Structured Decompositions: Structural and Algorithmic Compositionality

We introduce structured decompositions. These are category-theoretic data structures which simlutaneously generalize notions from graph theory (including tree-width, layered tree-width, co-tree-width and graph decomposition width) geometric group theory (specifically Bass-Serre theory) and dynamical systems (e.g. hybrid dynamical systems). Furthermore, structured decompositions allow us to generalize these aforementioned combinatorial invariants, which have played a central role in the study of structural and algorithmic compositionality in both graph theory and parameterized complexity, to new settings. For example, in any category with enough colimits they describe algorithmically useful structural compositionality: as an application of our theory we prove an algorithmic meta-theorem for the Sub_P-composition problem. In concrete terms, when instantiated in the category of graphs, this meta-theorem yields compositional algorithms for NP-hard problems such as: Maximum Bipartite Subgraph, Maximum Planar Subgraph and Longest Path.