Course detail

Complexity (in English)

FIT-SLOaAcad. year: 2023/2024

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.

Language of instruction

English

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. Mgr. Adam Rogalewicz, Ph.D.

Department

Department of Intelligent Systems (UITS)

Offered to foreign students

Of all faculties

Entry knowledge

Formal language theory and theory of computability on master level.

Rules for evaluation and completion of the course

  • 3 projects - 10 points each (recommended minimal gain is 15 points)

  • Final exam is performed in written form. The maximal amount of points one can get is 70 points



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

Study aids

Not applicable.

Prerequisites and corequisites

Basic literature

Arora, S., Barak, B.: Computational Complexity: A Modern Approach, Cambridge University Press, 2009, ISBN: 0521424267. Dostupné online. (EN)
Ding-Zhu Du, Ker-I Ko: Theory of Computational Complexity, 2nd Edition, Wiley 2014, ISBN: 978-1-118-30608-6 (EN)
Gruska, J.: Foundations of Computing, International Thomson Computer Press, 1997, ISBN 1-85032-243-0 (EN)
Hopcroft, J.E. et al: Introduction to Automata Theory, Languages, and Computation, Addison Wesley, 2001, ISBN 0-201-44124-1 (EN)
Papadimitriou, C. H.: Computational Complexity, Addison-Wesley, 1994, ISBN 0201530821 (EN)

Recommended reading

Bovet, D.P., Crescenzi, P.: Introduction to the Theory of Complexity, Prentice Hall International Series in Computer Science, 1994, ISBN 0-13915-380-2 (EN)
Goldreich, O.: Computational Complexity: A Conceptual Perspective, Cambridge University Press, 2008, ISBN 0-521-88473-X (EN)
Kozen, D.C.: Theory of Computation, Springer, 2006, ISBN 1-846-28297-7 (EN)

Elearning

eLearning: currently opened course

Classification of course in study plans

  • Programme IT-MSC-2 Master's

    branch MGMe , 0 year of study, summer semester, compulsory-optional

  • Programme IT-MSC-2 Master's

    branch MIN , 0 year of study, summer semester, compulsory-optional
    branch MMM , 0 year of study, summer semester, compulsory-optional
    branch MBS , 0 year of study, summer semester, elective
    branch MPV , 0 year of study, summer semester, elective
    branch MIS , 1 year of study, summer semester, elective
    branch MGM , 0 year of study, summer semester, elective
    branch MBI , 0 year of study, summer semester, elective
    branch MSK , 0 year of study, summer semester, elective

  • Programme MIT-EN Master's 0 year of study, summer semester, compulsory-optional

  • Programme MITAI Master's

    specialization NISY , 0 year of study, summer semester, elective
    specialization NSPE , 0 year of study, summer semester, elective
    specialization NBIO , 0 year of study, summer semester, elective
    specialization NSEN , 0 year of study, summer semester, elective
    specialization NVIZ , 0 year of study, summer semester, elective
    specialization NGRI , 0 year of study, summer semester, elective
    specialization NADE , 0 year of study, summer semester, elective
    specialization NISD , 0 year of study, summer semester, elective
    specialization NMAT , 0 year of study, summer semester, compulsory
    specialization NSEC , 0 year of study, summer semester, elective
    specialization NISY up to 2020/21 , 0 year of study, summer semester, elective
    specialization NCPS , 0 year of study, summer semester, elective
    specialization NHPC , 0 year of study, summer semester, elective
    specialization NNET , 0 year of study, summer semester, elective
    specialization NMAL , 0 year of study, summer semester, elective
    specialization NVER , 0 year of study, summer semester, elective
    specialization NIDE , 0 year of study, summer semester, elective
    specialization NEMB , 0 year of study, summer semester, elective
    specialization NEMB up to 2021/22 , 0 year of study, summer semester, elective

  • Programme IT-MGR-1H Master's

    specialization MGH , 0 year of study, summer semester, recommended course

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. Mgr. Adam Rogalewicz, Ph.D.
Ing. Ondřej Lengál, Ph.D.

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 hod., compulsory

Teacher / Lecturer

doc. Mgr. Adam Rogalewicz, Ph.D.

Syllabus

3 projects dedicated on different aspects of the complexity theory.

Elearning

eLearning: currently opened course