Hans L. Bodlaender ; Carla Groenland ; Hugo Jacob - On the parameterized complexity of computing tree-partitions

dmtcs:12540 - Discrete Mathematics & Theoretical Computer Science, February 16, 2025, vol. 26:3 - https://doi.org/10.46298/dmtcs.12540
On the parameterized complexity of computing tree-partitionsArticle

Authors: Hans L. Bodlaender ; Carla Groenland ; Hugo Jacob

    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 $O(k^7)$ for $G$ or reports that $G$ has tree-partition-width more than $k$, in time $k^{O(1)}n^2$. 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. Next, 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. Finally, for the related parameter weighted tree-partition-width, we give a similar approximation algorithm (with ratio now $O(k^{15})$) and show XALP-completeness for the special case where vertices and edges have weight 1.


    Volume: vol. 26:3
    Section: Discrete Algorithms
    Published on: February 16, 2025
    Accepted on: July 29, 2024
    Submitted on: November 12, 2023
    Keywords: Computer Science - Discrete Mathematics

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