Parameterized complexity of isometric path partition: treewidth and diameter

2025-08-07

AI Summary

This paper investigates the parameterized complexity of the Isometric Path Partition problem with respect to treewidth. We establish, for the first time, that the problem is W[1]-hard even on pathwidth-bounded graphs, thereby identifying treewidth as a fundamental parameter inducing inherent computational hardness. We design two dynamic programming algorithms leveraging tree decompositions and metric structure: one runs in (n^{O(mathrm{tw})}) time, and the other in (mathrm{diam}^{O(mathrm{tw}^2)} cdot n^{O(1)}) time. Matching (almost tight) lower bounds are provided, ruling out significant runtime improvements under standard complexity assumptions. Our key innovation lies in integrating diameter constraints and the intrinsic metric properties of isometric paths into the DP state definition—departing from conventional topological approaches to path partitioning. This yields a novel paradigm for parameterized algorithms on metric graph partitioning problems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Intelligent Robots: Motion and Path Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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. In this paper, 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 $ extrm{diam}^{O(mathrm{tw}^2)} cdot n^{O(1)}$, where $ extrm{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 $ extrm{diam}^{o(mathrm{tw}^2/(log^3(mathrm{tw})))} cdot n^{O(1)}$, unless the Randomized-ETH fails.

Problem

• Study Isometric Path Partition complexity by treewidth
• Prove W[1]-hardness for treewidth parameterization
• Design dynamic programming algorithm for high treewidth

Innovation

• Dynamic programming for treewidth parameterization
• W[1]-hardness proof for pathwidth parameterization
• Diameter-based algorithm with high treewidth dependency