Abstract
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.
Highlights
Consider the following problem: given a tree decomposition of a graph of some width k, possibly suboptimal, we would like to compute an optimum-width tree decomposition of the graph
In this work we have constructed an mso transduction that, given a constant-width tree decomposition of a graph, computes a tree decomposition of this graph of optimum width. This transduction can be conveniently composed with the mso transduction given in [BP16] to prove that given a graph of constant treewidth, some optimum-width tree decomposition can be computed by means of an mso transduction
There, we have proved that if a class of graphs of treewidth at most k is recognizable, it can be defined in mso with modular counting predicates
Summary
Consider the following problem: given a tree decomposition of a graph of some width k, possibly suboptimal, we would like to compute an optimum-width tree decomposition of the graph. As a corollary of our main result, we show (Corollary 2.2) that an mso transduction can compute an optimum-width tree decomposition, even if the input is only the graph and not a (possibly suboptimal) tree decomposition. Small alternation is the key property allowing an optimum-width tree decomposition to be captured by an mso transduction or by a dynamic programming algorithm that works on the input suboptimal decomposition. This part of the proof essentially corresponds to the machinery of typical sequences of Bodlaender and Kloks. We formalize the intuition given by the Conflict Lemma in mso, constructing the mso transduction promised in our main result
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
