Prerequisite courses: Algorithms, Computation Theory

Aims

The aim of the course is to introduce the theory of computational
complexity. The course will explain measures of the complexity of
problems and of algorithms, based on time and space used on abstract
models. Important complexity classes will be defined, and the notion
of completeness established through a thorough study of
NP-completeness. Applications to cryptographic protocols will be
considered.

Lectures

Algorithms and problems.
Complexity of algorithms and of problems. Lower and upper bounds.
Examples: sorting and travelling salesman.

Time and space.
Models of computation and measures of complexity. Time and space
complexity on a Turing machine. Decidability and complexity.

Time complexity.
Time complexity classes. Polynomial time problems and algorithms. P
and NP.

Non-determinism.
Non-deterministic machines. The class NP redefined. Non-deterministic
algorithms for reachability and satisfiability.

NP-completeness.
Reductions and completeness. NP-completeness of satisfiability.

More NP-complete problems.
Graph-theoretic problems. Hamiltonian cycle and clique.

More NP-complete problems.
Sets, numbers and scheduling. Matching, set covering and bin packing.

coNP.
Validity of boolean formulas and its
completeness. NPcoNP. Primality and factorisation.

Cryptographic complexity.
One-way functions. The class UP.

Space complexity.
Deterministic and non-deterministic space complexity classes. The
reachability method. Savitch's theorem.

Hierarchy.
The time and space hierarchy theorems and complete problems.

Protocols.
Interactive proofs. Zero knowledge.

Objectives

At the end of the course students should

be able to analyse practical problems and classify them
according to their complexity

be familiar with the phenomenon of NP-completeness, and be able
to identify problems that are NP-complete

be aware of a variety of complexity classes and their
interrelationships

understand the role of complexity analysis in cryptographic
protocols

Recommended books

* Papadimitriou, Ch.H. (1994). Computational complexity. Addison-Wesley.
Sipser, M. (1997). Introduction to the theory of computation. PWS.