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（一种在最大度有界图上与树宽等价的图参数）的参数化复杂性。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，给定一个具有 $n$ 个顶点的图 $G$ 和一个整数 $k$，可以在 $k^{O(1)}n^2$ 时间内，为 $G$ 构造一个宽度为 $O(k^7)$ 的树划分，或者报告 $G$ 的树划分宽度大于 $k$。我们可以通过牺牲对 $k$ 或 $n$ 的依赖度，在近似因子上略有改进。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，这意味着它对所有 $t$ 都是 $W[t]$-难的。我们由此推导出计算多米诺树宽的问题也是 XALP-完全的。接着，我们调整了关于参数树划分宽度和拓扑子图关系的一些已知结果，并用它们来比较树划分宽度与树割宽度。最后，对于相关的参数加权树划分宽度，我们给出了一个类似的近似算法（其近似比现为 $O(k^{15})$），并证明了在顶点和边权重均为 1 的特殊情况下，该问题是 XALP-完全的。