Description:

The students will gain knowledge and understanding necessary deployment of combinatorial heuristics at a professional level. They will be able not only to select and implement but also to apply and evaluate heuristics for practical problems.

Contents:

1. Discrete optimization, examples of practical tasks. Combinatorial problems. Algorithm complexity, problem complexity.
2. Models of computation. The classes P and NP. Polynomial hierarchy.
3. The notion of completeness. Complexity comparison techniques. The classes NP-complete, NP-hard and NPI.
4. Communication and circuit complexity.
5. The classes PO and NPO and their structure. Deterministic approximation algorithms. Classification of approximative problems. Pseudopolynomial algorithms. Randomization and randomized algorithms.
6. Practical deployment of heuristic and exact algorithms. Experimental evaluation.
7. State space and search space, exact methods.
8. Local methods: heuristics.
9. Simulated annealing.
10. Simulated evolution: taxonomy, genetic algorithms.
11. Advanced genetic algorithms: competent GA, fast messy GA, Stochastic optimization: models and applications. Bayesian optimization.
12. Tabu search.
13. Global methods, taxonomy of decomposition-based methods. Exact and heuristic global methods, the Davis-Putnam procedure seen as a global method.

Seminar contents:

1. Seminar: terminology, examples of complexity.
2. Seminar: examples of state space.
3. Homework consultation when required, self-study: dynamic programming revision.
4. Solved problems session: the classes P and NP, complexity proofs, problems beyond NP.
5. Solved problems session: completeness, reductions.
6. Homework consultation when required.
7. Homework consultation when required.
8. Homework consultation when required.
9. Midterm test.
10. Homework consultation when required.
11. Solved problems session: advanced heuristics, applications.
12. Homework consultation when required.
13. Homework consultation when required.
14. Backup test term, evaluation.

Recommended literature:

1. Arora, S. : Computational Complexity: A Modern Approach. Cambridge University Press, 2017. ISBN 978-1316612156.
2. Hromkovič, J. : Algorithmics for Hard Problems: Introduction to Combinatorial Optimization, Randomization, Approximation, and Heuristics 2nd Edition. Springer, 2004. ISBN 978 3540441342.
3. Kučera, L. : Kombinatorické algoritmy. SNTL, 1993.
4. Ausiello, G. - Crescenzi, P. - Kann, V. - Gambosi, G. - Spaccamela, A. M. : Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, 2003. ISBN 3540654313.

Keywords:

Combinatorial optimization, problem complexity, classes of complexity, approximability, randomized algorithm, experimental evaluation, simulated annealing, genetic algorithm, tabu search, dynamic programming, global methods.

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar