# Groebner Basis Techniques for Constraint Satisfaction Problems

31st March 2006 to 30th March 2009

Many of the problems we use computers to solve have the same general form: we want the computer to find values for a collection
of variables which satisfy various constraints. The constraints restrict the combinations of values allowed for some subsets
of the variables. Many difficult computational problems fit this general framework. For example, the problem of scheduling
a collection of building tasks in a sensible order, or putting together a timetable for a school or university. The same kinds
of problems arise in choosing the frequencies for mobile phone transmitters, choosing the best routes for a fleet of delivery
vehicles, or trying to match a newly-discovered protein structure against a database. It has turned out to be very useful
for some purposes to view all these different problems as basically the same kind of problem: they can all be seen as constraint
satisfaction problems. Doing this has led to the development of special programming languages for this kind of problem, and
some very general techniques which allow us to analyse such problems and solve them as efficiently as possible. Some of the
most interesting ideas have come from linking problems of this kind to mathematical ideas, such as graph theory, or the idea
of an algebra. In this proposal we are seeking to build a new link between constraint satisfaction problems and the area of
mathematics that deals with polynomial equations. The combinations allowed by a constraint can be represented as the roots
of a polynomial equation, and then the solutions that satisfy a whole set of constraints correspond to the roots of a whole
collection of polynomial equations. Mathematicians have developed tools for solving polynomial equations, and manipulating
them into simpler forms. We want to see how these ideas can be used to manipulate constraint satisfaction problems. Also,
we want to see whether the techniques developed by computer scientists for tackling constraint satisfaction problems and analysing
their structure can be used in some new ways to analyse problems involving polynomials. We think that bringing the mathematical
ideas together with the computational techniques will give us some new insights into the mathematical ideas, and will help
to develop better ways to tackle constraint satisfaction problems.

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