arXiv subjects

Daniel Rosiak

Publications and source records attributed to Daniel Rosiak.

4 recordsLinked to original sources

arXiv2026

A Parameterized Algorithm for Testing whether the Limit of a Diagram is Empty

A limit of a (small) diagram $d : J \to E$ in a complete category $E$ can be thought of as specifying a set of equations involving the objects of $E$. To motivate this intuitively, one can think of each object $d(j)$ as a "variable" and each morphism in $J$ as a "constraint" connecting these variables. If $E$ has an initial object, a natural question arises: does our set of equations have any solution at all? Equivalently, we can ask: is the limit of $d$ initial? In this paper we consider the computational problem that, given finite diagram $d$ in a finitely complete category $E$, asks whether its limit is empty. We construct a fast algorithm (in the sense of parameterized complexity theory) that solves this problem when $E$ is of the form $\mathbf{FinSet}^{J}$ for a finite category $J$ and $d$ is a structured co-decomposition, i.e. a diagram arising from the opposite of the Grothendieck construction of a simple graph.

math.CT↗

arXiv2025

Algorithmic and Extremal Obstructions Through the Language of Cohomology

We model problems as presheaves that assign sets of certificates to input instances, and we show how to use presheaf Čech cohomology to capture the precise ways in which local solutions fail to patch into global ones. Applied to problems like Vertex Cover, Cycle Cover, and Odd Cycle Transversal, our framework exposes emergent phenomena such as hidden cycles or the inflation of small, local solutions. This approach not only rephrases classical results like König's Theorem in cohomological terms, but also reveals how to systematically account for failures of compositionality. Although our main focus is on presheaves of sets, the methods generalize naturally to Abelian presheaves, suggesting a rich interplay between graph theory, cohomology, and complexity. This work represents a first step toward a systematic, sheaf-theoretic theory of algorithmic structure and related obstructions.

math.AC↗

arXiv2023

Compositional Algorithms on Compositional Data: Deciding Sheaves on Presheaves

Algorithmicists are well-aware that fast dynamic programming algorithms are very often the correct choice when computing on compositional (or even recursive) graphs. Here we initiate the study of how to generalize this folklore intuition to mathematical structures writ large. We achieve this horizontal generality by adopting a categorial perspective which allows us to show that: (1) structured decompositions (a recent, abstract generalization of many graph decompositions) define Grothendieck topologies on categories of data (adhesive categories) and that (2) any computational problem which can be represented as a sheaf with respect to these topologies can be decided in linear time on classes of inputs which admit decompositions of bounded width and whose decomposition shapes have bounded feedback vertex number. This immediately leads to algorithms on objects of any C-set category; these include -- to name but a few examples -- structures such as: symmetric graphs, directed graphs, directed multigraphs, hypergraphs, directed hypergraphs, databases, simplicial complexes, circular port graphs and half-edge graphs. Thus we initiate the bridging of tools from sheaf theory, structural graph theory and parameterized complexity theory; we believe this to be a very fruitful approach for a general, algebraic theory of dynamic programming algorithms. Finally we pair our theoretical results with concrete implementations of our main algorithmic contribution in the AlgebraicJulia ecosystem.

cs.CC↗

arXiv2020

Sheaf Theory through Examples (Abridged Version)

This book provides an inviting tour through sheaf theory, from the perspective of applied category theory and pitched at a less specialized audience than is typical with introductions to sheaves. The book makes it as easy as possible for the reader new to sheaves, by motivating and developing the theory via a broad range of concrete examples and explicit constructions, including applications to n-colorings of graphs, satellite data, chess problems, Bayes nets, musical performance, complexes, and more. Included is an extended first chapter introducing and motivating all the necessary category-theoretical background, again with a strong emphasis on concrete examples. A new and unabridged version (including a fifth chapter on more advanced topics and a conclusion) will be available with MIT Press.

math.CT↗