On the parameterized complexity of computing tree-partitions

2022-06-23

International Symposium on Parameterized and Exact Computation

AI Summary

This work investigates the computational complexity and approximation algorithms for tree-partition-width. Regarding problem hardness, we establish that computing tree-partition-width exactly is XALP-complete—resolving a long-standing open question—and derive as a corollary the XALP-completeness of domino treewidth. We further characterize structural relationships between tree-partition-width and tree-cut width. Methodologically, we design the first polynomial-time O(k⁷)-approximation algorithm, running in kᴼ(¹)n² time, and prove that the problem is W[t]-hard for all t, revealing its intrinsic parameterized intractability. Collectively, our results precisely situate tree-partition-width within the sparse graph parameterization landscape: they resolve fundamental gaps concerning both exact solvability (via XALP-completeness) and efficient approximability (via the first nontrivial approximation guarantee), thereby providing a comprehensive theoretical foundation for this structural graph parameter.

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 on the approximation factor or the dependence on $n$ by sacrificing the dependence on $k$. 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.

Problem

• Tree Partitioning
• Width Metrics
• Approximation Algorithms

Innovation

• Approximation Algorithm
• Tree Partition Width
• XALP Completeness