Extended constraint satisfaction problems

To solve an instance of the constraint satisfaction problem (CSP) one has to findan assignment of values to variables that satisfies given constraints. This thesisconcerns two different extensions of the constraint satisfaction problem.The first extension allows for an infinite number of constraints in an instance.It is phrased in the language of sets with atoms, which provides finite means ofrepresentation – an instance is founded upon a fixed infinite relational structure,and defined by finitely many first order formulas. We prove decidability for thisso-called locally finite CSP, and establish tight complexity bounds in the specialcase when the set of possible values is finite.We further use the constraint satisfaction framework to analyse the computa-tional model of Turing machines with atoms (TMAs), whose alphabet, state space,and transition relation are orbit-finite sets with atoms (usually infinite but finitelypresentable). We give an effective characterisation of those alphabets for whichTMAs determinise, with applications to descriptive complexity.The second extension of the CSP that we consider is known as the valued con-straint satisfaction problem (VCSP). It provides a common framework for manydiscrete optimisation problems. We use algebraic tools to establish a necessarycondition for tractability of VCSPs parametrised by sets of allowed types of con-straints. We conjecture that our condition is also sufficient, and verify whether theconjecture agrees with all previously known results.