Related papers: Counting independent terms in big-oh notation
Robust estimation is much more challenging in high dimensions than it is in one dimension: Most techniques either lead to intractable optimization problems or estimators that can tolerate only a tiny fraction of errors. Recent work in…
We survey large value problems, including the large value problem for Dirichlet polynomials, the restriction problem, and problems from computer science. We describe known techniques and open problems, drawing on perspectives from all three…
We review results of papers written on the topic of polynomial amoebas with an emphasis on computational aspects of the topic. The polynomial amoebas have a lot of applications in various domains of science. Computation of the amoeba for a…
We survey results on the formalization and independence of mathematical statements related to major open problems in computational complexity theory. Our primary focus is on recent findings concerning the (un)provability of complexity…
An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large enough. Applications to the syndrome computation…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
Optimization seeks extremal points in a function. When there are superextensively many optima, optimization algorithms are liable to get stuck. Under these conditions, generic algorithms tend to find marginal optima, which have many nearly…
The works presented in this habilitation concern the algorithmics of polynomials. This is a central topic in computer algebra, with numerous applications both within and outside the field - cryptography, error-correcting codes, etc. For…
In many high-dimensional problems,polynomial-time algorithms fall short of achieving the statistical limits attainable without computational constraints. A powerful approach to probe the limits of polynomial-time algorithms is to study the…
In this paper we examine the problem of inference in Bayesian Networks with discrete random variables that have very large or even unbounded domains. For example, in a domain where we are trying to identify a person, we may have variables…
This paper proposes a thought experiment to search for efficient bounded algorithms of NPC problems by machine enumeration. The key contributions are: -- On Universal Turing Machines, a program's time complexity should be characterized as:…
Techniques for the evaluation of complex polynomials with one and two variables are introduced. Polynomials arise in may areas such as control systems, image and signal processing, coding theory, electrical networks, etc., and their…
One approach to confronting computational hardness is to try to understand the contribution of various parameters to the running time of algorithms and the complexity of computational tasks. Almost no computational tasks in real life are…
This is a chapter in the Encyclopedia of Robotics. It is devoted to the study of complexity of complete (or exact) algorithms for robot motion planning. The term ``complete'' indicates that an approach is guaranteed to find the correct…
Given two weighted automata, we consider the problem of whether one is big-O of the other, i.e., if the weight of every finite word in the first is not greater than some constant multiple of the weight in the second. We show that the…
Optimization of complex functions, such as the output of computer simulators, is a difficult task that has received much attention in the literature. A less studied problem is that of optimization under unknown constraints, i.e., when the…
New algorithms are devised for finding the maxima of multidimensional point samples, one of the very first problems studied in computational geometry. The algorithms are very simple and easily coded and modified for practical needs. The…
We exhibit a probabilistic algorithm which computes a rational point of an absolutely irreducible variety over a finite field defined by a reduced regular sequence. Its time--space complexity is roughly quadratic in the logarithm of the…
Optimization problems with both control variables and environmental variables arise in many fields. This paper introduces a framework of personalized optimization to han- dle such problems. Unlike traditional robust optimization,…
We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an…