Introductory description

The focus of combinatorial optimisation is on finding the "optimal" object (i.e. an object that maximises or minimises a particular function) from a finite set of mathematical objects. Problems of this type arise frequently in real world settings and throughout pure and applied mathematics, operations research and theoretical computer science. Typically, it is impractical to apply an exhaustive search as the number of possible solutions grows rapidly with the "size" of the input to the problem. The aim of combinatorial optimisation is to find more clever methods (i.e. algorithms) for exploring the solution space.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

This module provides an introduction to combinatorial optimisation. Our main focus is on several fundamental problems arising in graph theory and algorithms developed to solve them. These include problems related to shortest paths, minimum weight spanning trees, matchings, network flows, cliques, colourings and matroids. We will also discuss "intractable" (e.g. NP-hard) problems.

Learning outcomes

By the end of the module, students should be able to:

• Upon completion of the module the students should be able to apply various combinatorial structures (graphs, matroids, etc.) and basic algorithmic techniques (breadth-first search, depth-first search, etc.) to describe and solve fundamental problems of combinatorial optimization (shortest path, travelling salesman, etc.).

Indicative reading list

Main Reference:

B. Korte and J. Vygen, Combinatorial Optimization: Theory and Algorithms, Springer, 6th Edition, 2018. E-book available through the Warwick Library; click the link.

Other Resources:

A. Bondy and U. S. R. Murty, Graph Theory with Applications, Elsevier, 1976. Available through the link.

W.J. Cook, William H. Cunningham, W. R. Pulleybank, and A. Schrijver, Combinatorial Optimization, Wiley-Interscience Series in Discrete Mathematics, 1998.

C.H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Dover Publications, 1998.

Subject specific skills

The module will help the students to develop an algorithmic style of thinking.

Transferable skills

Upon completion of the module the students should be able to formalise real-life optimisation problems and apply formal methods to solve them.