1,900 research outputs found
A Natural Generalization of Bounded Tree-Width and Bounded Clique-Width
We investigate a new width parameter, the fusion-width of a graph. It is a
natural generalization of the tree-width, yet strong enough that not only
graphs of bounded tree-width, but also graphs of bounded clique-width,
trivially have bounded fusion-width. In particular, there is no exponential
growth between tree-width and fusion-width, as is the case between tree-width
and clique-width. The new parameter gives a good intuition about the
relationship between tree-width and clique-width.Comment: To appear in the proceedings of Latin 2014. Springer LNCS 839
Hypergraph Acyclicity and Propositional Model Counting
We show that the propositional model counting problem #SAT for CNF- formulas
with hypergraphs that allow a disjoint branches decomposition can be solved in
polynomial time. We show that this class of hypergraphs is incomparable to
hypergraphs of bounded incidence cliquewidth which were the biggest class of
hypergraphs for which #SAT was known to be solvable in polynomial time so far.
Furthermore, we present a polynomial time algorithm that computes a disjoint
branches decomposition of a given hypergraph if it exists and rejects
otherwise. Finally, we show that some slight extensions of the class of
hypergraphs with disjoint branches decompositions lead to intractable #SAT,
leaving open how to generalize the counting result of this paper
Grad and classes with bounded expansion I. decompositions
We introduce classes of graphs with bounded expansion as a generalization of
both proper minor closed classes and degree bounded classes. Such classes are
based on a new invariant, the greatest reduced average density (grad) of G with
rank r, grad r(G). For these classes we prove the existence of several
partition results such as the existence of low tree-width and low tree-depth
colorings. This generalizes and simplifies several earlier results (obtained
for minor closed classes)
Shrub-depth: Capturing Height of Dense Graphs
The recent increase of interest in the graph invariant called tree-depth and
in its applications in algorithms and logic on graphs led to a natural
question: is there an analogously useful "depth" notion also for dense graphs
(say; one which is stable under graph complementation)? To this end, in a 2012
conference paper, a new notion of shrub-depth has been introduced, such that it
is related to the established notion of clique-width in a similar way as
tree-depth is related to tree-width. Since then shrub-depth has been
successfully used in several research papers. Here we provide an in-depth
review of the definition and basic properties of shrub-depth, and we focus on
its logical aspects which turned out to be most useful. In particular, we use
shrub-depth to give a characterization of the lower levels of the
MSO1 transduction hierarchy of simple graphs
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
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