Parameterized complexity of isometric path partition: treewidth and diameter

TL;DR

This paper studies the parameterized complexity of the Isometric Path Partition problem, showing it is W[1]-hard with respect to treewidth, and provides algorithms with high dependency on treewidth and diameter, establishing complexity bounds.

Contribution

It proves the W[1]-hardness of Isometric Path Partition parameterized by treewidth and develops algorithms with specific complexity bounds, highlighting the problem's computational difficulty.

Findings

Isometric Path Partition is W[1]-hard when parameterized by treewidth.

A dynamic programming algorithm with runtime n^{O(tw)} is designed.

No significantly faster algorithm exists unless the randomized ETH fails.

Abstract

We investigate the parameterized complexity of the Isometric Path Partition problem when parameterized by the treewidth ($\mathrm{tw}$) of the input graph, arguably one of the most widely studied parameters. Courcelle's theorem shows that graph problems that are expressible as MSO formulas of constant size admit FPT algorithms parameterized by the treewidth of the input graph. This encompasses many natural graph problems. However, many metric-based graph problems, where the solution is defined using some metric-based property of the graph (often the distance) are not expressible as MSO formulas of constant size. These types of problems, Isometric Path Partition being one of them, require individual attention and often draw the boundary for the success story of parameterization by treewidth. We prove that Isometric Path Partition is $W[1]$-hard when parameterized by treewidth (in fact, even pathwidth), answering the question by Dumas et al. [SIDMA, 2024], Fernau et al. [CIAC, 2023], and confirming the aforementioned tendency. We complement this hardness result by designing a tailored dynamic programming algorithm running in $n^{O(\mathrm{tw})}$ time. This dynamic programming approach also results in an algorithm running in time $\textrm{diam}^{O(\mathrm{tw}^2)} \cdot n^{O(1)}$, where $\textrm{diam}$ is the diameter of the graph. Note that the dependency on treewidth is unusually high, as most problems admit algorithms running in time $2^{O(\mathrm{tw})}\cdot n^{O(1)}$ or $2^{O(\mathrm{tw} \log (\mathrm{tw}))}\cdot n^{O(1)}$. However, we rule out the possibility of a significantly faster algorithm by proving that Isometric Path Partition does not admit an algorithm running in time $\textrm{diam}^{o(\mathrm{tw}^2/(\log^3(\mathrm{tw})))} \cdot n^{O(1)}$, unless the Randomized-ETH fails.