The Fine Grained Complexity Of Constraint Satisfaction Problems
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Applications of Partial Polymorphisms in (Fine-Grained) Complexity of Constraint Satisfaction Problems
Author: Biman Roy
language: en
Publisher: Linköping University Electronic Press
Release Date: 2020-03-23
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 ? (CSP(?)) is the following problem: given a set of variables restricted by a set of constraints based on the relations ?, is there an assignment to thevariables that satisfies all constraints? We refer to the set ? as aconstraint language. The inverse CSPproblem over ? (Inv-CSP(?)) asks the opposite: given a relation R, does there exist a CSP(?) instance with R as its set of models? When ? is a Boolean language, then we use the term SAT(?) instead of CSP(?) and Inv-SAT(?) instead of Inv-CSP(?). 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(?) problems. An NP-complete CSP(?) problem is said to be easier than an NP-complete CSP(?) problem if the worst-case time complexity of CSP(?) 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(?) is easier than CSP(?) when the strong partial clone corresponding to ? contains the strong partial clone of ?. An NP-complete CSP(?) 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(?) problems. An ultraconservative CSP(?) is a special class of CSP problems where the constraint language containsall unary relations. We additionally show that no NP-complete CSP(?) 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 ?, Inv-SAT(?) 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(?) is coNP-complete if and only if the clone corresponding to ? 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(?) that are parameterized bya constraint language ?(such as Inv-SAT), there are strong connections between the complexity of X(?) 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.
The Fine-grained Complexity of Constraint Satisfaction Problems
"Constraint satisfaction problems (CSPs) provide a unified framework for studying a wide variety of computational problems naturally arising in combinatorics, artificial intelligence and database theory. To any finite domain D and any constraint language [Gamma] (a finite set of relations over D), we associate the constraint satisfaction problem CSP([Gamma]): an instance of CSP([Gamma]) consists of a list of variables x1,x2,...,xn and a list of constraints of the form "(x7,x2,...,x5) [symbol] R" for some relation R in [Gamma]. The goal is to determine whether the variables can be assigned values in D such that all constraints are simultaneously satisfied. The computational complexity of CSP([Gamma]) is entirely determined by the structure of the constraint language [Gamma] and, thus, one wishes to identify classes of [Gamma] such that CSP([Gamma]) belongs to a particular complexity class. In recent years, combined logical and algebraic approaches to understand the complexity of CSPs within the complexity class P have been especially fruitful. In particular, precise algebraic conditions on [Gamma] have been conjectured to be sufficient and necessary for the membership of CSP([Gamma]) in the complexity classes L and NL (under standard complexity theoretic assumptions, e.g. L different from NL). These algebraic conditions are known to be necessary, and from the algorithmic side, a promising body of evidence is fast-growing. The main tools to establish membership of CSPs in L and NL are the logic programming fragments symmetric and linear Datalog, respectively. This thesis is centered around the above algebraic conjecture for CSPs in L, and most of the technical work is devoted to establishing the membership of several large classes of CSPs in L. Among other results, we characterize all graphs for which the list homomorphism problem is in L, a well-studied and natural class of CSPs. We also extend this result to obtain a complete characterization of the complexity of the list homomorphism for graphs. We develop new tool (dualities for symmetric Datalog) to show membership of CSPs in L, prove an L-NL dichotomy for the list homomorphism problem for oriented paths, provide results about the structure and polymorphisms of Maltsev digraphs, and also contribute to the conjecture of Dalmau that every CSP in NL is in fact in linear Datalog." --
Complexity of Infinite-Domain Constraint Satisfaction
Author: Manuel Bodirsky
language: en
Publisher: Cambridge University Press
Release Date: 2021-06-10
Constraint Satisfaction Problems (CSPs) are natural computational problems that appear in many areas of theoretical computer science. Exploring which CSPs are solvable in polynomial time and which are NP-hard reveals a surprising link with central questions in universal algebra. This monograph presents a self-contained introduction to the universal-algebraic approach to complexity classification, treating both finite and infinite-domain CSPs. It includes the required background from logic and combinatorics, particularly model theory and Ramsey theory, and explains the recently discovered link between Ramsey theory and topological dynamics and its implications for CSPs. The book will be of interest to graduate students and researchers in theoretical computer science and to mathematicians in logic, combinatorics, and dynamics who wish to learn about the applications of their work in complexity theory.