In this new and fast growing area, we are confronted with the number of very fundamental questions for which the classical theory of computation knows only hardly the answers. It remains for example an important open question how much randomness is necessary to accomplish certain levels of efficiency in algorithms, and also how efficient the deterministic simulation of some classes of algorithms is possible. The proposal will address the above mentioned problems and concentrate on:
% - Design of efficient (both sequential and parallel) randomised algorithms for some selected combinatorial, algebraic and geometric problems
- Foundations of randomised complexity of computational problems
- Randomised approximation problems
- Computation with limited randomness resources (de-randomisation methods)
- Computational learning theory, theory of Neural Networks (VC Dimension of Neural Networks) and Applications
- Quantum Computation
The selection of the partners for this Working Group was based on the grounds of the relevance of their research towards the above topics (this has also motivated the extension of the group by the Weizmann Institute, Rehovot).
In recent years randomised algorithms, approximation algorithms and randomised complexity theory have become major subjects of study in the design of computer algorithms and in the theory of computation.
For some computational problems, it appears now that randomised or pseudo-randomised algorithms are more efficient than the deterministic ones (in terms of hardware size, running time, circuits depths, transparent descriptions, etc.). The very striking examples recently are the new randomised approximation algorithms for enumerating the number of perfect matchings in certain classes of graphs or for some enumeration problems in the boolean algebra and the finite fields, new approximation schemes for a number of NP-hard problems estimating the volume of convex bodies, and the quantum algorithms for factoring integers. Solutions to these problems have applications ranging from the circuit design, optimisation and coding theory to statistic mechanic and quantum field theory.