252 research outputs found
Obstructions for Matroids of Path-Width at most k and Graphs of Linear Rank-Width at most k
International audienceEvery minor-closed class of matroids of bounded branch-width can be characterized by a minimal list of excluded minors, but unlike graphs, this list could be infinite in general. However, for each fixed finite field , the list contains only finitely many -representable matroids, due to the well-quasi-ordering of -representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these -representable excluded minors in general. We consider the class of matroids of path-width at most for fixed . We prove that for a finite field , every -representable excluded minor for the class of matroids of path-width at most~ has at most elements. We can therefore compute, for any integer and a fixed finite field , the set of -representable excluded minors for the class of matroids of path-width , and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an -represented matroid is at most . We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most has at most vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs
Linear rank-width of distance-hereditary graphs I. A polynomial-time algorithm
Linear rank-width is a linearized variation of rank-width, and it is deeply
related to matroid path-width. In this paper, we show that the linear
rank-width of every -vertex distance-hereditary graph, equivalently a graph
of rank-width at most , can be computed in time , and a linear layout witnessing the linear rank-width can be computed with
the same time complexity. As a corollary, we show that the path-width of every
-element matroid of branch-width at most can be computed in time
, provided that the matroid is given by an
independent set oracle.
To establish this result, we present a characterization of the linear
rank-width of distance-hereditary graphs in terms of their canonical split
decompositions. This characterization is similar to the known characterization
of the path-width of forests given by Ellis, Sudborough, and Turner [The vertex
separation and search number of a graph. Inf. Comput., 113(1):50--79, 1994].
However, different from forests, it is non-trivial to relate substructures of
the canonical split decomposition of a graph with some substructures of the
given graph. We introduce a notion of `limbs' of canonical split
decompositions, which correspond to certain vertex-minors of the original
graph, for the right characterization.Comment: 28 pages, 3 figures, 2 table. A preliminary version appeared in the
proceedings of WG'1
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
On the Optimality of Pseudo-polynomial Algorithms for Integer Programming
In the classic Integer Programming (IP) problem, the objective is to decide
whether, for a given matrix and an -vector , there is a non-negative integer -vector such that . Solving
(IP) is an important step in numerous algorithms and it is important to obtain
an understanding of the precise complexity of this problem as a function of
natural parameters of the input.
The classic pseudo-polynomial time algorithm of Papadimitriou [J. ACM 1981]
for instances of (IP) with a constant number of constraints was only recently
improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder
[ArXiv 2018]. We continue this line of work and show that under the Exponential
Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal.
We also show that when the matrix is assumed to be non-negative, a
component of Papadimitriou's original algorithm is already nearly optimal under
ETH.
This motivates us to pick up the line of research initiated by Cunningham and
Geelen [IPCO 2007] who studied the complexity of solving (IP) with non-negative
matrices in which the number of constraints may be unbounded, but the
branch-width of the column-matroid corresponding to the constraint matrix is a
constant. We prove a lower bound on the complexity of solving (IP) for such
instances and obtain optimal results with respect to a closely related
parameter, path-width. Specifically, we prove matching upper and lower bounds
for (IP) when the path-width of the corresponding column-matroid is a constant.Comment: 29 pages, To appear in ESA 201
Towards a matroid-minor structure theory
This paper surveys recent work that is aimed at generalising the results and techniques of the Graph Minors Project of Robertson and Seymour to matroids
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