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.

翻译：我们研究了计算树划分宽度的参数化复杂性，该图参数等价于有界最大度图上的树宽度。一方面，我们可以高效地获得树划分宽度的近似值：我们证明存在一种算法，对于给定的$n$顶点图$G$和整数$k$，在时间$k^{O(1)}n^2$内，能为$G$构造宽度为$O(k^7)$的树划分，或报告$G$的树划分宽度大于$k$。通过牺牲对$k$或$n$的依赖关系，我们可以略微改进近似因子。另一方面，我们证明了精确计算树划分宽度的问题是XALP完全的，这意味它对所有$t$都是$W[t]$-困难的。我们推导出计算多米诺树宽度问题的XALP完全性。接下来，我们改编一些关于参数树划分宽度与拓扑子图关系的已知结果，并利用它们将树划分宽度与树割宽度进行比较。最后，针对相关参数加权树划分宽度，我们给出了类似的近似算法（比率现在为$O(k^{15})$），并证明了在顶点和边权重均为1的特例中该问题是XALP完全的。