Overview. The main character in the stories I tell is Complexity and I mean this in the broadest sense: I am interested in classifying different kinds of complexity (Structural, algorithmic, behavioral, etc.) and in understanding how complexity arises. The main focus of my work is the emergence of complexity and how this relates to the mathematics of Compositionality (the philosophical contention that the structure or meaning of the whole depends on that of its parts). In this context, the stories I am interested in are often set in the land of computational complexity where combinatorial explosion guides and motivates deep insights in graph structure theory, parameterized complexity theory and finite model theory. I use category theory to better understand how the profound tools developed in the above areas relate to each other and to understand how far they can be pushed and generalized.
Additive Invariants of Open Petri Nets [Bumpus, Libkind, Lopez Garcia,Sorkatti, Tenka]We classify all additive invariants of open Petri nets: these are invariants valued in the natural numbers which are additive with respect to sequential and parallel composition of open Petri nets. In particular, we prove two classification theorems: one for open Petri nets and one for monically open Petri nets (i.e. open Petri nets whose interfaces are specified by monic maps). Our results can be summarized as follows. The additive invariants of open Petri nets are completely determined by their values on a particular class of single-transition Petri nets. However, for monically open Petri nets, the additive invariants are determined by their values on transitionless Petri nets and all single-transition Petri nets. Our results confirm a conjecture of John Baez (stated during the AMS’ 2022 Mathematical Research Communities workshop).
Compositional Algorithms on Compositional Data: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.
Deciding Sheaves on Presheaves [Althaus, Bumpus, Fairbanks, Rosiak]
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.
Structured Decompositions: Structural and Algorithmic Compositionality [Bumpus, Kocsis, Master]
We introduce structured decompositions: category-theoretic generalizations of many combinatorial invariants — including tree-width, layered tree-width, co-tree-width and graph decomposition width — which have played a central role in the study of structural and algorithmic compositionality in both graph theory and parameterized complexity. Structured decompositions allow us to generalize combinatorial invariants to new settings (for example decompositions of matroids) in which they describe algorithmically useful structural compositionality. As an application of our theory we prove an algorithmic meta theorem for the Sub_P-composition problem which, when instantiated in the category of graphs, yields compositional algorithms for NP-hard problems such as: Maximum Bipartite Subgraph, Maximum Planar Subgraph and Longest Path.
We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex as one that is contained in all c-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of non-essential vertices in the solution.
Degree of Satisfiability in Heyting Algebras [Bumpus, Kocsis]
Given some finite structure M and property p, it is a natural to study the
degree of satisfiability of p in M; i.e. to ask: what is probability that uniformly randomly chosen elements in M satisfy p? In group theory, a well-known result of Gustafson states that the equation xy = yx has a finite satisfiability gap: its degree of satisfiability is either 1 (in Abelian groups) or no larger than 5/8 . Degree of satisfiability has proven useful in the study of (finite and infinite) group-like and ring-like algebraic structures, but finite satisfiability gap questions have not been considered in lattice-like, order-theoretic settings yet. Here we investigate degree of satisfiability questions in the context of Heyting algebras and intuitionistic logic. We classify all equations in one free variable with respect to finite satisfiability gap, and determine which common principles of classical logic in multiple free variables have finite satisfiability gap. In particular we prove that, in a finite non-Boolean Heyting algebra, the probability that a randomly chosen element satisfies x ∨¬x = T is no larger than 2/3 . Finally, we generalize our results to infinite Heyting algebras, and present their applications to point-set topology, black-box algebras, and the philosophy of logic.
We introduce a natural temporal analogue of Eulerian circuits and prove that, in contrast with the static case, it is NP-hard to determine whether a given temporal graph is temporally Eulerian even if strong restrictions are placed on the structure of the underlying graph and each edge is active at only three times. However, we do obtain an FPT-algorithm with respect to a new parameter called interval-membership-width which restricts the times assigned to different edges; we believe that this parameter will be of independent interest for other temporal graph problems. Our techniques also allow us to resolve two open question of Akrida, Mertzios and Spirakis [CIAC 2019] concerning a related problem of exploring temporal stars.
Spined categories: generalizing tree-width beyond graphs. [Bumpus, Kocsis] Link to the European Journal of Combinatorics (EJC) version. Talk recording here.
Problems that are NP-hard in general are often tractable on inputs that have a recursive structure. For instance consider classes defined in terms of `graph decompositions’ such as of bounded tree- or clique-width graphs. Given the algorithmic success of graph decompositions, it is natural to seek analogues of these notions in other settings. What should a `tree-width-k’ digraph or lattice or temporal graph even look like? Since most decomposition notions are defined in terms of the internal structure of the decomposed object, generalizing a given notion of decomposition to a larger class of objects tends to be an arduous task. Here we show how this difficulty can be reduced significantly by finding a characteristic property formulated purely in terms of the category that the decomposed objects inhabit, which defines the decomposition independently of the internal structure. We introduce an abstract characterisation of tree-width (called the triangulation functor) as a vast generalisation of Halin’s definition of tree-width as the maximal graph parameter sharing certain properties with the Hadwiger number and chromatic number. Our uniform construction of tree-width provides a roadmap to the discovery of new tree-width-like parameters simply by collecting the relevant objects into our new notion of a spined category. (You can also find a 3-page extended abstract related to this work here.)
Directed branch-width: A directed analogue of tree-width [Bumpus, Meeks, Pettersson] (submitted)
We introduce a new digraph width measure called directed branch-width. To do this, we generalize a characterization of graph classes of bounded tree-width in terms of their line graphs to digraphs. We show that problems such as directed Hamilton path and Max-Cut (which are hard when parameterized by other known directed width measures) are in FPT when parameterized by directed branch-width. More generally, we obtain an algorithmic meta-theorem for the model-checking problem for a restricted variant of MSO2-logic on classes of bounded directed branch-width.
In a past life
During my undergraduate degree at the University of Stirling, I undertook multiple posts as a research intern. These projects all had industrial and policy applications within the field of mathematical modeling for biology using techniques from evolutionary computing and metaheuristics under the supervision of: Anthony O’Hare, Jessica Enright and Adam Kleczkowski.