Constructing Algorithms for Constraint Satisfaction and Related Problems Methods and Applications

Abstract: In this thesis, we will discuss the construction of algorithms for solving Constraint Satisfaction Problems (CSPs), and describe two new ways of approaching them. Both approaches are based on the idea that it is sometimes faster to solve a large number of restricted problems than a single, large, problem. One of the strong points of these methods is that the intuition behind them is fairly simple, which is a definite advantage over many techniques currently in use.The first method, the covering method, can be described as follows: We want to solve a CSP with n variables, each having a domain with d elements. We have access to an algorithm which solves problems where the domain has size e < d, and we want to cover the original problem using a number of restricted instances, in such a way that the solution set is preserved. There are two ways of doing this, depending on the amount of work we are willing to invest; either we construct an explicit covering and end up with a deterministic algorithm for the problem, or we use a probabilistic reasoning and end up with a probabilistic algorithm.The second method, called the partitioning method, relaxes the demand on the underlying algorithm. Instead of having a single algorithm for solving problems with domain less than d, we allow an arbitrary number of them, each solving the problem for a different domain size. Thus by splitting, or partitioning, the domain of the large problem, we again solve a large number of smaller problems before arriving at a solution.Armed with these new techniques, we study a number of different problems; the decision problems (d, l)-CSP and k-Colourability, together with their counting counterparts, as well as the optimisation problems Max Ind CSP, Max Value CSP, Max CSP, and Max Hamming CSP. Among the results, we find a very fast, polynomial space algorithm for determining k-colourability of graphs.