Combinatorial optimization

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Lecturer:

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Department of Digital Design

Synopsis:

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.

Requirements:

Syllabus of lectures:

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. The classes PO and NPO and their structure. Deterministic approximation algorithms. Classification of approximative problems. Pseudopolynomial algorithms. Randomization and randomized algorithms.

5. Communication and circuit complexity

6. Practical deployment of heuristic and exact algorithms. Experimental evaluation.

7. Local methods: state space and search space, exact methods, heuristics.

8. Simulated annealing.

9. Simulated evolution: taxonomy, genetic algorithms.

10. Advanced genetic algorithms: competent GA, fast messy GA, Stochastic optimization: models and applications. Bayesian optimization.

11. Tabu search.

12. Global methods, taxonomy of decomposition-based methods. Exact and heuristic global methods, the Davis-Putnam procedure seen as a global method.

13. Reserved

Syllabus of tutorials:

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.

Study Objective:

Many practical tasks are computationally infeasible. The course is about fast algorithms for such problems, both exact and heuristic. Heuristic algorithms tend to be simple to program, but difficult to configure and deploy. Their successful application requires a deeper understanding of their operation and complexity theory. A series of individual works guides the student from simple techniques to solutions of practically significant problems.

Study materials:

1. Arora, S. : Computational Complexity: A Modern Approach. Cambridge University Press, 2017. ISBN 978-1316612156.

2. Hromkovic, 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.

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Master specialization Computer Science, in Czech, 2018-2019 (compulsory course in the program)