Related papers: Complexity Classes and Completeness in Algebraic G…
We investigate the arithmetic nature of P-recursive sequences through the lens of their D-finite generating functions. Building on classical tools from differential algebra, we revisit the integrality criterion for Motzkin-type sequences…
Planar graphs can be represented as intersection graphs of different types of geometric objects in the plane, e.g., circles (Koebe, 1936), line segments (Chalopin \& Gon{\c{c}}alves, 2009), \textsc{L}-shapes (Gon{\c{c}}alves et al, 2018).…
We present an overview of the notions of exact sequences of Hopf algebras and tensor categories and their connections. We also present some examples illustrating their main features; these include simple fusion categories and a natural…
Starting from any unital colored PROP $P$, we define a category $P(P)$ of shapes called $P$-propertopes. Presheaves on $P(P)$ are called $P$-propertopic sets. For $0 \leq n \leq \infty$ we define and study $n$-time categorified $P$-algebras…
The field of numerical algebraic geometry consists of algorithms for numerically solving systems of polynomial equations. When the system is exact, such as having rational coefficients, the solution set is well-defined. However, for a…
This article belongs to a series on geometric complexity theory (GCT), an approach to the P vs. NP and related problems through algebraic geometry and representation theory. The basic principle behind this approach is called the flip. In…
Let $G$ be a finitely presented group. A new complexity called \textit{Karoubi-Weibel complexity} or \textit{covering type}, is defined for $G$. The construction is inspired by recent work of Karoubi and Weibel \cite{KW}, initially applied…
As it follows from G\"odel's incompleteness theorems, any consistent formal system of axioms and rules of inference should imply a true unprovable statement. Actually, this fundamental principle can be efficiently applicable in…
In this paper, we define and study variants of several complexity classes of decision problems that are defined via some criteria on the number of accepting paths of an NPTM. In these variants, we modify the acceptance criteria so that they…
In this note we introduce a notion of a generically (strongly generically) NP-complete problem and show that the randomized bounded version of the halting problem is strongly generically NP-complete.
In this paper we discuss some special generalizations of equationally Noetherian property which naturally arise in the universal algebraic geometry. We introduce weakly equationally Noetherian, qw-compact, uw-compact, and weakly uw-compact…
In the recent literature of Artificial Intelligence, an intensive research effort has been spent, for various algebras of qualitative relations used in the representation of temporal and spatial knowledge, on the problem of classifying the…
Do complexity classes have many-one complete sets if and only if they have Turing-complete sets? We prove that there is a relativized world in which a relatively natural complexity class-namely a downward closure of NP, \rsnnp - has…
In this note we study the computational complexity of feedback arc set counting problems in directed graphs, highlighting some subtle yet common properties of counting classes. Counting the number of feedback arc sets of cardinality $k$ and…
There are many methods developed to approximate a cloud of vectors embedded in high-dimensional space by simpler objects: starting from principal points and linear manifolds to self-organizing maps, neural gas, elastic maps, various types…
In this paper we study two problems related to the drawing of level graphs, that is, T-LEVEL PLANARITY and CLUSTERED-LEVEL PLANARITY. We show that both problems are NP-complete in the general case and that they become polynomial-time…
The complexity of a graph is the number of its labeled spanning trees. In this work complexity is studied in settings that admit regular graphs. An exact formula is established linking complexity of the complement of a regular graph to…
We develop a theory of complexity for numerical computations that takes into account the condition of the input data and allows for roundoff in the computations. We follow the lines of the theory developed by Blum, Shub, and Smale for…
A compact T-algebra is an initial T-algebra whose inverse is a final T-coalgebra. Functors with this property are said to be algebraically compact. This is a very strong property used in programming semantics which allows one to interpret…
Order types are a well known abstraction of combinatorial properties of a point set. By Mn\"ev's universality theorem for each semi-algebraic set $V$ there is an order type with a realization space that is \emph{stably equivalent} to $V$.…