| Summary: | The Valued Constraint Satisfaction Problem (VCSP) is a framework which captures many natural decision and optimisation problems. An instance of the VCSP consists of a set of variables, which are to be assigned labels from a finite domain, and a collection of local constraints, each specified by a weighted relation mapping labellings of the variables in the constraint's scope to values. The objective is to minimise the total value from all constraints. The VCSP is commonly parameterised by a language, i.e. a set of weighted relations that are available for use in the constraints. Languages are classified according to the computational complexity of the VCSP as tractable, for which the problem can be solved in polynomial time, and intractable, for which the problem is NP-hard. The recently proved VCSP dichotomy theorem established a classification of all languages into these two categories. Additionally, various structural restrictions can be imposed to limit the set of admissible instances further, thus potentially changing the complexity of the VCSP. Our first contribution relates to the algebraic approach to the VCSP, which proved instrumental in recent advances in the field. We generalise the Galois connection between weighted relational clones and weighted clones so that it applies to infinite sets as well. Second, we study a structural restriction requiring that the incidence graph be planar. In this setting, we establish a complexity classification of conservative languages (i.e. languages containing all {0,1}-valued unary weighted relations) and a necessary tractability condition for Boolean languages (i.e. languages over a two-element domain). Third, we study the surjective variant of the VCSP, in which labellings are required to assign every domain element to at least one variable. We establish a complexity classification of Boolean languages, which encompasses a new tractable class of problems.
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