6 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
A Linear Fixed Parameter Tractable Algorithm for Connected Pathwidth
International audienceThe graph parameter of {\sl pathwidth} can be seen as a measure of the topological resemblance of a graph to a path. A popular definition of pathwidth is given in terms of {\sl node search} where we are given a system of tunnels (represented by a graph) that is contaminated by some infectious substance and we are looking for a search strategy that, at each step, either places a searcher on a vertex or removes a searcher from a vertex and where an edge is cleaned when both endpoints are simultaneously occupied by searchers. It was proved that the minimum number of searchers required for a successful cleaning strategy is equal to the pathwidth of the graph plus one.Two desired characteristics for a cleaning strategy is to be {\sl monotone} (no recontamination occurs) and {\sl connected} (clean territories always remain connected). Under these two demands, the number of searchers is equivalent to a variant of pathwidth called {\em connected pathwidth}. We prove that connected pathwidth is fixed parameter tractable, in particular we design a time algorithm that checks whether the connected pathwidth of is at most This resolves an open question by [{\sl Dereniowski, Osula, and Rz{\k{a}}{\.{z}}ewski, Finding small-width connected path-decompositions in polynomial time. Theor. Comput. Sci., 794:85–100, 2019}\,]. For our algorithm, we enrich the {\sl typical sequence technique} that is able to deal with the connectivity demand. Typical sequences have been introduced in [{\sl Bodlaender and Kloks. Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms, 21(2):358–402, 1996}\,] for the design of linear parameterized algorithms for treewidth and pathwidth. While this technique has been later applied to other parameters, none of its advancements was able to deal with the connectivity demand, as it is a ``global’’ demand that concerns an unbounded number of parts of the graph of unbounded size. The proposed extension is based on an encoding of the connectivity property that is quite versatile and may be adapted so to deliver linear parameterized algorithms for the connected variants of other width parameters as well. An immediate consequence of our result is a time algorithm for the monotone and connected version of the edge search number
Tight Lower Bounds for Problems Parameterized by Rank-Width
We show that there is no 2o(k2)nO(1) time algorithm for Independent Set on n-vertex graphs with rank-width k, unless the Exponential Time Hypothesis (ETH) fails. Our lower bound matches the 2O(k2)nO(1) time algorithm given by Bui-Xuan, Telle, and Vatshelle [Discret. Appl. Math., 2010] and it answers the open question of Bergougnoux and Kanté [SIAM J. Discret. Math., 2021]. We also show that the known 2O(k2)nO(1) time algorithms for Weighted Dominating Set, Maximum Induced Matching and Feedback Vertex Set parameterized by rank-width k are optimal assuming ETH. Our results are the first tight ETH lower bounds parameterized by rank-width that do not follow directly from lower bounds for n-vertex graphs
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
Optimizing tree decompositions in MSO
The classic algorithm of Bodlaender and Kloks [J. Algorithms, 1996] solves
the following problem in linear fixed-parameter time: given a tree
decomposition of a graph of (possibly suboptimal) width k, compute an
optimum-width tree decomposition of the graph. In this work, we prove that this
problem can also be solved in mso in the following sense: for every positive
integer k, there is an mso transduction from tree decompositions of width k to
tree decompositions of optimum width. Together with our recent results [LICS
2016], this implies that for every k there exists an mso transduction which
inputs a graph of treewidth k, and nondeterministically outputs its tree
decomposition of optimum width. We also show that mso transductions can be
implemented in linear fixed-parameter time, which enables us to derive the
algorithmic result of Bodlaender and Kloks as a corollary of our main result