190 research outputs found
The fine classification of conjunctive queries and parameterized logarithmic space
We perform a fundamental investigation of the complexity of conjunctive query evaluation from the perspective of parameterized complexity. We classify sets of boolean conjunctive queries according to the complexity of this problem. Previous work showed that a set of conjunctive queries is fixed-parameter tractable precisely when the set is equivalent to a set of queries having bounded treewidth. We present a fine classification of query sets up to parameterized logarithmic space reduction. We show that, in the bounded treewidth regime, there are three complexity degrees and that the properties that determine the degree of a query set are bounded pathwidth and bounded tree depth. We also engage in a study of the two higher degrees via logarithmic space machine characterizations and complete problems. Our work yields a significantly richer perspective on the complexity of conjunctive queries and, at the same time, suggests new avenues of research in parameterized complexity
The parameterized space complexity of model-checking bounded variable first-order logic
The parameterized model-checking problem for a class of first-order sentences
(queries) asks to decide whether a given sentence from the class holds true in
a given relational structure (database); the parameter is the length of the
sentence. We study the parameterized space complexity of the model-checking
problem for queries with a bounded number of variables. For each bound on the
quantifier alternation rank the problem becomes complete for the corresponding
level of what we call the tree hierarchy, a hierarchy of parameterized
complexity classes defined via space bounded alternating machines between
parameterized logarithmic space and fixed-parameter tractable time. We observe
that a parameterized logarithmic space model-checker for existential bounded
variable queries would allow to improve Savitch's classical simulation of
nondeterministic logarithmic space in deterministic space .
Further, we define a highly space efficient model-checker for queries with a
bounded number of variables and bounded quantifier alternation rank. We study
its optimality under the assumption that Savitch's Theorem is optimal
One hierarchy spawns another: graph deconstructions and the complexity classification of conjunctive queries
We study the problem of conjunctive query evaluation relative to a class of queries. This problem is formulated here as the relational homomorphism problem relative to a class of structures A, in which each instance must be a pair of structures such that the first structure is an element of A. We present a comprehensive complexity classification of these problems, which strongly links graph-theoretic properties of A to the complexity of the corresponding homomorphism problem. In particular, we define a binary relation on graph classes, which is a preorder, and completely describe the resulting hierarchy given by this relation. This relation is defined in terms of a notion that we call graph deconstruction and that is a variant of the well-known notion of tree decomposition. We then use this hierarchy of graph classes to infer a complexity hierarchy of homomorphism problems that is comprehensive up to a computationally very weak notion of reduction, namely, a parameterized version of quantifier-free, first-order reduction. In doing so, we obtain a significantly refined complexity classification of homomorphism problems as well as a unifying, modular, and conceptually clean treatment of existing complexity classifications. We then present and develop the theory of Ehrenfeucht-Fraïssé-style pebble games, which solve the homomorphism problems where the cores of the structures in A have bounded tree depth. This condition characterizes those classical homomorphism problems decidable in logarithmic space, assuming a hypothesis from parameterized space complexity. Finally, we use our framework to classify the complexity of model checking existential sentences having bounded quantifier rank
Parameterised Counting in Logspace
Logarithmic space-bounded complexity classes such as L and NL play a central role in space-bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space-bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space-bounded models developed by Elberfeld, Stockhusen and Tantau. They defined the operators paraW and paraβ for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators paraW and paraβ by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, paraW[1] and paraβtail. Then, we consider counting versions of all four operators and apply them to the class L. We obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0, 1)-matrices is # paraβtailL-hard and can be written as the difference of two functions in # paraβtailL. These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of # paraβtailL under parameterised logspace parsimonious reductions coincides with # paraβL. In other words, in the setting of read-once access to nondeterministic bits, tail-nondeterminism coincides with unbounded nondeterminism modulo parameterised reductions. Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Finally, we want to emphasise the significance of this topic by providing a promising outlook highlighting several open problems and directions for further research
Parameterised Counting in Logspace
Logarithmic space bounded complexity classes such as L and NL play a central role in space bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015).
In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space bounded models developed by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). They defined the operators para_W and para_? for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators para_W and para_? by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, para_{W[1]} and para_{?tail}. Then, we consider counting versions of all four operators applied to logspace and obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0,1)-matrices is #para_{?tail} L-hard and can be written as the difference of two functions in #para_{?tail} L. These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of #para_{?tail} L under parameterised logspace parsimonious reductions coincides with #para_? L, that is, modulo parameterised reductions, tail-nondeterminism with read-once access is the same as read-once nondeterminism.
Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions.
Also, we show that the counting classes defined can naturally be characterised by parameterised variants of classes based on branching programs in analogy to the classical counting classes.
Finally, we underline the significance of this topic by providing a promising outlook showing several open problems and options for further directions of research
Semantic Width and the Fixed-Parameter Tractability of Constraint Satisfaction Problems
Constraint satisfaction problems (CSPs) are an important formal framework for
the uniform treatment of various prominent AI tasks, e.g., coloring or
scheduling problems. Solving CSPs is, in general, known to be NP-complete and
fixed-parameter intractable when parameterized by their constraint scopes. We
give a characterization of those classes of CSPs for which the problem becomes
fixed-parameter tractable.
Our characterization significantly increases the utility of the CSP framework
by making it possible to decide the fixed-parameter tractability of problems
via their CSP formulations.
We further extend our characterization to the evaluation of unions of
conjunctive queries, a fundamental problem in databases. Furthermore, we
provide some new insight on the frontier of PTIME solvability of CSPs.
In particular, we observe that bounded fractional hypertree width is more
general than bounded hypertree width only for classes that exhibit a certain
type of exponential growth.
The presented work resolves a long-standing open problem and yields powerful
new tools for complexity research in AI and database theory.Comment: Full and extended version of the IJCAI2020 paper with the same titl
The tractability frontier of graph-like first-order query sets
We study first-order model checking, by which we refer to the problem of deciding whether or not a given first-order sentence is satisfied by a given finite structure. In particular, we aim to understand on which sets of sentences this problem is tractable, in the sense of parameterized complexity theory. To this end, we define the notion of a graph-like sentence set, which definition is inspired by previous work on first-order model checking wherein the permitted connectives and quantifiers were restricted. Our main theorem is the complete tractability classification of such graphlike sentence sets, which is (to our knowledge) the first complexity classification theorem concerning a class of sentences that has no restriction on the connectives and quantifiers. To present and prove our classification, we introduce and develop a novel complexity-theoretic framework which is built on parameterized complexity and includes new notions of reduction
Fast Parallel Fixed-Parameter Algorithms via Color Coding
Fixed-parameter algorithms have been successfully applied to solve numerous
difficult problems within acceptable time bounds on large inputs. However, most
fixed-parameter algorithms are inherently \emph{sequential} and, thus, make no
use of the parallel hardware present in modern computers. We show that parallel
fixed-parameter algorithms do not only exist for numerous parameterized
problems from the literature -- including vertex cover, packing problems,
cluster editing, cutting vertices, finding embeddings, or finding matchings --
but that there are parallel algorithms working in \emph{constant} time or at
least in time \emph{depending only on the parameter} (and not on the size of
the input) for these problems. Phrased in terms of complexity classes, we place
numerous natural parameterized problems in parameterized versions of AC. On
a more technical level, we show how the \emph{color coding} method can be
implemented in constant time and apply it to embedding problems for graphs of
bounded tree-width or tree-depth and to model checking first-order formulas in
graphs of bounded degree
Fine-Grained Complexity of Regular Path Queries
A regular path query (RPQ) is a regular expression q that returns all node pairs (u, v) from a graph database that are connected by an arbitrary path labelled with a word from L(q). The obvious algorithmic approach to RPQ evaluation (called PG-approach), i. e., constructing the product graph between an NFA for q and the graph database, is appealing due to its simplicity and also leads to efficient algorithms. However, it is unclear whether the PG-approach is optimal. We address this question by thoroughly investigating which upper complexity bounds can be achieved by the PG-approach, and we complement these with conditional lower bounds (in the sense of the fine-grained complexity framework). A special focus is put on enumeration and delay bounds, as well as the data complexity perspective. A main insight is that we can achieve optimal (or near optimal) algorithms with the PG-approach, but the delay for enumeration is rather high (linear in the database). We explore three successful approaches towards enumeration with sub-linear delay: super-linear preprocessing, approximations of the solution sets, and restricted classes of RPQs
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