On the Parameterized Complexity of Computing Tree-Partitions

Conference paper (2022)

Authors

Hans L. Bodlaender Universiteit Utrecht
C.E. Groenland Universiteit Utrecht
Hugo Jacob ENS Paris-Saclay
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Approximation Algorithms Treewidth Parameterized algorithms Parameterized Complexity Domino Treewidth Tree partitions Tree-partition-width
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Published Date

2022

Language

English

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Abstract

We study the parameterized complexity of computing the tree-partition-width, a graph parameter equivalent to treewidth on graphs of bounded maximum degree. On one hand, we can obtain approximations of the tree-partition-width efficiently: we show that there is an algorithm that, given an n-vertex graph G and an integer k, constructs a tree-partition of width O(k7) for G or reports that G has tree-partition width more than k, in time kO(1)n2. We can improve slightly on the approximation factor by sacrificing the dependence on k, or on n. On the other hand, we show the problem of computing tree-partition-width exactly is XALP-complete, which implies that it is W[t]-hard for all t. We deduce XALP-completeness of the problem of computing the domino treewidth. Finally, we adapt some known results on the parameter tree-partition-width and the topological minor relation, and use them to compare tree-partition-width to tree-cut width.