Applications of Partial Polymorphisms in (Fine-Grained) Complexity of Constraint Satisfaction Problems

Abstract

In this thesis we study the worst-case complexity of constraint satisfaction problems and some of its variants. We use methods from universal algebra: in particular, algebras of total functions and partial functions that are respectively known as clones and strong partial clones. The constraint satisfaction problem parameterized by a set of relations Γ (CSP(Γ)) is the following problem: given a set of variables restricted by a set of constraints based on the relations Γ, is there an assignment to the variables that satisfies all constraints? We refer to the set Γ as a constraint language. The inverse CSP problem over Γ (I-CSP(Γ)) asks the opposite: given a relation R, does there exist a CSP(Γ) instance with R as its set of models? When Γ is a Boolean language, then we use the term SAT(Γ) instead of CSP(Γ) and I-SAT(Γ) instead of I-CSP(Γ). Fine-grained complexity is an approach in which we zoom inside a complexity class and classify the problems in it based on their worst-case time complexities. We start by investigating the fine-grained complexity of NP-complete CSP(Γ) problems. An NP-complete CSP(Γ) problem is said to be easier than an NP-complete CSP(∆) problem if the worst-case time complexity of CSP(Γ) is not higher than 5. Victor Lagerkvist and Biman Roy. The Inclusion Structure of Boolean Weak Bases.