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]
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.