DM553: Complexity and Computability

Students taking the course are expected to:

• Have knowledge of basic datastructures
• Have
  knowledge of basic algorithms for manipulating (representations of)
  sets of numbers and graphs as well as implementations of such in an
  imperative programming language.
• Be able to use basic mathematical argumentation, including
• proof by induction, proof by contradiction and logic expressions
• Be familiar with the use of combinatorial and probabilistic techniques to develop algorithms.

• Apply formalisms of formal languages in order to formulate decision problems precisely
• Construct
  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 possibility to develop an approximation algorithm for a given NP-hard optimization problem.
  
• Give lower bounds for the complexity of problem that are similar in nature to those studied in the course.
  

The course builds on the knowledge
acquired in the courses DM507 Algorithms and data structures and DM551
Algorithms and probability.

Together with courses
as above this course also provides a basis for doing a masters thesis on
algorithmic  and complexity theoretic subjects.

• 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 finite automata and regular expressions for simple languages.
  
• 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.
  

The learning objectives of the course is that the student demonstrates the ability to:

• Judge the complexity of (decision) problems.
• Judge the computational power of various models of computation.
• Construct finite automata and regular expressions for simple languages.
• 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.

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
• Repetition for the exam