87 research outputs found
Finding branch-decompositions of matroids, hypergraphs, and more
Given subspaces of a finite-dimensional vector space over a fixed finite
field , we wish to find a "branch-decomposition" of these subspaces
of width at most , that is a subcubic tree with leaves mapped
bijectively to the subspaces such that for every edge of , the sum of
subspaces associated with leaves in one component of and the sum of
subspaces associated with leaves in the other component have the intersection
of dimension at most . This problem includes the problems of computing
branch-width of -represented matroids, rank-width of graphs,
branch-width of hypergraphs, and carving-width of graphs.
We present a fixed-parameter algorithm to construct such a
branch-decomposition of width at most , if it exists, for input subspaces of
a finite-dimensional vector space over . Our algorithm is analogous
to the algorithm of Bodlaender and Kloks (1996) on tree-width of graphs. To
extend their framework to branch-decompositions of vector spaces, we developed
highly generic tools for branch-decompositions on vector spaces. The only known
previous fixed-parameter algorithm for branch-width of -represented
matroids was due to Hlin\v{e}n\'y and Oum (2008) that runs in time
where is the number of elements of the input -represented
matroid. But their method is highly indirect. Their algorithm uses the
non-trivial fact by Geelen et al. (2003) that the number of forbidden minors is
finite and uses the algorithm of Hlin\v{e}n\'y (2005) on checking monadic
second-order formulas on -represented matroids of small
branch-width. Our result does not depend on such a fact and is completely
self-contained, and yet matches their asymptotic running time for each fixed
.Comment: 73 pages, 10 figure
Integer programming models for the branchwidth problem
We consider the problem of computing the branchwidth and an optimal branch decomposition
of a graph. Branch decompositions and branchwidth were introduced in
1991 by Robertson and Seymour and were used in the proof of Graph Minors Theorem
(GMT), a well known conjecture (Wagner's conjecture) in graph theory. The
notions of branchwidth and branch decompositions have been proved to be useful for
solving many NP-hard problems that have applications in fields such as graph theory,
network design, sensor networks and biology. Branch decompositions have been
utilized for problems such as the traveling salesman problem by Cook and Seymour,
general minor containment and the branchwidth problem by Hicks by means of the
relevant branch decomposition-based algorithms.
Branch decomposition-based algorithms are fixed parameter tractable algorithms
obtained by combining dynamic programming techniques with branch decompositions.
The running time and space of these algorithms strongly depend on the width
of the utilized branch decomposition. Thus, finding optimal or close to optimal branch
decompositions is very important for the efficiency of the branch decomposition-based
algorithms. Motivated by the vastness of the fields of application, we aim to increase
the efficiency of the branch decomposition-based algorithms by investigating effective techniques to find optimal branch decompositions.
We present three integer programming models for the branchwidth problem.
Two similar formulations are based on the relationship of branchwidth problem with
a special case of the Steiner tree packing problem. The third formulation is based on
the notion of laminar separations. We utilize upper and lower bounds obtained by
heuristic algorithms, reduction techniques and cutting planes to increase the efficiency
of our models. We use all three models for the branchwidth problem on hypergraphs as
well. We compare the performance of three models both on graphs and hypergraphs.
Furthermore we use the third model for rank-width problem and also offer a
heuristic for finding good rank decompositions. We provide computational results for
this problem, which can be a basis of comparison for future formulations
Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
Approximating branchwidth on parametric extensions of planarity
The \textsl{branchwidth} of a graph has been introduced by Roberson and
Seymour as a measure of the tree-decomposability of a graph, alternative to
treewidth. Branchwidth is polynomially computable on planar graphs by the
celebrated ``Ratcatcher''-algorithm of Seymour and Thomas. We investigate an
extension of this algorithm to minor-closed graph classes, further than planar
graphs as follows: Let be a graph embeddedable in the projective plane
and be a graph embeddedable in the torus. We prove that every
-minor free graph contains a subgraph where the
difference between the branchwidth of and the branchwidth of is
bounded by some constant, depending only on and . Moreover, the
graph admits a tree decomposition where all torsos are planar. This
decomposition can be used for deriving an EPTAS for branchwidth: For
-minor free graphs, there is a function
and a -approximation algorithm
for branchwidth, running in time for every
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