80 research outputs found
Optimizing Tree Decompositions in MSO
The classic algorithm of Bodlaender and Kloks 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, 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
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
Definable decompositions for graphs of bounded linear cliquewidth
We prove that for every positive integer , there exists an
-transduction that given a graph of linear cliquewidth at most
outputs, nondeterministically, some cliquewidth decomposition of the graph
of width bounded by a function of . A direct corollary of this result is the
equivalence of the notions of -definability and recognizability
on graphs of bounded linear cliquewidth.Comment: 39 pages, 5 figures. The conference version of the manuscript
appeared in the proceedings of LICS 201
Does Treewidth Help in Modal Satisfiability?
Many tractable algorithms for solving the Constraint Satisfaction Problem
(CSP) have been developed using the notion of the treewidth of some graph
derived from the input CSP instance. In particular, the incidence graph of the
CSP instance is one such graph. We introduce the notion of an incidence graph
for modal logic formulae in a certain normal form. We investigate the
parameterized complexity of modal satisfiability with the modal depth of the
formula and the treewidth of the incidence graph as parameters. For various
combinations of Euclidean, reflexive, symmetric and transitive models, we show
either that modal satisfiability is FPT, or that it is W[1]-hard. In
particular, modal satisfiability in general models is FPT, while it is
W[1]-hard in transitive models. As might be expected, modal satisfiability in
transitive and Euclidean models is FPT.Comment: Full version of the paper appearing in MFCS 2010. Change from v1:
improved section 5 to avoid exponential blow-up in formula siz
Some Remarks on Deciding Equivalence for Graph-To-Graph Transducers
We study the following decision problem: given two mso transductions that input and output graphs of bounded treewidth, decide if they are equivalent, i.e. isomorphic inputs give isomorphic outputs. We do not know how to decide it, but we propose an approach that uses automata manipulating elements of a ring extended with division. The approach works for a variant of the problem, where isomorphism on output graphs is replaced by a relaxation of isomorphism
Hitting and Harvesting Pumpkins
The "c-pumpkin" is the graph with two vertices linked by c>0 parallel edges.
A c-pumpkin-model in a graph G is a pair A,B of disjoint subsets of vertices of
G, each inducing a connected subgraph of G, such that there are at least c
edges in G between A and B. We focus on covering and packing c-pumpkin-models
in a given graph: On the one hand, we provide an FPT algorithm running in time
2^O(k) n^O(1) deciding, for any fixed c>0, whether all c-pumpkin-models can be
covered by at most k vertices. This generalizes known single-exponential FPT
algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the
cases c=1,2 respectively. On the other hand, we present a O(log
n)-approximation algorithm for both the problems of covering all
c-pumpkin-models with a smallest number of vertices, and packing a maximum
number of vertex-disjoint c-pumpkin-models.Comment: v2: several minor change
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
Definable decompositions for graphs of bounded linear cliquewidth
We prove that for every positive integer k, there exists an
MSO_1-transduction that given a graph of linear cliquewidth at most k outputs,
nondeterministically, some cliquewidth decomposition of the graph of width
bounded by a function of k. A direct corollary of this result is the
equivalence of the notions of CMSO_1-definability and recognizability on graphs
of bounded linear cliquewidth
- …