# MA252-10 Combinatorial Optimisation

Department
Warwick Mathematics Institute
Level
Oleg Pikhurko
Credit value
10
Module duration
10 weeks
Assessment
Multiple
Study location
University of Warwick main campus, Coventry

##### 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 aims

To introduce students to basic concepts and techniques of combinatorial optimisation.

##### 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.).

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.

## Study time

Type Required
Lectures 30 sessions of 1 hour (30%)
Tutorials 9 sessions of 1 hour (9%)
Private study 31 hours (31%)
Assessment 30 hours (30%)
Total 100 hours
##### Private study description

Review lectured material and work on set exercises.

## Costs

No further costs have been identified for this module.

You do not need to pass all assessment components to pass the module.

##### Assessment group B
Weighting Study time
2 hour examination (Summer) 100% 30 hours
##### Assessment group R
Weighting Study time
In-person Examination - Resit 100%

Exam feedback.

## Courses

This module is Core option list A for:

• UMAA-GV17 Undergraduate Mathematics and Philosophy
• Year 2 of GV17 Mathematics and Philosophy
• Year 2 of GV17 Mathematics and Philosophy
• Year 2 of GV17 Mathematics and Philosophy
• Year 2 of UMAA-GV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations

This module is Core option list B for:

• Year 3 of UMAA-GV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations

This module is Core option list D for:

• Year 4 of UMAA-GV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations

This module is Option list A for:

• Year 2 of G4G1 Discrete Mathematics
• Year 2 of G4G1 Discrete Mathematics
• Year 2 of UCSA-G4G3 Undergraduate Discrete Mathematics
• UMAA-G105 Undergraduate Master of Mathematics (with Intercalated Year)
• Year 2 of G105 Mathematics (MMath) with Intercalated Year
• Year 4 of G105 Mathematics (MMath) with Intercalated Year
• Year 2 of G100 Mathematics
• Year 2 of G100 Mathematics
• Year 2 of G100 Mathematics
• Year 3 of G100 Mathematics
• Year 3 of G100 Mathematics
• Year 3 of G100 Mathematics
• Year 2 of G100 Mathematics
• Year 2 of G103 Mathematics (MMath)
• Year 2 of G103 Mathematics (MMath)
• Year 3 of G100 Mathematics
• Year 3 of G103 Mathematics (MMath)
• Year 3 of G103 Mathematics (MMath)
• Year 2 of UMAA-G1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
• Year 2 of UMAA-GL11 Undergraduate Mathematics and Economics
• Year 2 of UECA-GL12 Undergraduate Mathematics and Economics (with Intercalated Year)
• USTA-GG14 Undergraduate Mathematics and Statistics (BSc)
• Year 2 of GG14 Mathematics and Statistics
• Year 2 of GG14 Mathematics and Statistics
• UMAA-G101 Undergraduate Mathematics with Intercalated Year
• Year 2 of G101 Mathematics with Intercalated Year
• Year 4 of G101 Mathematics with Intercalated Year

This module is Option list B for:

• UPXA-GF13 Undergraduate Mathematics and Physics (BSc)
• Year 2 of GF13 Mathematics and Physics
• Year 2 of GF13 Mathematics and Physics
• UPXA-FG31 Undergraduate Mathematics and Physics (MMathPhys)
• Year 2 of FG31 Mathematics and Physics (MMathPhys)
• Year 2 of FG31 Mathematics and Physics (MMathPhys)
• USTA-Y602 Undergraduate Mathematics,Operational Research,Statistics and Economics
• Year 2 of Y602 Mathematics,Operational Research,Stats,Economics
• Year 2 of Y602 Mathematics,Operational Research,Stats,Economics