Ph D abstract - Fredrik Kuivinen
Algorithms and Hardness Results for Some Valued CSPs
In the Constraint Satisfaction Problem (CSP) one is supposed to find an assignment to a set of variables so that a set of given constraints are satisfied. Many problems, both practical and theoretical, can be modelled as CSPs. As these problems are computationally hard it is interesting to investigate what kind of restrictions on the problems implies computational tractability. In this thesis the computational complexity of restrictions of two optimisation problems which are related to the CSP is studied.
The first problem we investigate is Maximum Solution (Max Sol) where one is looking for a solution which satisfies all constraints and also maximises a linear objective function. The Maximum Solution problem is a generalisation of the well-known integer linear programming problem. In the case when the constraints are equations over an abelian group we obtain tight inapproximability results. We also study Max Sol for so-called maximal constraint languages and a partial classification theorem is obtained in this case. Finally, Max Sol over the boolean domain is studied in a setting where each variable only occurs a bounded number of times.
The second problem is the Maximum Constraint Satisfaction Problem (Max CSP). In this problem one is looking for an assignment which maximises the number of satisfied constraints. We first show that if the constraints are known to give rise to an NP-hard CSP, then one cannot get arbitrarily good approximate solutions in polynomial time, unless P = NP. We use this result to show a similar hardness result for the case when only one constraint relation is used. We also study the submodular function minimisation problem (SFM) on certain finite lattices. There is a strong link between Max CSP and SFM; new tractability results for SFM implies new tractability results for Max CSP. It is conjectured that SFM is the only reason for Max CSP to be tractable, but no one has yet managed to prove this. We obtain new tractability results for SFM on diamonds and evidence which supports the hypothesis that all modular lattices are tractable.