817 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
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
Approximating acyclicity parameters of sparse hypergraphs
The notions of hypertree width and generalized hypertree width were
introduced by Gottlob, Leone, and Scarcello in order to extend the concept of
hypergraph acyclicity. These notions were further generalized by Grohe and
Marx, who introduced the fractional hypertree width of a hypergraph. All these
width parameters on hypergraphs are useful for extending tractability of many
problems in database theory and artificial intelligence. In this paper, we
study the approximability of (generalized, fractional) hyper treewidth of
sparse hypergraphs where the criterion of sparsity reflects the sparsity of
their incidence graphs. Our first step is to prove that the (generalized,
fractional) hypertree width of a hypergraph H is constant-factor sandwiched by
the treewidth of its incidence graph, when the incidence graph belongs to some
apex-minor-free graph class. This determines the combinatorial borderline above
which the notion of (generalized, fractional) hypertree width becomes
essentially more general than treewidth, justifying that way its functionality
as a hypergraph acyclicity measure. While for more general sparse families of
hypergraphs treewidth of incidence graphs and all hypertree width parameters
may differ arbitrarily, there are sparse families where a constant factor
approximation algorithm is possible. In particular, we give a constant factor
approximation polynomial time algorithm for (generalized, fractional) hypertree
width on hypergraphs whose incidence graphs belong to some H-minor-free graph
class
Deciding first-order properties of nowhere dense graphs
Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez,
form a large variety of classes of "sparse graphs" including the class of
planar graphs, actually all classes with excluded minors, and also bounded
degree graphs and graph classes of bounded expansion.
We show that deciding properties of graphs definable in first-order logic is
fixed-parameter tractable on nowhere dense graph classes. At least for graph
classes closed under taking subgraphs, this result is optimal: it was known
before that for all classes C of graphs closed under taking subgraphs, if
deciding first-order properties of graphs in C is fixed-parameter tractable,
then C must be nowhere dense (under a reasonable complexity theoretic
assumption).
As a by-product, we give an algorithmic construction of sparse neighbourhood
covers for nowhere dense graphs. This extends and improves previous
constructions of neighbourhood covers for graph classes with excluded minors.
At the same time, our construction is considerably simpler than those. Our
proofs are based on a new game-theoretic characterisation of nowhere dense
graphs that allows for a recursive version of locality-based algorithms on
these classes. On the logical side, we prove a "rank-preserving" version of
Gaifman's locality theorem.Comment: 30 page
Distributed Dominating Set Approximations beyond Planar Graphs
The Minimum Dominating Set (MDS) problem is one of the most fundamental and
challenging problems in distributed computing. While it is well-known that
minimum dominating sets cannot be approximated locally on general graphs, over
the last years, there has been much progress on computing local approximations
on sparse graphs, and in particular planar graphs.
In this paper we study distributed and deterministic MDS approximation
algorithms for graph classes beyond planar graphs. In particular, we show that
existing approximation bounds for planar graphs can be lifted to bounded genus
graphs, and present (1) a local constant-time, constant-factor MDS
approximation algorithm and (2) a local -time
approximation scheme. Our main technical contribution is a new analysis of a
slightly modified variant of an existing algorithm by Lenzen et al.
Interestingly, unlike existing proofs for planar graphs, our analysis does not
rely on direct topological arguments.Comment: arXiv admin note: substantial text overlap with arXiv:1602.0299
Minors and dimension
It has been known for 30 years that posets with bounded height and with cover
graphs of bounded maximum degree have bounded dimension. Recently, Streib and
Trotter proved that dimension is bounded for posets with bounded height and
planar cover graphs, and Joret et al. proved that dimension is bounded for
posets with bounded height and with cover graphs of bounded tree-width. In this
paper, it is proved that posets of bounded height whose cover graphs exclude a
fixed topological minor have bounded dimension. This generalizes all the
aforementioned results and verifies a conjecture of Joret et al. The proof
relies on the Robertson-Seymour and Grohe-Marx graph structure theorems.Comment: Updated reference
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