On Some Combinatorial Optimization Problems Algorithms and Complexity

Abstract: This thesis is about the computational complexity of several classes of combinatorial optimization problems, all related to the constraint satisfaction problems.A constraint language consists of a domain and a set of relations on the domain. For each such language there is a constraint satisfaction problem (CSP). In this problem we are given a set of variables and a collection of constraints, each of which is constraining some variables with a relation in the language. The goal is to determine if domain values can be assigned to the variables in a way that satisfies all constraints. An important question is for which constraint languages the corresponding CSP can be solved in polynomial time. We study this kind of question for optimization problems related to the CSPs.The main focus is on extended minimum cost homomorphism problems. These are optimization versions of CSPs where instances come with an objective function given by a weighted sum of unary cost functions, and where the goal is not only to determine if a solution exists, but to find one of minimum cost. We prove a complete classification of the complexity for these problems on three-element domains. We also obtain a classification for the so-called conservative case.Another class of combinatorial optimization problems are the surjective maximum CSPs. These problems are variants of CSPs where a non-negative weight is attached to each constraint, and the objective is to find a surjective mapping of the variables to values that maximizes the weighted sum of satisfied constraints. The surjectivity requirement causes these problems to behave quite different from for example the minimum cost homomorphism problems, and many powerful techniques are not applicable. We prove a dichotomy for the complexity of the problems in this class on two-element domains. An essential ingredient in the proof is an algorithm that solves a generalized version of the minimum cut problem. This algorithm might be of independent interest.In a final part we study properties of NP-hard optimization problems. This is done with the aid of restricted forms of polynomial-time reductions that for example preserves solvability in sub-exponential time. Two classes of optimization problems similar to those discussed above are considered, and for both we obtain what may be called an easiest NP-hard problem. We also establish some connections to the exponential time hypothesis.