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Insights into (k, ρ)-Shortcutting Algorithms

Authors Alexander Leonhardt , Ulrich Meyer , Manuel Penschuck

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Author Details

Alexander Leonhardt
  • Goethe University Frankfurt, Germany
Ulrich Meyer
  • Goethe University Frankfurt, Germany
Manuel Penschuck
  • Goethe University Frankfurt, Germany

Funding

  • Penschuck, Manuel: Funded by the Deutsche Forschungsgemeinschaft (DFG) - {ME 2088/5-2} (FOR 2975 - Algorithms, Dynamics, and Information Flow in Networks).

Acknowledgements

The authors thank the anonymous reviewers for their insightful comments which greatly improved this paper.

Cite AsGet BibTex

Alexander Leonhardt, Ulrich Meyer, and Manuel Penschuck. Insights into (k, ρ)-Shortcutting Algorithms. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 84:1-84:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.84

Abstract

A graph is called a (k, ρ)-graph iff every node can reach ρ of its nearest neighbors in at most k hops. This property has proven useful in the analysis and design of parallel shortest-path algorithms [Blelloch et al., 2016; Dong et al., 2021]. Any graph can be transformed into a (k, ρ)-graph by adding shortcuts. Formally, the (k,ρ)-Minimum-Shortcut-Problem (kρ-MSP) asks to find an appropriate shortcut set of minimal cardinality. We show that kρ-MSP is NP-complete in the practical regime of k ≥ 3 and ρ = Θ(n^ε) for ε > 0. With a related construction, we bound the approximation factor of known kρ-MSP heuristics [Blelloch et al., 2016] from below and propose algorithmic countermeasures improving the approximation quality. Further, we describe an integer linear problem (ILP) that optimally solves kρ-MSP. Finally, we compare the practical performance and quality of all algorithms empirically.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Complexity
  • Approximation
  • Optimal algorithms
  • Parallel shortest path

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