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The Complexity of Packing Edge-Disjoint Paths

Authors Jan Dreier , Janosch Fuchs , Tim A. Hartmann , Philipp Kuinke , Peter Rossmanith , Bjoern Tauer, Hung-Lung Wang

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • parameterized complexity
  • embedding
  • packing
  • covering
  • Hamiltonian path
  • unary binpacking
  • path-perfect graphs

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Abstract

We introduce and study the complexity of Path Packing. Given a graph G and a list of paths, the task is to embed the paths edge-disjoint in G. This generalizes the well known Hamiltonian-Path problem.
Since Hamiltonian Path is efficiently solvable for graphs of small treewidth, we study how this result translates to the much more general Path Packing. On the positive side, we give an FPT-algorithm on trees for the number of paths as parameter. Further, we give an XP-algorithm with the combined parameters maximal degree, number of connected components and number of nodes of degree at least three. Surprisingly the latter is an almost tight result by runtime and parameterization. We show an ETH lower bound almost matching our runtime. Moreover, if two of the three values are constant and one is unbounded the problem becomes NP-hard.
Further, we study restrictions to the given list of paths. On the positive side, we present an FPT-algorithm parameterized by the sum of the lengths of the paths. Packing paths of length two is polynomial time solvable, while packing paths of length three is NP-hard. Finally, even the spacial case Exact Path Packing where the paths have to cover every edge in G exactly once is already NP-hard for two paths on 4-regular graphs.

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Jan Dreier, Janosch Fuchs, Tim A. Hartmann, Philipp Kuinke, Peter Rossmanith, Bjoern Tauer, and Hung-Lung Wang. The Complexity of Packing Edge-Disjoint Paths. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.IPEC.2019.10

Author Details

Jan Dreier
  • Dept. of Computer Science, RWTH Aachen University, Germany
Janosch Fuchs
  • Dept. of Computer Science, RWTH Aachen University, Germany
Tim A. Hartmann
  • Dept. of Computer Science, RWTH Aachen University, Germany
Philipp Kuinke
  • Dept. of Computer Science, RWTH Aachen University, Germany
Peter Rossmanith
  • Dept. of Computer Science, RWTH Aachen University, Germany
Bjoern Tauer
  • Dept. of Computer Science, RWTH Aachen University, Germany
Hung-Lung Wang
  • Computer Science and Information Engineering, National Taiwan Normal University, Taiwan

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