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Using a Geometric Lens to Find k Disjoint Shortest Paths

Authors Matthias Bentert, André Nichterlein , Malte Renken , Philipp Zschoche

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • graph algorithms
  • dynamic programming
  • W[1]-hardness
  • geometry

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Abstract

Given an undirected n-vertex graph and k pairs of terminal vertices (s_1,t_1), …, (s_k,t_k), the k-Disjoint Shortest Paths (k-DSP) problem asks whether there are k pairwise vertex-disjoint paths P_1, …, P_k such that P_i is a shortest s_i-t_i-path for each i ∈ [k]. Recently, Lochet [SODA '21] provided an algorithm that solves k-DSP in n^{O(k^{5^k})} time, answering a 20-year old question about the computational complexity of k-DSP for constant k. On the one hand, we present an improved O(kn^{16k ⋅ k! + k + 1})-time algorithm based on a novel geometric view on this problem. For the special case k = 2, we show that the running time can be further reduced to O(nm) by small modifications of the algorithm and a further refined analysis. On the other hand, we show that k-DSP is W[1]-hard with respect to k, showing that the dependency of the degree of the polynomial running time on the parameter k is presumably unavoidable.

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Matthias Bentert, André Nichterlein, Malte Renken, and Philipp Zschoche. Using a Geometric Lens to Find k Disjoint Shortest Paths. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.26

Author Details

Matthias Bentert
  • Faculty IV, Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
André Nichterlein
  • Faculty IV, Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
Malte Renken
  • Faculty IV, Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
Philipp Zschoche
  • Faculty IV, Algorithmics and Computational Complexity, Technische Universität Berlin, Germany

Funding

This work was started at the research retreat of the group Algorithmics and Computational Complexity in September 2019, held in Schloss Neuhausen (Prignitz, Germany).
  • Renken, Malte: Supported by the DFG project MATE (NI 369/17).

References

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