15 research outputs found
Tangles, tree-decompositions, and grids in matroids
A tangle in a matroid is an obstruction to small branch-width. In particular, the maximum order of a tangle is equal to the branch-width. We prove that: (i) there is a tree-decomposition of a matroid that “displays” all of the maximal tangles, and (ii) when M is representable over a finite field, each tangle of sufficiently large order “dominates” a large grid-minor. This extends results of Robertson and Seymour concerning Graph Minors
Branchwidth of graphic matroids.
Answering a question of Geelen, Gerards, Robertson and Whittle, we prove that the branchwidth of a bridgeless graph is equal to the branch- width of its cycle matroid. Our proof is based on branch-decompositions of hypergraph
Branch-decomposition heuristics for linear matroids
This thesis present two new heuristics which utilize classification and max-flow algorithm respectively to derive near-optimal branch-decompositions for linear matroids. In the literature, there are already excellent heuristics for graphs, however, no practical branch-decomposition methods for general linear matroids have been addressed yet. Introducing a "measure" which compares the "similarity" of elements of a linear matroid, this work reforms the linear matroid into a similarity graph. Then, two different methods, classification method and max-flow method, both basing on the similarity graph are developed into heuristics. Computational results using the classification method and the max-flow method on linear matroid instances are shown respectively
Tree automata and pigeonhole classes of matroids -- I
Hlineny's Theorem shows that any sentence in the monadic second-order logic
of matroids can be tested in polynomial time, when the input is limited to a
class of F-representable matroids with bounded branch-width (where F is a
finite field). If each matroid in a class can be decomposed by a subcubic tree
in such a way that only a bounded amount of information flows across displayed
separations, then the class has bounded decomposition-width. We introduce the
pigeonhole property for classes of matroids: if every subclass with bounded
branch-width also has bounded decomposition-width, then the class is
pigeonhole. An efficiently pigeonhole class has a stronger property, involving
an efficiently-computable equivalence relation on subsets of the ground set. We
show that Hlineny's Theorem extends to any efficiently pigeonhole class. In a
sequel paper, we use these ideas to extend Hlineny's Theorem to the classes of
fundamental transversal matroids, lattice path matroids, bicircular matroids,
and H-gain-graphic matroids, where H is any finite group. We also give a
characterisation of the families of hypergraphs that can be described via tree
automata: a family is defined by a tree automaton if and only if it has bounded
decomposition-width. Furthermore, we show that if a class of matroids has the
pigeonhole property, and can be defined in monadic second-order logic, then any
subclass with bounded branch-width has a decidable monadic second-order theory.Comment: Slightly extending the main theorem to cover a more expressive logi
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
Algorithmic Meta-Theorems
Algorithmic meta-theorems are general algorithmic results applying to a whole
range of problems, rather than just to a single problem alone. They often have
a "logical" and a "structural" component, that is they are results of the form:
every computational problem that can be formalised in a given logic L can be
solved efficiently on every class C of structures satisfying certain
conditions. This paper gives a survey of algorithmic meta-theorems obtained in
recent years and the methods used to prove them. As many meta-theorems use
results from graph minor theory, we give a brief introduction to the theory
developed by Robertson and Seymour for their proof of the graph minor theorem
and state the main algorithmic consequences of this theory as far as they are
needed in the theory of algorithmic meta-theorems