Postgraduate Course: Integer and Combinatorial Optimization (MATH11192)
Course Outline
School  School of Mathematics 
College  College of Science and Engineering 
Credit level (Normal year taken)  SCQF Level 11 (Postgraduate) 
Availability  Not available to visiting students 
SCQF Credits  10 
ECTS Credits  5 
Summary  In many optimization problems, the solution is found among a set of finite elements. However, exhaustive search is usually prohibitive and, thus, specialized mathematical techniques must be used to explore the solution space in an efficient way. This course will study exact and heuristic methods for solving several of the most important integer and combinatorial optimization problems. 
Course description 
In many optimization problems, the solution is found among a set of finite elements. Typical such problems are routing problems, matching problems or scheduling problems. However, the space search can be very large (combinatorial explosion) and, as a consequence, exhaustive search is usually prohibitive. Therefore, specialized mathematical techniques must be used to explore the solution space in an efficient way. In order to study these techniques, it is important to understand fundamental notions from integer programming and graphs theory (total unimodularity, matching, spanning tree, etc.) as well as general techniques (lagrangean relaxation, branchandcut, metaheuristics).
This course will study exact and heuristic methods for solving several of the most important integer and combinatorial optimization problems. We will first cover some basic notions in integer programming and graph theory. Later, they will be applied to the study of specific problems and solution algorithms.
1. Integer Programming. Total Unimodularity. Valid Inequalities and Preprocessing.
2. Solution Algorithms. BranchandCut. Lagrangean Relaxation. Metaheuristics.
3. Matching Problems. The Assignment Problem.
4. Network Problems. Spanning Trees.
5. Covering Problems.
6. The Traveling Salesman Problem. Heuristics for the TSP.
7. Other Applications: Knapsack Problems, Scheduling Problems.

Course Delivery Information

Academic year 2024/25, Not available to visiting students (SS1)

Quota: None 
Course Start 
Semester 2 
Timetable 
Timetable 
Learning and Teaching activities (Further Info) 
Total Hours:
100
(
Lecture Hours 20,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
71 )

Assessment (Further Info) 
Written Exam
50 %,
Coursework
50 %,
Practical Exam
0 %

Additional Information (Assessment) 
There will be two assessments, each worth 25% of the final mark. There will be one week for submitting each assessment. The first assessment will be published on Week 5 and the second on Week 10. 
Feedback 
Feedback in writing. 
Exam Information 
Exam Diet 
Paper Name 
Hours & Minutes 

Main Exam Diet S2 (April/May)  Integer and Combinatorial Optimization (MATH11192)  1:30  
Learning Outcomes
On completion of this course, the student will be able to:
 Identify optimization problems which require integer variables.
 Formulate mathematical models for integer and combinatorial optimization problems.
 Identify and apply suitable solution techniques for integer and combinatorial optimization problems.
 Demonstrate an understanding of the theoretical results behind integer and combinatorial optimization, e.g., by proving unfamiliar results or applying them to unfamiliar problems.

Reading List
Combinatorial Optimization: Theory and Algorithms. B. Korte and J. Vygen. 5th edition. Springer (2012).
Combinatorial Optimization. W. Cook, W.H. Cunningham, W.R. Pulleyblank, A. Schrijver. Wiley (1998).
Integer and Combinatorial Optimization. G.L. Nemhauser and L.A. Wolsey. Wiley (1988).
Integer Programming. L.A. Wolsey. Wiley (1998). 
Additional Information
Graduate Attributes and Skills 
Not entered 
Keywords  ICO,Combinatorial Optimization,Integer Programming,Algorithms 
Contacts
Course organiser  Dr Sergio Garcia Quiles
Tel: (0131 6)50 5038
Email: Sergio.GarciaQuiles@ed.ac.uk 
Course secretary  Miss Gemma Aitchison
Tel: (0131 6)50 9268
Email: Gemma.Aitchison@ed.ac.uk 

