Compositional Algorithms on Compositional Data: Deciding Sheaves on Presheaves

TL;DR

This paper develops a categorial framework linking sheaf theory and graph decompositions to enable linear-time algorithms for a wide class of computational problems on complex data structures.

Contribution

It introduces a general approach using Grothendieck topologies on categories of data to design efficient algorithms for problems represented as sheaves.

Findings

Sheaf-based algorithms run in linear time for bounded decompositions.

Applicable to various data structures like graphs, hypergraphs, and complexes.

Bridges sheaf theory, graph theory, and complexity for a unified algorithmic framework.

Abstract

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.