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Consider an ideal $I \subset R = \bC[x_1,...,x_n]$ defining a complex affine variety $X \subset \bC^n$. We describe the components associated to $I$ by means of {\em numerical primary decomposition} (NPD). The method is based on the…
Until now, Computer Scientists have concerned themselves with identifying efficient algorithms for solving the general case of some problem -- that is finding one which performs well when the size of the input tends to infinity. In this…
We consider the general problem of blocking all solutions of some given combinatorial problem with only few elements. For example, the problem of destroying all maximum cliques of a given graph by forbidding only few vertices. Problems of…
We will find a lower bound on the recognition complexity of the theories that are nontrivial relative to some equivalence relation (this relation may be equality), namely, each of these theories is consistent with the formula, whose sense…
State-of-the-art NLP methods achieve human-like performance on many tasks, but make errors nevertheless. Characterizing these errors in easily interpretable terms gives insight into whether a classifier is prone to making systematic errors,…
A promising approach to achieve computational supremacy over the classical von Neumann architecture explores classical and quantum hardware as Ising machines. The minimisation of the Ising Hamiltonian is known to be NP-hard problem for…
Treating a conjecture, P^#P != NP, on the separation of complexity classes as an axiom, an implication is found in three manifold topology with little obvious connection to complexity theory. This is reminiscent of Harvey Friedman's work on…
In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like "input", "size" and…
Principal Component Analysis is a key technique for reducing the complexity of high-dimensional data while preserving its fundamental data structure, ensuring models remain stable and interpretable. This is achieved by transforming the…
While many approaches were developed for obtaining worst-case complexity bounds for first-order optimization methods in the last years, there remain theoretical gaps in cases where no such bound can be found. In such cases, it is often…
We provide a lower bound showing that the $O(1/k)$ convergence rate of the NoLips method (a.k.a. Bregman Gradient) is optimal for the class of functions satisfying the $h$-smoothness assumption. This assumption, also known as relative…
Many recent studies on first-order methods (FOMs) focus on \emph{composite non-convex non-smooth} optimization with linear and/or nonlinear function constraints. Upper (or worst-case) complexity bounds have been established for these…
Many theorems about Kolmogorov complexity rely on existence of combinatorial objects with specific properties. Usually the probabilistic method gives such objects with better parameters than explicit constructions do. But the probabilistic…
Variational analysis provides the theoretical foundations and practical tools for constructing optimization algorithms without being restricted to smooth or convex problems. We survey the central concepts in the context of a concrete but…
A choice of first-order variables for the characteristic problem of the linearized Einstein equations is found which casts the system into manifestly well-posed form. The concept of well-posedness for characteristic problems invoked is that…
The assembly index of assembly theory quantifies the minimal number of composition steps required to construct an object from elementary components. The study proves that the decision version of the assembly index problem is NP-complete,…
We survey the average-case complexity of problems in NP. We discuss various notions of good-on-average algorithms, and present completeness results due to Impagliazzo and Levin. Such completeness results establish the fact that if a certain…
We study nominal anti-unification, which is concerned with computing least general generalizations for given terms-in-context. In general, the problem does not have a least general solution, but if the set of atoms permitted in…
State minimization of combinatorial filters is a fundamental problem that arises, for example, in building cheap, resource-efficient robots. But exact minimization is known to be NP-hard. This paper conducts a more nuanced analysis of this…
In the past few years powerful generalizations to the Euclidean k-means problem have been made, such as Bregman clustering [7], co-clustering (i.e., simultaneous clustering of rows and columns of an input matrix) [9,18], and tensor…