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Parameterized Complexity of Directed Spanner Problems.

Fedor V Fomin1, Petr A Golovach1, William Lochet1

  • 1Department of Informatics, University of Bergen, PB 7803, 5020 Bergen, Norway.

Algorithmica
|July 26, 2022
PubMed
Summary
This summary is machine-generated.

This study explores parameterized complexity for directed graph t-spanners. Directed Multiplicative Spanner is efficiently solvable, while Directed Additive Spanner is proven to be W[1]-hard, even for DAGs.

Keywords:
Directed graphsGraph spannersKernelizationParameterized complexity

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Area of Science:

  • Theoretical Computer Science
  • Graph Theory
  • Algorithm Analysis

Background:

  • Spanner problems are crucial for network design and analysis.
  • Parameterized complexity offers a finer-grained approach to studying NP-hard problems.
  • Directed graph spanners are less understood than their undirected counterparts.

Purpose of the Study:

  • To initiate the parameterized complexity study of minimum t-spanner problems on directed graphs.
  • To analyze the computational complexity of Directed Multiplicative Spanner and Directed Additive Spanner.
  • To establish hardness results for directed additive t-spanners.

Main Methods:

  • Parameterized complexity analysis.
  • Algorithm design and analysis for spanner construction.
  • Reductions from known hard problems to prove W[1]-hardness.

Main Results:

  • Directed Multiplicative Spanner admits a polynomial kernel of size O(k^2) and can be solved in randomized FPT time.
  • The weighted variant of Directed Multiplicative Spanner is solvable in FPT time on directed acyclic graphs.
  • Directed Additive Spanner is W[1]-hard when parameterized by k, even for directed acyclic graphs.

Conclusions:

  • The parameterized complexity of directed t-spanners differs significantly between multiplicative and additive distortion.
  • Efficient algorithms exist for the multiplicative version, but the additive version remains computationally challenging.
  • The hardness result for Directed Additive Spanner highlights the impact of directedness on spanner complexity.