# DM553: Complexity and Computability

## Comment

## Entry requirements

## Academic preconditions

- Have knowledge of basic algorithms for manipulating (representations of) sets of numbers and graphs, along with analysis of algorithms.
- Be able to use basic mathematical argumentation, including proof by induction, proof by contradiction and logical expressions
- Be familiar with the use of combinatorial techniques to develop algorithms.

## Course introduction

- Apply formalisms of formal languages in order to formulate decision problems precisely.
- Work with finite automata, regular expressions, push-down automata and context-free grammars as elements in an algorithmic solution of more complicated problems.
- Decide the complexity of new problems based on knowledge of the complexity of important examples of problems from the course.
- Judge whether a given problem may be solved by a computer or is undecidable.
- Argue that problems are NP-complete.
- Judge the possibilities for developing an approximation or fixed parameter algorithm for a given NP-hard optimization problem.
- Give lower bounds for the complexity of problems that are similar in nature to those studied in the course.

- Give the competence to analyze complexity of (decision) problems.
- Give knowledge about the computational power of different models of computation.
- Enable the student to construct push-down automata and context-free grammars for simple languages.
- Equip the students with important tools to prove that a given language cannot be recognized by a finite automation, a push-down automaton or a Turing machine.
- Enable the student to prove lower bounds for the complexity of algorithms for a given problem.
- Enable the student to develop new approximation algorithms.
- Give the student important tools for proving that a given decision problem is NP-complete or undecidable.

## Expected learning outcome

- Judge the complexity of (decision) problems.
- Judge the computational power of various models of computation.
- Construct push-down automata and context-free grammars for simple languages.
- Prove that a given language, which in nature resembles those from the course, cannot be recognized by a finite automaton, a push-down automaton or a Turing machine.
- Prove lower bounds for the complexity of algorithms for a given problem which in nature resembles those from the course.
- Design new approximation algorithms for a given problem which in nature resembles those from the course.
- Prove that a given decision problem which in nature resembles those from the course is NP-complete or undecidable.
- Define fixed parameter tractability and explain an example of this.
- Give clear, precise definitions and proofs for the above.

## Content

- Finite automata and push-down automata
- Context-free languages and grammars
- Turing machines
- Decidability
- Problem reductions
- Lower bounds (information theoretical and adversary arguments)
- The complexity classes P and NP
- The theory of NP-completeness
- Approximation algorithms
- Parameterized complexity

## Literature

## Examination regulations

## Exam element a)

## Timing

## Prerequisites

Type | Prerequisite name | Prerequisite course |
---|---|---|

Examination part | Prerequisites for participating in the exam a) | N330048101, DM553: Complexity and Computability |

## Tests

## Oral exam

## EKA

## Assessment

## Grading

## Identification

## Language

## Duration

## Examination aids

To be announced during the course

## ECTS value

## Additional information

The exam consists of an oral exam and a number of assignments handed in during the course. The final grade is given on the basis of an overall assessment of assignments and the oral exam. The external examiner will be able to see the assignments.

The re-exam is an oral exam. External examiner, Danish 7 mark scale.

## Prerequisites for participating in the exam a)

## Timing

## Tests

## Written assignment

## EKA

## Assessment

## Grading

## Identification

## Language

## Examination aids

## ECTS value

## Additional information

Prerequisites for participating in the exam consists handing in a document with name and confirmation of participation in the oral exam. The prerequisite examination is a prerequisite for participation in exam element a).

## Indicative number of lessons

## Teaching Method

- Intro phase (lectures) - 40 hours
- Training phase: 30 hours

Note that part of the lectures will be given as video lectures. The participants will watch these and prepare themselves for dicussion about the topics in class.

In the intro phase, concepts, theories and models are introduced and put into perspective.

In the training phase, students train their skills through exercises and dig deeper into the subject matter.

Educational activities

- Self study of the teaching materials
- Solving weekly assignments in order to discuss these at the tutorials
- Written assignments as part of the exam
- Selfguided followup on the intro and tutorial classes
- Review for the exam