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Approximate Tree Decompositions of Planar Graphs in Linear Time
| Citations: | 5 - 0 self |
BibTeX
@MISC{Kammer_approximatetree,
author = {Frank Kammer and Torsten Tholey},
title = {Approximate Tree Decompositions of Planar Graphs in Linear Time},
year = {}
}
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Abstract
Many algorithms have been developed for NP-hard problems on graphs with small treewidth k. For example, all problems that are expressible in linear extended monadic second order can be solved in linear time on graphs of bounded treewidth. It turns out that the bottleneck of many algorithms for NP-hard problems is the computation of a tree decomposition of width O(k). In particular, by the bidimensional theory, there are many linear extended monadic second order problems that can be solved on n-vertex planar graphs with treewidth k in a time linear in n and subexponential in k if a tree decomposition of width O(k) can be found in such a time. We present the first algorithm that, on n-vertex planar graphs with treewidth k, finds a tree decomposition of width O(k) in such a time. In more detail, our algorithm has a running time of O(nk3 log k). The previous best algorithm with a running time subexponential in k was the algorithm of Gu and Tamaki [12] with a running time of O(n1+ɛ log n) and an approximation ratio 1.5 + 1/ɛ for any ɛ> 0. The running time of our algorithm is also better than the running time of O(f(k) · n log n) of Reed’s algorithm [18] for general graphs, where f is a function exponential in k. Key words: tree decomposition, treewidth, branchwidth, rank-width, planar graph, ℓ-outerplanar, linear
Keyphrases
tree decomposition planar graph linear time running time approximate tree decomposition np-hard problem n-vertex planar graph many algorithm function exponential bounded treewidth general graph linear extended monadic second order nk3 log first algorithm running time subexponential bidimensional theory monadic second order problem n1 log small treewidth key word approximation ratio
