$ and $Q$ be two polygonal chains with…
We introduce a new class of quadratic functions based on a hierarchy of linear time-varying (LTV) dynamical systems. These quadratic functions in the higher order space can be also seen as a non-homogeneous polynomial Lyapunov functions for…
The proximal point algorithm (PPA) has been well studied in the literature. In particular, its linear convergence rate has been studied by Rockafellar in 1976 under certain condition. We consider a generalized PPA in the generic setting of…
In recent years, we have seen several approaches to the graph isomorphism problem based on "generic" mathematical programming or algebraic (Gr\"obner basis) techniques. For most of these, lower bounds have been established. In fact, it has…
We design a Quasi-Polynomial time deterministic approximation algorithm for computing the integral of a multi-dimensional separable function, supported by some underlying hyper-graph structure, appropriately defined. Equivalently, our…
Point counting estimates are a key stepping stone to various results in metric Diophantine approximation. In this paper we use the quantitative non-divergence estimates originally developed by Kleinbock and Margulis to improve lower bounds…
We looked into the algorithm for calculating Betti numbers presented by Lloyd, Garnerone, and Zanardi (LGZ). We present a new algorithm in the same spirit as LGZ with the intent of clarifying quantum algorithms for computing Betti numbers.…
Coffman and Sethi proposed a heuristic algorithm, called LD, for multi-processor scheduling, to minimize makespan over flowtime-optimal schedules. LD algorithm is a natural extension of a very well-known list scheduling algorithm, Longest…
The lefthanded Lov\'asz local lemma (LLLL) is a generalization of the Lov\'asz local lemma (LLL), a powerful technique from the probabilistic method. We prove a computable version of the LLLL and use it to effectivize a collection of…
In approximation theory, logarithmic derivatives of complex polynomials are called simple partial fractions (SPF) as suggested by Eu.P. Dolzhenko. Many solved and unsolved extremal problems related to SPF are traced back to works of G.…
We provide efficient replicable algorithms for the problem of learning large-margin halfspaces. Our results improve upon the algorithms provided by Impagliazzo, Lei, Pitassi, and Sorrell [STOC, 2022]. We design the first…
The Lov\'{a}sz Local Lemma (LLL) is a probabilistic tool which shows that, if a collection of "bad" events $\mathcal B$ in a probability space are not too likely and not too interdependent, then there is a positive probability that no…
We study the task of learning Generalized Linear models (GLMs) in the agnostic model under the Gaussian distribution. We give the first polynomial-time algorithm that achieves a constant-factor approximation for \textit{any} monotone…
We show that Mahler's classification of real numbers $\zeta$ with respect to the growth of the sequence $(w_{n}(\zeta))_{n\geq 1}$ is equivalently induced by certain natural assumptions on the decay of the sequence…
We design a deterministic subexponential time algorithm that takes as input a multivariate polynomial $f$ computed by a constant-depth circuit over rational numbers, and outputs a list $L$ of circuits (of unbounded depth and possibly with…
Lattice reduction is a NP-hard problem well known in computer science and cryptography. The Lenstra-Lenstra-Lovasz (LLL) algorithm based on the calculation of orthogonal Gram-Schmidt (GS) bases is efficient and gives a good solution in…