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Graph layout problems parameterized by vertex cover. (English) Zbl 1183.68424
Hong, Seok-Hee (ed.) et al., Algorithms and computation. 19th international symposium, ISAAC 2008, Gold Coast, Australia, December 15–17, 2008. Proceedings. Berlin: Springer (ISBN 978-3-540-92181-3/pbk). Lecture Notes in Computer Science 5369, 294-305 (2008).
Summary: In the framework of parameterized complexity, one of the most commonly used structural parameters is the treewidth of the input graph. The reason for this is that most natural graph problems turn out to be fixed parameter tractable when parameterized by treewidth. However, Graph Layout problems are a notable exception. In particular, no fixed parameter tractable algorithms are known for the Cutwidth, Bandwidth, Imbalance and Distortion problems parameterized by treewidth. In fact, Bandwidth remains NP-complete even restricted to trees. A possible way to attack graph layout problems is to consider structural parameterizations that are stronger than treewidth. In this paper we study graph layout problems parameterized by the size of the minimum vertex cover of the input graph. We show that all the mentioned problems are fixed parameter tractable. Our basic ingredient is a classical algorithm for Integer Linear Programming when parameterized by dimension, designed by Lenstra and later improved by Kannan. We hope that our results will serve to re-emphasize the importance and utility of this algorithm.
For the entire collection see [Zbl 1154.68014].

68R10 Graph theory (including graph drawing) in computer science
68Q25 Analysis of algorithms and problem complexity
90C35 Programming involving graphs or networks
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