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

Abstract: In this thesis we study the worst-case complexity ofconstraint 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 satisfactionproblem 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 thevariables that satisfies all constraints? We refer to the set Γ as aconstraint language. The inverse CSPproblem over Γ (Inv-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 Inv-SAT(Γ) instead of Inv-CSP(Γ).Fine-grained complexity is an approach in which we zoom inside a complexity class and classify theproblems 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 thanthe worst-case time complexity of CSP(∆). We first analyze the NP-complete SAT problems that are easier than monotone 1-in-3-SAT (which can be represented by SAT(R) for a certain relation R), and find out that there exists a continuum of such problems. For this, we use the connection between constraint languages and strong partial clones and exploit the fact that CSP(Γ) is easier than CSP(∆) when the strong partial clone corresponding to  Γ contains the strong partial clone of ∆. An NP-complete CSP(Γ) problem is said to be the easiest with respect to a variable domain D if it is easier than any other NP-complete CSP(∆) problem of that domain. We show that for every finite domain there exists an easiest NP-complete problem for the ultraconservative CSP(Γ) problems. An ultraconservative CSP(Γ) is a special class of CSP problems where the constraint language containsall unary relations. We additionally show that no NP-complete CSP(Γ) problem can be solved insub-exponential time (i.e. in2^o(n) time where n is the number of variables) given that theexponentialtime hypothesisis true.Moving to classical complexity, we show that for any Boolean constraint language Γ, Inv-SAT(Γ) is either in P or it is coNP-complete. This is a generalization of an earlier dichotomy result, which was only known to be true for ultraconservative constraint languages. We show that Inv-SAT(Γ) is coNP-complete if and only if the clone corresponding to Γ contains essentially unary functions only. For arbitrary finite domains our results are not conclusive, but we manage to prove that theinversek-coloring problem is coNP-complete for each k>2. We exploit weak bases to prove many of theseresults. A weak base of a clone C is a constraint language that corresponds to the largest strong partia clone that contains C. It is known that for many decision problems X(Γ) that are parameterized bya constraint language Γ(such as Inv-SAT), there are strong connections between the complexity of X(Γ) and weak bases. This fact can be exploited to achieve general complexity results. The Boolean domain is well-suited for this approach since we have a fairly good understanding of Boolean weak bases. In the final result of this thesis, we investigate the relationships between the weak bases in the Boolean domain based on their strong partial clones and completely classify them according to the setinclusion. To avoid a tedious case analysis, we introduce a technique that allows us to discard a largenumber of cases from further investigation.