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Parameterized Complexity of Isometric Path Partition: Treewidth and Diameter

Authors Dibyayan Chakraborty , Oscar Defrain , Florent Foucaud , Mathieu Mari , Prafullkumar Tale

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ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Isometric path partition
  • parameterized complexity
  • treewidth
  • diameter
  • Randomized ETH

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Abstract

In the Isometric Path Partition problem, the input is a graph G with n vertices and an integer k, and the objective is to determine whether the vertices of G can be partitioned into k vertex-disjoint shortest paths. We investigate the parameterized complexity of the problem when parameterized by the treewidth (tw) of the input graph, arguably one of the most widely studied parameters. Courcelle’s theorem [Information & Computation, 1990] 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 show that Isometric Path Partition is W[1]-hard when parameterized by treewidth (in fact, even pathwidth (pw)), answering the question by Dumas et al. [SIDMA, 2024], Fernau et al. [TCS, 2025], and confirming the aforementioned tendency. We complement this hardness result by designing a tailored dynamic programming algorithm running in n^{O(tw)} time. This dynamic programming approach also results in an algorithm running in time diam^{O(tw²)} ⋅ n^{O(1)}, where diam is the diameter of the graph. It is known that Isometric Path Partition remains NP-hard on graphs of diameter 2; hence, the combination of both parameters is necessary to obtain a tractable algorithm. Note that the dependency on treewidth is unusually high, as most problems that are FPT for treewidth admit algorithms running in time 2^{O(tw)}⋅ n^{O(1)} or 2^{O(tw log (tw))}⋅ n^{O(1)}. However, we rule out the possibility of a significantly faster algorithm, showing that Isometric Path Partition does not admit an algorithm running in time diam^{o(pw²/(log³(pw)))} ⋅ n^{O(1)}, assuming the Randomized-ETH.

Cite As Get BibTex

Dibyayan Chakraborty, Oscar Defrain, Florent Foucaud, Mathieu Mari, and Prafullkumar Tale. Parameterized Complexity of Isometric Path Partition: Treewidth and Diameter. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WG.2026.11

Author Details

Dibyayan Chakraborty
  • School of Computer Science, University of Leeds, UK
Oscar Defrain
  • LIS, Aix-Marseille University, CNRS, France
Florent Foucaud
  • Université Clermont Auvergne, CNRS, Mines Saint-Étienne, Clermont Auvergne INP, LIMOS, 63000 Clermont-Ferrand, France
Mathieu Mari
  • LIRMM, Université de Montpellier, CNRS, Montpellier, France
Prafullkumar Tale
  • Indian Institute of Science Education and Research Pune, India

Funding

  • Foucaud, Florent: ANR project GRALMECO (ANR-21-CE48-0004), French government IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25), International Research Center "Innovation Transportation and Production Systems" of the I-SITE CAP 20-25, and the CNRS IRL ReLaX.
  • Tale, Prafullkumar: Supported by ARG-MATRICS Grant titled MaPLE (ARGM/2026/0089).

Acknowledgements

The authors would like to thank the organizers and participants of the 2024 Workshop of the AGC team of LIMOS at Saint-Jacques-d'Ambur for the relaxed yet inspiring atmosphere, where this research was initiated.

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