Combinatorial Optimization with Environmental Applications, 8 credits (TAOP86)

Kombinatorisk optimering med miljötillämpningar, 8 hp

Main field of study

Mathematics Applied Mathematics


First cycle

Course type

Programme course


Kaj Holmberg

Director of studies or equivalent

Ingegerd Skoglund
Course offered for Semester Period Timetable module Language Campus VOF
6CITE Information Technology, M Sc in Engineering 5 (Autumn 2017) 1 2 Swedish Linköping o

Main field of study

Mathematics, Applied Mathematics

Course level

First cycle

Advancement level


Course offered for

  • Information Technology, M Sc in Engineering

Specific information

Is not allowed in the diploma together with TAOP33.

Entry requirements

Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshold requirements for progression within the programme, or corresponding.


Linear algebra, Discrete structures, Data structures and algorithms

Intended learning outcomes

The course deals with mathematical tools for formulating, solving and
analyzing combinatorial optimization problems, often based on
different network and graph structures. Sustainable development and
environmental aspects are prominent aspects in the applications that
are discussed. An important point is the ability to choose and use
the most efficient algorithm for each specific problem structure. The
algorithms are intended to be suitable for large scale problems and
implementation on computer.
After finishing the course, the student shall be able to:
describe important types of combinatorial optimization problems.
formulate combinatorial optimization problems as mathematical models,possibly with graph terminology, and determine the difficulty of the problems with the help of complexity theory.
explain the design of and the principles behind efficient solution methods and choose and use the methods for solving different types of combinatorial optimization problems.
use available software for solving optimization problems.
take part of development of software for optimization problems.
develop heuristics for certain structured combinatorial optimization problems.
explain and use basic concepts, such as local and global optimality, convexity, extreme point, complexity, duality, heuristic, branch-and-bound, cutting planes, and basic graph theory, especially trees and cycles of different kinds.
give examples of how combinatorial optimization can be used to promote sustainable development and improve the environment.

Course content

Introduction to optimization, problem formulation, graphical solution,
computational complexity, problem complexity. The simplex method, linear duality and sensitivity analysis. Basic graph theory and
overview of different optimization problems in graphs. Models and
methods for finding minimal spanning tree, minimum cost traveling
salesman tour, minimum cost postman tour, shortest path, minimum cost
assignment, minimum cost flow and maximal flow. Methods for integer
programming, especially branch-and-bound, cutting planes and dynamic
programming. Heuristics for hard combinatorial optimization problems.
Examples on applications that concern different aspects within
sustainable development, for instance concerning a scenario that is
common for several courses.

Teaching and working methods

The course is given as seminars, computer exercises and work in PBL
groups. The seminars can be seen as a mixture of lectures and
exercises, and treats theory, methods and models. Time is also spend
on exercises in model formulation and problem solving. The computer
exercises contain both implementation of optimization algorithms and
solution of combinatorial optimization problems with the help of
available software.


BAS1Work in PBL groupU, G2 credits
LAB1LaborationsU, G2 credits
TEN1Written examinationU, 3, 4, 54 credits


Four-grade scale, LiU, U, 3, 4, 5


Matematiska institutionen

Director of Studies or equivalent

Ingegerd Skoglund


Kaj Holmberg

Course website and other links

Education components

Preliminary scheduled hours: 68 h
Recommended self-study hours: 145 h

Course literature

Kaj Holmberg: Optimering (Liber, 2010).
Kaj Holmberg: Optimering (Liber, 2010).
BAS1 Work in PBL group U, G 2 credits
LAB1 Laborations U, G 2 credits
TEN1 Written examination U, 3, 4, 5 4 credits

Regulations (apply to LiU in its entirety)

The university is a government agency whose operations are regulated by legislation and ordinances, which include the Higher Education Act and the Higher Education Ordinance. In addition to legislation and ordinances, operations are subject to several policy documents. The Linköping University rule book collects currently valid decisions of a regulatory nature taken by the university board, the vice-chancellor and faculty/department boards.

LiU’s rule book for education at first-cycle and second-cycle levels is available at 

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