418 research outputs found
Local tree-width, excluded minors, and approximation algorithms
The local tree-width of a graph G=(V,E) is the function ltw^G: N -> N that
associates with every natural number r the maximal tree-width of an
r-neighborhood in G. Our main graph theoretic result is a decomposition theorem
for graphs with excluded minors that essentially says that such graphs can be
decomposed into trees of graphs of bounded local tree-width.
As an application of this theorem, we show that a number of combinatorial
optimization problems, such as Minimum Vertex Cover, Minimum Dominating Set,
and Maximum Independent Set have a polynomial time approximation scheme when
restricted to a class of graphs with an excluded minor
Fixed-parameter tractability, definability, and model checking
In this article, we study parameterized complexity theory from the
perspective of logic, or more specifically, descriptive complexity theory.
We propose to consider parameterized model-checking problems for various
fragments of first-order logic as generic parameterized problems and show how
this approach can be useful in studying both fixed-parameter tractability and
intractability. For example, we establish the equivalence between the
model-checking for existential first-order logic, the homomorphism problem for
relational structures, and the substructure isomorphism problem. Our main
tractability result shows that model-checking for first-order formulas is
fixed-parameter tractable when restricted to a class of input structures with
an excluded minor. On the intractability side, for every t >= 0 we prove an
equivalence between model-checking for first-order formulas with t quantifier
alternations and the parameterized halting problem for alternating Turing
machines with t alternations. We discuss the close connection between this
alternation hierarchy and Downey and Fellows' W-hierarchy.
On a more abstract level, we consider two forms of definability, called Fagin
definability and slicewise definability, that are appropriate for describing
parameterized problems. We give a characterization of the class FPT of all
fixed-parameter tractable problems in terms of slicewise definability in finite
variable least fixed-point logic, which is reminiscent of the Immerman-Vardi
Theorem characterizing the class PTIME in terms of definability in least
fixed-point logic.Comment: To appear in SIAM Journal on Computin
Computing with Tangles
Tangles of graphs have been introduced by Robertson and Seymour in the
context of their graph minor theory. Tangles may be viewed as describing
"k-connected components" of a graph (though in a twisted way). They play an
important role in graph minor theory. An interesting aspect of tangles is that
they cannot only be defined for graphs, but more generally for arbitrary
connectivity functions (that is, integer-valued submodular and symmetric set
functions).
However, tangles are difficult to deal with algorithmically. To start with,
it is unclear how to represent them, because they are families of separations
and as such may be exponentially large. Our first contribution is a data
structure for representing and accessing all tangles of a graph up to some
fixed order.
Using this data structure, we can prove an algorithmic version of a very
general structure theorem due to Carmesin, Diestel, Harman and Hundertmark (for
graphs) and Hundertmark (for arbitrary connectivity functions) that yields a
canonical tree decomposition whose parts correspond to the maximal tangles.
(This may be viewed as a generalisation of the decomposition of a graph into
its 3-connected components.
Canonisation and Definability for Graphs of Bounded Rank Width
We prove that the combinatorial Weisfeiler-Leman algorithm of dimension
is a complete isomorphism test for the class of all graphs of rank
width at most . Rank width is a graph invariant that, similarly to tree
width, measures the width of a certain style of hierarchical decomposition of
graphs; it is equivalent to clique width. It was known that isomorphism of
graphs of rank width is decidable in polynomial time (Grohe and Schweitzer,
FOCS 2015), but the best previously known algorithm has a running time
for a non-elementary function . Our result yields an isomorphism
test for graphs of rank width running in time . Another
consequence of our result is the first polynomial time canonisation algorithm
for graphs of bounded rank width. Our second main result is that fixed-point
logic with counting captures polynomial time on all graph classes of bounded
rank width.Comment: 32 page
Infinite Probabilistic Databases
Probabilistic databases (PDBs) are used to model uncertainty in data in a quantitative way. In the standard formal framework, PDBs are finite probability spaces over relational database instances. It has been argued convincingly that this is not compatible with an open-world semantics (Ceylan et al., KR 2016) and with application scenarios that are modeled by continuous probability distributions (Dalvi et al., CACM 2009).
We recently introduced a model of PDBs as infinite probability spaces that addresses these issues (Grohe and Lindner, PODS 2019). While that work was mainly concerned with countably infinite probability spaces, our focus here is on uncountable spaces. Such an extension is necessary to model typical continuous probability distributions that appear in many applications. However, an extension beyond countable probability spaces raises nontrivial foundational issues concerned with the measurability of events and queries and ultimately with the question whether queries have a well-defined semantics.
It turns out that so-called finite point processes are the appropriate model from probability theory for dealing with probabilistic databases. This model allows us to construct suitable (uncountable) probability spaces of database instances in a systematic way. Our main technical results are measurability statements for relational algebra queries as well as aggregate queries and Datalog queries
The succinctness of first-order logic on linear orders
Succinctness is a natural measure for comparing the strength of different logics. Intuitively, a logic L_1 is more succinct than another logic L_2 if all properties that can be expressed in L_2 can be expressed in L_1 by formulas of (approximately) the same size, but some properties can be expressed in L_1 by (significantly) smaller formulas.
We study the succinctness of logics on linear orders. Our first theorem is concerned with the finite variable fragments of first-order logic. We prove that:
(i) Up to a polynomial factor, the 2- and the 3-variable fragments of first-order logic on linear orders have the same succinctness. (ii) The 4-variable fragment is exponentially more succinct than the 3-variable fragment. Our second main result compares the succinctness of first-order logic on linear orders with that of monadic second-order logic. We prove that the fragment of monadic second-order logic that has the same expressiveness as first-order logic on linear orders is non-elementarily more succinct than first-order logic
The Complexity of Datalog on Linear Orders
We study the program complexity of datalog on both finite and infinite linear
orders. Our main result states that on all linear orders with at least two
elements, the nonemptiness problem for datalog is EXPTIME-complete. While
containment of the nonemptiness problem in EXPTIME is known for finite linear
orders and actually for arbitrary finite structures, it is not obvious for
infinite linear orders. It sharply contrasts the situation on other infinite
structures; for example, the datalog nonemptiness problem on an infinite
successor structure is undecidable. We extend our upper bound results to
infinite linear orders with constants.
As an application, we show that the datalog nonemptiness problem on Allen's
interval algebra is EXPTIME-complete.Comment: 21 page
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