Related papers: The complexity of MinRank
In this note, we provide proven estimates for the complexity of the SupportMinors Modeling, mostly confirming the heuristic complexity estimates contained in the original article.
We study the complexity of solving the \emph{generalized MinRank problem}, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at most $r$. A natural algebraic representation of this problem gives rise to a…
Complexity theory provides a wealth of complexity classes for analyzing the complexity of decision and counting problems. Despite the practical relevance of enumeration problems, the tools provided by complexity theory for this important…
Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond ${\bf NP\neq co NP}$. These conjectures formally connect computational complexity with the difficulty of…
The MinRank problem is a simple linear algebra problem: given matrices with coefficients in a field, find a non trivial linear combination of the matrices that has a small rank. There are several algebraic modeling of the problem. The main…
Building upon works of Hironaka, Bierstone-Milman, Villamayor and Wlodarczyk, we give an a priori estimate for the complexity of the simplified Hironaka algorithm. As a consequence of this result, we show that there exists canonical…
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…
Is it possible to find a shortest description for a binary string? The well-known answer is "no, Kolmogorov complexity is not computable." Faced with this barrier, one might instead seek a short list of candidates which includes a laconic…
We investigate the computational complexity of tensor rank, a concept that plays fundamental role in different topics of modern applied mathematics. For tensors over any integral domain, we prove that the rank problem is polynomial time…
Finding the most probable explanation for observed variables in a Bayesian network is a notoriously intractable problem, particularly if there are hidden variables in the network. In this paper we examine the complexity of a related…
The discrete logarithm problem (DLP) generalizes to the constrained DLP, where the secret exponent $x$ belongs to a set known to the attacker. The complexity of generic algorithms for solving the constrained DLP depends on the choice of the…
We initiate a study of algorithms with a focus on the computational complexity of individual elements, and introduce the fragile complexity of comparison-based algorithms as the maximal number of comparisons any individual element takes…
In this paper, we will present a generalization for a minimization problem from I. Daubechies, M. Defrise, and C. Demol [3]. This generalization is useful for solving many practical problems in which more than one constraint are involved.…
We discuss the frequent pattern mining problem in a general setting. From an analysis of abstract representations, summarization and frequent pattern mining, we arrive at a generalization of the problem. Then, we show how the problem can be…
We prove some new rank selection theorems for balanced simplicial complexes. Specifically, we prove that rank selected subcomplexes of balanced simplicial complexes satisfying Serre's condition $(S_{\ell})$ retain $(S_{\ell})$. We also…
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…
We show that determining the rank of a tensor over a field has the same complexity as deciding the existential theory of that field. This implies earlier NP-hardness results by H{\aa}stad~\cite{H90}. The hardness proof also implies an…
We show that computing the minimum rank of a sign pattern matrix is NP hard. Our proof is based on a simple but useful connection between minimum ranks of sign pattern matrices and the stretchability problem for pseudolines arrangements. In…
Effective complexity measures the information content of the regularities of an object. It has been introduced by M. Gell-Mann and S. Lloyd to avoid some of the disadvantages of Kolmogorov complexity, also known as algorithmic information…
We give a broad survey of recent results in Enumerative Combinatorics and their complexity aspects.