Generalizing graph decompositions

Bumpus, Benjamin Merlin (2021) Generalizing graph decompositions. PhD thesis, University of Glasgow.

Abstract

The Latin aphorism ‘divide et impera’ conveys a simple, but central idea in mathematics and computer science: ‘split your problem recursively into smaller parts, attack the parts, and conquer the whole’. There is a vast literature on how to do this on graphs. But often we need to compute on other structures (decorated graphs or perhaps algebraic objects such as groups) for which we do not have a wealth of decomposition methods. This thesis attacks this problem head on: we propose new decomposition methods in a variety of settings.

In the setting of directed graphs, we introduce a new tree-width analogue called directed branch-width. We show that parameterizing by directed branch-width allows us to obtain linear-time algorithms for problems such as directed Hamilton Path and Max-Cut which are intractable by any other known directed analogue of tree-width. In fact, the algorithmic success of our new measure is more far-reaching: by proving algorithmic meta-theorems parameterized by directed branch-width, we deduce linear-time algorithms for all problems expressable in a variant of monadic second-order logic.

Moving on from directed graphs, we then provide a meta-answer to the broader question of obtaining tree-width analogues for objects other than simple graphs. We do so introducing the theory of spined categories and triangualtion functors which constitutes a vast category-theoretic abstraction of a definition of tree-width due to Halin. Our theory acts as a black box for the definition and discovery of tree-width-like parameters in new settings: given a spined category as input, it yields an appropriate tree-width analogue as output.

Finally we study temporal graphs: these are graphs whose edges appear and disappear over time. Many problems on temporal graphs are intractable even when their topology is severely restricted (such as being a tree or even a star); thus, to be able to conquer, we need decompositions that take temporal information into account. We take these considerations to heart and define a purely temporal width measure called interval-membership-width which allows us to employ dynamic programming (i.e. divide and conquer) techniques on temporal graphs whose times are sufficiently well-structured, regardless of the underlying topology.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: Q Science > QA Mathematics > QA75 Electronic computers. Computer science
• Colleges/Schools: College of Science and Engineering > School of Computing Science
• Supervisor's Name: Meeks, Dr. Kitty and Manlove, Professor David
• Date of Award: 2021
• Depositing User: Theses Team
• Unique ID: glathesis:2021-82496
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 15 Oct 2021 12:52
• Last Modified: 16 Nov 2022 14:51
• Thesis DOI: 10.5525/gla.thesis.82496
• URI: https://theses.gla.ac.uk/id/eprint/82496