Distances to Lattice Points and Integer Linear Programming


My current research is in the field of Integer Optimisation. This mathematical discipline looks at optimisation or feasibility problems where some or all of the variables are restricted to integers. Because integer optimisation is concerned with discrete values, it is unsurprising that it is widely used to model problems in many real world scenarios including operational research tasks such as scheduling and manufacturing.

I currently focus on 'integer linear programming' (ILP) problems. These problems have the property that both the constraints and objective functions are linear. It is well known that in general, finding a solution to an ILP is NP-complete. Despite this, one can obtain an approximation within polynomial time by solving the problem's related 'linear program' (LP). I aim to obtain optimal worst-case upper bounds for the distance between an optimal solution to the LP to some nearby feasible integer solution, which solves the ILP. 

Publications & Preprints

  • Distances to Lattice Points in Rational Polyhedra (PhD Thesis)

A. Williams (2022)

[Online Research @ Cardiff]

  • Intelligent Security Control Based on the New Criterion of Edwards and Montgomery Curves, Isogenous  of These Curves Supersingularity

  • R. Skuratovskii, A. Williams and Y. Osadchyy (2021)

[Proceedings of International Conference on Advanced Computing Applications]​

  • Distance-Sparsity Transference for Vertices of Corner Polyhedra

I. Aliev, M. Celaya, M. Henk and A. Williams (2021)

[SIAM Journal on Optimization] ; [arXiv.org]​

  • Irreducible bases and subgroups of a wreath product in applying to diffeomorphism groups acting on the Möbius band

R. Skuratovskii and A. Williams (2020)

[Rendiconti del Circolo Matematico di Palermo Series 2 (Springer)

  • Some Approach to Key Exchange Protocol Based on Non-Commutative Groups

R. Skuratovskii and A. Williams (2020)

[International Journal of Mathematical Models and Methods in Applied Sciences] ; [arXiv.org]

  • A Solution of the Inverse Problem to Doubling of Twisted Edwards Curve Point over Finite Field

R. Skuratovskii and A. Williams (2019)

[Wydawnictwo Naukowe Akademii Techniczno-Humanistycznej w Bielsku-Białej]

  • Minimal Generating Set and a Structure of the Wreath Product of Groups, and the Fundamental Group of the Orbit Morse Function

R. Skuratovskii and A. Williams (2019)

[Вісник Донецького національного університету. Серія А: Природничі науки] ; [arXiv.org]

Conferences & Workshops

Contributed Talk:

  • British Early Career Mathematicians' Colloquium | University of Birmingham (2021)

  • Young Researchers in Mathematics | University of Bristol (2021)

  • International Conference of Young Mathematicians | Online Kiev, Ukraine (2021)

  • British Early Career Mathematicians' Colloquium | University of Birmingham (2020)

  • Algebraic and Geometric Methods of Analysis | Odessa, Ukraine (2020)

  • Intellectual Systems and Information Technologies | Odessa, Ukraine (2019)

  • British Combinatorial Conference | University of Birmingham (2019)

  • SIAM-UKIE National Student Chapter Conference | University of Manchester (2019)

  • International Conference of Young Mathematicians | Institute of Mathematics NAS of Ukraine (2019)

  • Welsh Mathematics Colloquium | Gregynog Hall (2019)

  • British Young Mathematicians Colloquium | University of Birmingham (2019)

Contributed Poster:

  • ​SIAM-UKIE National Student Chapter Conference | University of Manchester (2019)

  • SIAM-IMA Student Chapter Conference | Oxford University (2019)


  • IPCO and Summer School | London School of Economics (2020)

  • SCIP Online Workshop 2020 | University of Exeter (2020)

  • Mixed Integer Programming Workshop 2020 | Online Edition (2020)

  • LMS Computer Science Colloquium “Mathematics of Security” | London (2019)

  • NATCOR Combinatorial Optimisation | University of Southampton (2019)

  • NATCOR Stochastic Modelling | Lancaster University (2019)

  • Computational Optimisation in Action | Edinburgh University (2018) 

  • NATCOR Convex Optimisation | Edinburgh University (2018) 

  • Young Researchers in Mathematics | Southampton University (2018)

Society Membership

  • Society for Industrial and Applied Mathematics (Discrete Mathematics, Linear Algebra, Optimisation)

  • South Wales Operational Research Discussion Society (SWORDS)


  • International Journal for Mathematics Teaching and Learning | Reviewer

  • Canadian Journal of Science, Mathematics and Technology Education (Springer) | Reviewer