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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
Summary
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.
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.
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