Parameterised algorithms for counting subgraphs, matchings, and monochromatic partitions

Ryan, Jessica Laurette (2023) Parameterised algorithms for counting subgraphs, matchings, and monochromatic partitions. PhD thesis, University of Glasgow.

Abstract

Counting the number of solutions to a computational problem is at least as hard as deciding whether a solution exists. In fact, it is often much harder. However, in theory, as well as in practice, it is often of interest to determine the number of solutions to computational problems. In this thesis, we take advantage of the structure of some hard computational counting problems to develop efficient parameterised algorithms for the kinds of problem instances which we expect to see in practice.

The subgraph counting problem asks for the number of times that a “pattern graph” appears as a subgraph of a larger “host” graph. The subgraph counting problem is computationally hard for general pairs of host and pattern graphs. Our first result describes an efficient algorithm for counting small subgraphs in host graphs with a bounded number of high-degree vertices. Our work is motivated by practical applications of subgraph counting which involve counting copies of small pattern graphs in large host graphs with this structure.

Stable matching problems arise when we wish to match together a set of agents in such a way that no pair of agents would mutually prefer to deviate from the assignment. The problem of counting stable matchings is computationally hard even in the most basic stable marriage setting where agents’ preference lists are strict and complete. Here, we study stable matching problems in the setting where agents belong to groups of similar agents called “types”. We describe efficient parameterised algorithms for counting stable matchings in a number of different settings parameterised by the number of agent types.

Our final result concerns the problem of partitioning a large edge-coloured host graph into a small number of monochromatic subgraphs. Monochromatic partitioning problems are wellstudied for specific classes of host graphs. Here, we consider the complexity of monochromatic partitioning problems for more general classes of host graphs. Specifically, we provide an efficient algorithm for counting partitions of edge-coloured graphs which are “tree like” into monochromatic paths.

• 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: 2023
• Depositing User: Theses Team
• Unique ID: glathesis:2023-83568
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 04 May 2023 10:20
• Last Modified: 05 May 2023 11:06
• URI: https://theses.gla.ac.uk/id/eprint/83568