**Aims: **To present a detailed introduction to one of the central concepts of combinatorial algorithmics: NP-completeness; to extend this concept to real numbers computations; to study foundations of a more general problem of proving lower complexity bounds.
**Learning Outcomes: **
1. To be able to recognise NP-hard problems and prove the appropriate reductions.
2. To cope with NP-complete problems.
3. To know some fundamental methods of proving lower complexity bounds.
**Content: ** NP-completeness: Deterministic and Non-deterministic Turing Machines; class NP; versions of reducibility; NP-hard and NP-complete problems. Proof of NP-completeness of satisfiability problem for Boolean formulae.
Other NP-complete problems: clique, vertex cover, travelling salesman, subgraph isomorphism, etc. Polynomial-time approximation algorithms for travelling salesman and some other NP-complete graph problems.
Real Number Turing machines: Definitions; completeness of real roots existence problem for 4-degree polynomials.
Lower complexity bounds: Algebraic computation trees and their complexities; complexity of distinctness problem and of knapsack.
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