794 research outputs found
Evaluating Datalog via Tree Automata and Cycluits
We investigate parameterizations of both database instances and queries that
make query evaluation fixed-parameter tractable in combined complexity. We show
that clique-frontier-guarded Datalog with stratified negation (CFG-Datalog)
enjoys bilinear-time evaluation on structures of bounded treewidth for programs
of bounded rule size. Such programs capture in particular conjunctive queries
with simplicial decompositions of bounded width, guarded negation fragment
queries of bounded CQ-rank, or two-way regular path queries. Our result is
shown by translating to alternating two-way automata, whose semantics is
defined via cyclic provenance circuits (cycluits) that can be tractably
evaluated.Comment: 56 pages, 63 references. Journal version of "Combined Tractability of
Query Evaluation via Tree Automata and Cycluits (Extended Version)" at
arXiv:1612.04203. Up to the stylesheet, page/environment numbering, and
possible minor publisher-induced changes, this is the exact content of the
journal paper that will appear in Theory of Computing Systems. Update wrt
version 1: latest reviewer feedbac
Certainty Closure: Reliable Constraint Reasoning with Incomplete or Erroneous Data
Constraint Programming (CP) has proved an effective paradigm to model and
solve difficult combinatorial satisfaction and optimisation problems from
disparate domains. Many such problems arising from the commercial world are
permeated by data uncertainty. Existing CP approaches that accommodate
uncertainty are less suited to uncertainty arising due to incomplete and
erroneous data, because they do not build reliable models and solutions
guaranteed to address the user's genuine problem as she perceives it. Other
fields such as reliable computation offer combinations of models and associated
methods to handle these types of uncertain data, but lack an expressive
framework characterising the resolution methodology independently of the model.
We present a unifying framework that extends the CP formalism in both model
and solutions, to tackle ill-defined combinatorial problems with incomplete or
erroneous data. The certainty closure framework brings together modelling and
solving methodologies from different fields into the CP paradigm to provide
reliable and efficient approches for uncertain constraint problems. We
demonstrate the applicability of the framework on a case study in network
diagnosis. We define resolution forms that give generic templates, and their
associated operational semantics, to derive practical solution methods for
reliable solutions.Comment: Revised versio
Tree-width for first order formulae
We introduce tree-width for first order formulae \phi, fotw(\phi). We show
that computing fotw is fixed-parameter tractable with parameter fotw. Moreover,
we show that on classes of formulae of bounded fotw, model checking is fixed
parameter tractable, with parameter the length of the formula. This is done by
translating a formula \phi\ with fotw(\phi)<k into a formula of the k-variable
fragment L^k of first order logic. For fixed k, the question whether a given
first order formula is equivalent to an L^k formula is undecidable. In
contrast, the classes of first order formulae with bounded fotw are fragments
of first order logic for which the equivalence is decidable.
Our notion of tree-width generalises tree-width of conjunctive queries to
arbitrary formulae of first order logic by taking into account the quantifier
interaction in a formula. Moreover, it is more powerful than the notion of
elimination-width of quantified constraint formulae, defined by Chen and Dalmau
(CSL 2005): for quantified constraint formulae, both bounded elimination-width
and bounded fotw allow for model checking in polynomial time. We prove that
fotw of a quantified constraint formula \phi\ is bounded by the
elimination-width of \phi, and we exhibit a class of quantified constraint
formulae with bounded fotw, that has unbounded elimination-width. A similar
comparison holds for strict tree-width of non-recursive stratified datalog as
defined by Flum, Frick, and Grohe (JACM 49, 2002).
Finally, we show that fotw has a characterization in terms of a cops and
robbers game without monotonicity cost
Optimal subsampling designs
Subsampling is commonly used to overcome computational and economical bottlenecks in the analysis of finite populations and massive datasets. Existing methods are often limited in scope and use optimality criteria (e.g., A-optimality) with well-known deficiencies, such as lack of invariance to the measurement-scale of the data and parameterisation of the model. A unified theory of optimal subsampling design is still lacking. We present a theory of optimal design for general data subsampling problems, including finite population inference, parametric density estimation, and regression modelling. Our theory encompasses and generalises most existing methods in the field of optimal subdata selection based on unequal probability sampling and inverse probability weighting. We derive optimality conditions for a general class of optimality criteria, and present corresponding algorithms for finding optimal sampling schemes under Poisson and multinomial sampling designs. We present a novel class of transformation- and parameterisation-invariant linear optimality criteria which enjoy the best of two worlds: the computational tractability of A-optimality and invariance properties similar to D-optimality. The methodology is illustrated on an application in the traffic safety domain. In our experiments, the proposed invariant linear optimality criteria achieve 92-99% D-efficiency with 90-95% lower computational demand. In contrast, the A-optimality criterion has only 46% and 60% D-efficiency on two of the examples
Counting Problems on Quantum Graphs: Parameterized and Exact Complexity Classifications
Quantum graphs, as defined by LovĂĄsz in the late 60s, are formal linear combinations of simple graphs with finite support. They allow for the complexity analysis of the problem of computing finite linear combinations of homomorphism counts, the latter of which constitute the foundation of the structural hardness theory for parameterized counting problems: The framework of parameterized counting complexity was introduced by Flum and Grohe, and McCartin in 2002 and forms a hybrid between the classical field of computational counting as founded by Valiant in the late 70s and the paradigm of parameterized complexity theory due to Downey and Fellows which originated in the early 90s.
The problem of computing homomorphism numbers of quantum graphs subsumes general motif counting problems and the complexity theoretic implications have only turned out recently in a breakthrough regarding the parameterized subgraph counting problem by Curticapean, Dell and Marx in 2017.
We study the problems of counting partially injective and edge-injective homomorphisms, counting induced subgraphs, as well as counting answers to existential first-order queries. We establish novel combinatorial, algebraic and even topological properties of quantum graphs that allow us to provide exhaustive parameterized and exact complexity classifications, including necessary, sufficient and mostly explicit tractability criteria, for all of the previous problems.Diese Arbeit befasst sich mit der Komplexit atsanalyse von mathematischen Problemen die als Linearkombinationen von Graphhomomorphismenzahlen darstellbar sind. Dazu wird sich sogenannter Quantengraphen bedient, bei denen es sich um formale Linearkombinationen von Graphen handelt und welche von Lov asz Ende der 60er eingef uhrt wurden. Die Bestimmung der Komplexit at solcher Probleme erfolgt unter dem von Flum, Grohe und McCartin im Jahre 2002 vorgestellten Paradigma der parametrisierten Z ahlkomplexit atstheorie, die als Hybrid der von Valiant Ende der 70er begr undeten klassischen Z ahlkomplexit atstheorie und der von Downey und Fellows Anfang der 90er eingef uhrten parametrisierten Analyse zu verstehen ist. Die Berechnung von Homomorphismenzahlen zwischen Quantengraphen und Graphen subsumiert im weitesten Sinne all jene Probleme, die das Z ahlen von kleinen Mustern in gro en Strukturen erfordern. Aufbauend auf dem daraus resultierenden Durchbruch von Curticapean, Dell und Marx, das Subgraphz ahlproblem betre end, behandelt diese Arbeit die Analyse der Probleme des Z ahlens von partiell- und kanteninjektiven Homomorphismen, induzierten Subgraphen, und Tre ern von relationalen Datenbankabfragen die sich als existentielle Formeln ausdr ucken lassen. Insbesondere werden dabei neue kombinatorische, algebraische und topologische Eigenschaften von Quantengraphen etabliert, die hinreichende, notwendige und meist explizite Kriterien f ur die Existenz e zienter Algorithmen liefern
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