Course detail

Complexity (in English)

FIT-SLOaAcad. year: 2019/2020

Turing machines as a basic computational model for computational complexity analysis, time and space complexity on Turing machines. Alternative models of computation, RAM and RASP machines and their relation to Turing machines in the context of complexity. Asymptotic complexity estimations, complexity classes based on time- and space-constructive functions, typical examples of complexity classes. Properties of complexity classes: importance of determinism and non-determinism in the area of computational complexity, Savitch theorem, relation between non-determinism and determinism, closure w.r.t. complement of complexity classes, proper inclusion between complexity classes. Selected advanced properties of complexity classes: Blum theorem, gap theorem. Reduction in the context of complexity and the notion of complete classes. Examples of complete problems for different complexity classes. Deeper discussion of P and NP classes with a special attention on NP-complete problems (SAT problem, etc.). Relationship between decision and optimization problems. Methods for computational solving of hard optimization problems: deterministic approaches, approximation, probabilistic algorithms. Relation between complexity and cryptography.  Deeper discussion of PSPACE complete problems, complexity of formal verification methods.

Offered to foreign students

Of all faculties

Learning outcomes of the course unit

Understanding theoretical and practical limits of arbitrary computational systems. Ability to use a selected methods for computationally hard problems.

Prerequisites

Formal language theory and theory of computability on master level.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Arora, S., Barak, B.: Computational Complexity: A Modern Approach, Cambridge University Press, 2009, ISBN: 0521424267. Available online.
Bovet, D.P., Crescenzi, P.: Introduction to the Theory of Complexity, Prentice Hall International Series in Computer Science, 1994, ISBN 0-13915-380-2
Goldreich, O.: Computational Complexity: A Conceptual Perspective, Cambridge University Press, 2008, ISBN 0-521-88473-X
Kozen, D.C.: Theory of Computation, Springer, 2006, ISBN 1-846-28297-7
Gruska, J.: Foundations of Computing, International Thomson Computer Press, 1997, ISBN 1-85032-243-0
Papadimitriou, C. H.: Computational Complexity, Addison-Wesley, 1994, ISBN 0201530821
Hopcroft, J.E. et al: Introduction to Automata Theory, Languages, and Computation, Addison Wesley, 2001, ISBN 0-201-44124-1

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

  • 3 projects - 10 points each (recommended minimal gain is 15 points).
  • Final exam: max. 70 points

Language of instruction

English

Work placements

Not applicable.

Aims

Familiarize students with the complexity theory, which is necessary to understand practical limits of algorithmic problem solving on physical computational systems.
Familiarize students with a selected methods for solving hard computational problems.

Classification of course in study plans

  • Programme IT-MGR-2 Master's

    branch MBI , any year of study, summer semester, 5 credits, elective
    branch MPV , any year of study, summer semester, 5 credits, elective
    branch MGM , any year of study, summer semester, 5 credits, elective

  • Programme IT-MGR-2 Master's

    branch MGMe , any year of study, summer semester, 5 credits, compulsory-optional

  • Programme IT-MGR-2 Master's

    branch MSK , any year of study, summer semester, 5 credits, elective
    branch MBS , any year of study, summer semester, 5 credits, elective
    branch MIN , any year of study, summer semester, 5 credits, compulsory-optional
    branch MMM , any year of study, summer semester, 5 credits, compulsory-optional

  • Programme MITAI Master's

    specialization NADE , any year of study, summer semester, 5 credits, elective
    specialization NBIO , any year of study, summer semester, 5 credits, elective
    specialization NGRI , any year of study, summer semester, 5 credits, elective
    specialization NNET , any year of study, summer semester, 5 credits, elective
    specialization NVIZ , any year of study, summer semester, 5 credits, elective
    specialization NCPS , any year of study, summer semester, 5 credits, elective
    specialization NSEC , any year of study, summer semester, 5 credits, elective
    specialization NEMB , any year of study, summer semester, 5 credits, elective
    specialization NHPC , any year of study, summer semester, 5 credits, elective
    specialization NISD , any year of study, summer semester, 5 credits, elective
    specialization NIDE , any year of study, summer semester, 5 credits, elective
    specialization NISY , any year of study, summer semester, 5 credits, elective
    specialization NMAL , any year of study, summer semester, 5 credits, elective
    specialization NMAT , any year of study, summer semester, 5 credits, compulsory
    specialization NSEN , any year of study, summer semester, 5 credits, elective
    specialization NVER , any year of study, summer semester, 5 credits, elective
    specialization NSPE , any year of study, summer semester, 5 credits, elective

  • Programme IT-MGR-1H Master's

    branch MGH , any year of study, summer semester, 5 credits, recommended

  • Programme IT-MGR-2 Master's

    branch MIS , 1. year of study, summer semester, 5 credits, elective

Type of course unit

 

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

  1. Introduction. Complexity, time and space complexity.
  2. Matematical models of computation, RAM, RASP machines and their relation with Turing machines.
  3. Asymptotic estimations, complexity classes, determinism and non-determinism from the point of view of complexity.
  4. Relation between time and space, closure of complexity classes w.r.t. complementation, proper inclusion of complexity classes.
  5. Blum theorem. Gap theorem.
  6. Reduction, notion of complete problems, well known examples of complete problems.
  7. Classes P and NP. NP-complete problems. SAT problem.
  8. Travelling salesman problem, Knapsack problem and other important NP-complete problems
  9. NP optimization problems and their deterministic solution: pseudo-polynomial algorithms, parametric complexity
  10. Approximation algorithms.
  11. Probabilistic algorithms, probabilistic complexity classes.
  12. Complexity and cryptography
  13. PSPACE-complete problems. Complexity and formal verification.

Project

26 hours, compulsory

Teacher / Lecturer

Syllabus

3 projects dedicated on different aspects of the complexity theory.