# Discrete Mathematics and Data Science Research Team

The Discrete Mathematics and Data Science (DM&DS) interdisciplinary team considers discrete and data related interdisciplinary research topics in the broad sense.

The key areas of interest run across the intersection of common research areas of the MATHS and COMSC groups. We are interested in combinatorial, geometric, computational, algorithmic, and optimisation aspects of the mathematical research, as well as in mathematical foundations for topics in Computer Science.

## Aims

Our activities are focused on promotion and enhancement of collaborations between researchers from the Schools of Mathematics and Computer Science and Informatics through organisation of seminars, workshops, and discussion groups.

We aim to bring together researchers working in different areas of Discrete Mathematics and Data Science, as well as in a number of other areas within Computer Science and related Mathematics.

Our team member's research interests currently include:

- Discrete Geometry and Geometry of Numbers
- Integer Programming
- Optimisation in Graphs and Networks
- Design and Analysis of Algorithms
- Algorithm Engineering
- Data Analysis and Data Mining
- Machine Learning
- Topological Data Analysis

## Academic staff

## Dr Andrei Gagarin

Lecturer in Mathematics

- Email:
- gagarina@cardiff.ac.uk
- Telephone:
- +44 (0)29 2068 8850

## Dr Angela Mihai

Senior Lecturer in Applied Mathematics

- Email:
- mihaila@cardiff.ac.uk
- Telephone:
- +44 (0)29 2087 5570

## Dr Thomas Woolley

Lecturer in Applied Mathematics

- Email:
- woolleyt1@cardiff.ac.uk
- Telephone:
- 02920 870618

## Professor Anatoly Zhigljavsky

Chair in Statistics

- Email:
- zhigljavskyaa@cardiff.ac.uk
- Telephone:
- +44 (0)29 2087 5076

## Dr Andrey Pepelyshev

Senior Lecturer

- Email:
- pepelyshevan@cardiff.ac.uk
- Telephone:
- +44 (0)29 2087 5530

## Seminars

All seminars are held at 15:10-16:00 in Room M/1.02, Senghennydd Road, Cardiff unless stated otherwise.

### Programme Coordinators

Dr Iskander Aliev (MATHS), Dr P. Corcoran (COMPS) and Dr A. Gagarin (MATHS/COMPS)

Date | Speaker | Seminar |
---|---|---|

18 March 2019 | Joe Paat |
Proximity between an integer program (IP) and a linear program (LP) measures the distance between an optimal IP solution and the closest optimal LP solution. In this talk, we consider proximity as a function that depends on the right hand side vector of the IP and LP. We analyse how this proximity function is distributed and create a spectrum of probabilistic-like results regarding its value. This work uses ideas from group theory and Ehrhart theory, and it improves upon a recent result of Eisenbrand and Weismantel in the average case. This is joint work with Timm Oertel and Robert Weismantel. The proximity functions for IPs. |

25 February 2019 | Oded Lachish (Birkbeck, University of London) |
In the area of property testing, a central goal is to design algorithms, called tests, that decide, with high probability, whether a word over a finite alphabet is in a given property or far from the property. A property is a subset of all the possible words over the alphabet. For instance, the word can be a book, and the property can be the set of all the books that are written in English; a book is 0.1 far from being written in English if at least 0.1 of its words are not in English. The 0.1 is called the distance parameter and it can be any value in [0,1]. The input of a test is the distance parameter, the length of the input word and access to an oracle that answers queries of the sort: please give me the i'th letter in the word. The quality of a test is measured by it query complexity, which is the maximum number of queries it uses as a function of the input word length and the distance parameter, ideally this number does not depend on the input length. Tests that achieve this ideal for specific properties have been discovered for numerous properties. In general, tests that achieve the ideal for different properties differ in the manner in which they select their queries. That is, the choice of queries depends on the property. In this talk, we will see that for the price of a significant increase in the number of queries it is possible to get rid of this dependency. We will also give scenarios in which this trade-off is beneficial. |

11 December 2018 |
Reasoning with relational data, learning, and dealing with uncertainty are central to many aspects of Artificial Intelligence. Their combination is studied under a variety of names, and a broad range of languages and tools have been developed. Probabilistic logic programming achieves this combination by extending the representation and reasoning capabilities of logic programming to settings with uncertain data. This talk provides a gentle introduction to the field, and also touches upon applications and challenges. | |

20 November 2018 Please note that this seminar is at 14:10. |
Given a graph G, a classic problem is how to determine whether it is possible to draw G in the plane (on the sphere) with no edge crossings. Such drawing of G in the plane would be a planar embedding. The torus is the sphere with a handle, i.e. an orientable topological surface of genus 1, which is closest to the sphere. A similar problem is how to determine whether it is possible to draw G on the torus with no edge crossings, i.e. to obtain a toroidal embedding of G. A toroidality testing algorithm usually starts with a (non-planar) sub-graph of G isomorphic to a subdivision of K5 or K3,3 and tries to extend one of its embeddings on the torus to an embedding of the whole graph G in all possible ways. We have shown a modification of this approach - non-planar graphs which don't contain certain types of K3,3-subdivisions are much easier to decide on their toroidality by decomposing them in accordance with the 'edges' of K5 in the K5-subdivision, testing planarity of resulting components, and eventually considering rearrangements of planar embeddings. This provides an efficient method to handle the case of an initial K5-subdivision sub-graph in the graph. For general non-planar graphs containing K3,3-subdivisions, we show some particular examples to decide on their toroidality in an ad-hoc way. | |

16 October 2018 Please note that this seminar is at 14:10. |
Formal argumentation theory has been one of the main topics in the area of non-monotonic reasoning for the last two decades. The idea is, roughly, construct arguments (which are defeasible inferences) from an underlying knowledge base (which consists of inference rules). Some of these arguments will then attack other arguments (for instance by having an opposite conclusion). The resulting directed graph (in which the vertices represent the arguments and the edges represent the attack relation) is called an argumentation framework. Given such an argumentation framework, one then needs a graph-theoretical principle to determine which set (or sets) of arguments is justified. This principle (and there are many of them) is called an argumentation semantics. Once the set (or sets) of justified arguments has been determined, the justified conclusions (that is, the resulting logical inference) will be the conclusions of the justified arguments.
| |

9 October 2018 Please note that this seminar is at 14:10. |
We show that the scalability challenges of Global Optimisation (GO) algorithms can be overcome for functions with low effective dimensionality, which are constant along certain linear subspaces. Such functions can often be found in applications, for example, in hyper-parameter optimisation for neural networks, heuristic algorithms for combinatorial optimisation problems and complex engineering simulations. We propose the use of random subspace embeddings within a(ny) global minimisation algorithm, extending the approach in Wang et al (2013). Using tools from random matrix theory and conic integral geometry, we investigate the success rates of our low-dimensional embeddings of the original problem, in both a static and adaptive formulation, and show their independence on the (large) ambient dimension of the problem. We illustrate our algorithmic proposals and theoretical findings numerically, using state of the art global solvers. This work is joint with Adilet Otemissov (Turing Institute, London and Oxford University). |

## Past events

Discrete Mathematics and Data Science Research Team Seminars 2017-18

Discrete Mathematics and Data Science Research Team Seminars 2016-17