123 research outputs found
Are there any good digraph width measures?
Several different measures for digraph width have appeared in the last few
years. However, none of them shares all the "nice" properties of treewidth:
First, being \emph{algorithmically useful} i.e. admitting polynomial-time
algorithms for all \MS1-definable problems on digraphs of bounded width. And,
second, having nice \emph{structural properties} i.e. being monotone under
taking subdigraphs and some form of arc contractions. As for the former,
(undirected) \MS1 seems to be the least common denominator of all reasonably
expressive logical languages on digraphs that can speak about the edge/arc
relation on the vertex set.The latter property is a necessary condition for a
width measure to be characterizable by some version of the cops-and-robber game
characterizing the ordinary treewidth. Our main result is that \emph{any
reasonable} algorithmically useful and structurally nice digraph measure cannot
be substantially different from the treewidth of the underlying undirected
graph. Moreover, we introduce \emph{directed topological minors} and argue that
they are the weakest useful notion of minors for digraphs
FPT is Characterized by Useful Obstruction Sets
Many graph problems were first shown to be fixed-parameter tractable using
the results of Robertson and Seymour on graph minors. We show that the
combination of finite, computable, obstruction sets and efficient order tests
is not just one way of obtaining strongly uniform FPT algorithms, but that all
of FPT may be captured in this way. Our new characterization of FPT has a
strong connection to the theory of kernelization, as we prove that problems
with polynomial kernels can be characterized by obstruction sets whose elements
have polynomial size. Consequently we investigate the interplay between the
sizes of problem kernels and the sizes of the elements of such obstruction
sets, obtaining several examples of how results in one area yield new insights
in the other. We show how exponential-size minor-minimal obstructions for
pathwidth k form the crucial ingredient in a novel OR-cross-composition for
k-Pathwidth, complementing the trivial AND-composition that is known for this
problem. In the other direction, we show that OR-cross-compositions into a
parameterized problem can be used to rule out the existence of efficiently
generated quasi-orders on its instances that characterize the NO-instances by
polynomial-size obstructions.Comment: Extended abstract with appendix, as accepted to WG 201
Courcelle's Theorem - A Game-Theoretic Approach
Courcelle's Theorem states that every problem definable in Monadic
Second-Order logic can be solved in linear time on structures of bounded
treewidth, for example, by constructing a tree automaton that recognizes or
rejects a tree decomposition of the structure. Existing, optimized software
like the MONA tool can be used to build the corresponding tree automata, which
for bounded treewidth are of constant size. Unfortunately, the constants
involved can become extremely large - every quantifier alternation requires a
power set construction for the automaton. Here, the required space can become a
problem in practical applications.
In this paper, we present a novel, direct approach based on model checking
games, which avoids the expensive power set construction. Experiments with an
implementation are promising, and we can solve problems on graphs where the
automata-theoretic approach fails in practice.Comment: submitte
On The Power of Tree Projections: Structural Tractability of Enumerating CSP Solutions
The problem of deciding whether CSP instances admit solutions has been deeply
studied in the literature, and several structural tractability results have
been derived so far. However, constraint satisfaction comes in practice as a
computation problem where the focus is either on finding one solution, or on
enumerating all solutions, possibly projected to some given set of output
variables. The paper investigates the structural tractability of the problem of
enumerating (possibly projected) solutions, where tractability means here
computable with polynomial delay (WPD), since in general exponentially many
solutions may be computed. A general framework based on the notion of tree
projection of hypergraphs is considered, which generalizes all known
decomposition methods. Tractability results have been obtained both for classes
of structures where output variables are part of their specification, and for
classes of structures where computability WPD must be ensured for any possible
set of output variables. These results are shown to be tight, by exhibiting
dichotomies for classes of structures having bounded arity and where the tree
decomposition method is considered
A note on uniquely 10-colorable graphs
We prove for k at most 10, that every graph of chromatic number k with a
unique k-coloring admits a clique minor of order k
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