On the parameterized complexity of computing tree-partitions

TL;DR

This work addresses the parameterized complexity of computing tree-partition-width (tpw), a graph parameter closely related to treewidth on bounded-degree graphs. It provides a suite of approximation algorithms achieving widths $O(k^7)$ in time $k^{O(1)}n^2$ and demonstrates XALP-completeness for exact tpw computation, indicating strong hardness across all $W[t]$ classes. The paper also examines connections with domino treewidth and tree-cut width, showing tpw is polynomially tied to tree-cut width and exploring stable and weighted variants, with corresponding approximation and hardness results. By detailing the approximation framework, subdivision-based stability, and the weighted extension, the work offers both practical algorithms and deeper complexity insights for graph decompositions. The results advance our understanding of tpw as a tractable target for approximations while clarifying the limits of exact computation under XALP.

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 $O(k^7)$ 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.

Figures (4)

• Figure 1: Local structure of Tree-Chained Multicolored Independent Set. For each $ab \in E_T$, the subgraph of $G$ induced by $V_a \cup V_b$ is a Multicolored Independent Set instance.
• Figure 2: Local structure of our construction : a trunk with attached clique chains. All clique chains have to be folded at their endpoints to fit on the trunk as illustrated by the clique chain on the right.
• Figure 3: Local structure of the gadgets. (i) If two larger cliques from the clique chains are aligned with the edge constraint of the trunk (encoding that we chose the two endpoints of the edge), then the width is slightly too large locally. (ii) If two larger cliques from clique chains are aligned but not aligned with an edge constraint, this fits the expected width (there cannot be more than 2). (iii) If at most one larger clique from clique chains is aligned with an edge constraint of the trunk, it fits the expected width.
• Figure 4: An example of the subgraph for a vertex $v$ of weight 3; here $k=5$, and an illustration of how a tree-partition of $G$ is transformed to a tree-partition of $G'$.

Theorems & Definitions (70)

• Theorem 1.1
• Theorem 1.2
• Claim 3.1
• proof
• Claim 3.2
• proof
• Claim 3.3
• proof
• Lemma 3.4
• Corollary 3.5