3,055 research outputs found
An Algorithmic Metatheorem for Directed Treewidth
The notion of directed treewidth was introduced by Johnson, Robertson,
Seymour and Thomas [Journal of Combinatorial Theory, Series B, Vol 82, 2001] as
a first step towards an algorithmic metatheory for digraphs. They showed that
some NP-complete properties such as Hamiltonicity can be decided in polynomial
time on digraphs of constant directed treewidth. Nevertheless, despite more
than one decade of intensive research, the list of hard combinatorial problems
that are known to be solvable in polynomial time when restricted to digraphs of
constant directed treewidth has remained scarce. In this work we enrich this
list by providing for the first time an algorithmic metatheorem connecting the
monadic second order logic of graphs to directed treewidth. We show that most
of the known positive algorithmic results for digraphs of constant directed
treewidth can be reformulated in terms of our metatheorem. Additionally, we
show how to use our metatheorem to provide polynomial time algorithms for two
classes of combinatorial problems that have not yet been studied in the context
of directed width measures. More precisely, for each fixed , we show how to count in polynomial time on digraphs of directed
treewidth , the number of minimum spanning strong subgraphs that are the
union of directed paths, and the number of maximal subgraphs that are the
union of directed paths and satisfy a given minor closed property. To prove
our metatheorem we devise two technical tools which we believe to be of
independent interest. First, we introduce the notion of tree-zig-zag number of
a digraph, a new directed width measure that is at most a constant times
directed treewidth. Second, we introduce the notion of -saturated tree slice
language, a new formalism for the specification and manipulation of infinite
sets of digraphs.Comment: 41 pages, 6 figures, Accepted to Discrete Applied Mathematic
On Brambles, Grid-Like Minors, and Parameterized Intractability of Monadic Second-Order Logic
Brambles were introduced as the dual notion to treewidth, one of the most
central concepts of the graph minor theory of Robertson and Seymour. Recently,
Grohe and Marx showed that there are graphs G, in which every bramble of order
larger than the square root of the treewidth is of exponential size in |G|. On
the positive side, they show the existence of polynomial-sized brambles of the
order of the square root of the treewidth, up to log factors. We provide the
first polynomial time algorithm to construct a bramble in general graphs and
achieve this bound, up to log-factors. We use this algorithm to construct
grid-like minors, a replacement structure for grid-minors recently introduced
by Reed and Wood, in polynomial time. Using the grid-like minors, we introduce
the notion of a perfect bramble and an algorithm to find one in polynomial
time. Perfect brambles are brambles with a particularly simple structure and
they also provide us with a subgraph that has bounded degree and still large
treewidth; we use them to obtain a meta-theorem on deciding certain
parameterized subgraph-closed problems on general graphs in time singly
exponential in the parameter.
The second part of our work deals with providing a lower bound to Courcelle's
famous theorem, stating that every graph property that can be expressed by a
sentence in monadic second-order logic (MSO), can be decided by a linear time
algorithm on classes of graphs of bounded treewidth. Using our results from the
first part of our work we establish a strong lower bound for tractability of
MSO on classes of colored graphs
Large induced subgraphs via triangulations and CMSO
We obtain an algorithmic meta-theorem for the following optimization problem.
Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an
integer. For a given graph G, the task is to maximize |X| subject to the
following: there is a set of vertices F of G, containing X, such that the
subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X)
models \phi.
Some special cases of this optimization problem are the following generic
examples. Each of these cases contains various problems as a special subcase:
1) "Maximum induced subgraph with at most l copies of cycles of length 0
modulo m", where for fixed nonnegative integers m and l, the task is to find a
maximum induced subgraph of a given graph with at most l vertex-disjoint cycles
of length 0 modulo m.
2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\
containing a planar graph, the task is to find a maximum induced subgraph of a
given graph containing no graph from \Gamma\ as a minor.
3) "Independent \Pi-packing", where for a fixed finite set of connected
graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G
with the maximum number of connected components, such that each connected
component of G[F] is isomorphic to some graph from \Pi.
We give an algorithm solving the optimization problem on an n-vertex graph G
in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential
maximal cliques in G and f is a function depending of t and \phi\ only. We also
show how a similar running time can be obtained for the weighted version of the
problem. Pipelined with known bounds on the number of potential maximal
cliques, we deduce that our optimization problem can be solved in time
O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with
polynomial number of minimal separators
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
Parameterized Algorithms for Modular-Width
It is known that a number of natural graph problems which are FPT
parameterized by treewidth become W-hard when parameterized by clique-width. It
is therefore desirable to find a different structural graph parameter which is
as general as possible, covers dense graphs but does not incur such a heavy
algorithmic penalty.
The main contribution of this paper is to consider a parameter called
modular-width, defined using the well-known notion of modular decompositions.
Using a combination of ILPs and dynamic programming we manage to design FPT
algorithms for Coloring and Partitioning into paths (and hence Hamiltonian path
and Hamiltonian cycle), which are W-hard for both clique-width and its recently
introduced restriction, shrub-depth. We thus argue that modular-width occupies
a sweet spot as a graph parameter, generalizing several simpler notions on
dense graphs but still evading the "price of generality" paid by clique-width.Comment: to appear in IPEC 2013. arXiv admin note: text overlap with
arXiv:1304.5479 by other author
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