Parametric Shortest Paths in Planar Graphs

@article{Gajjar2019ParametricSP,
  title={Parametric Shortest Paths in Planar Graphs},
  author={Kshitij Gajjar and Jaikumar Radhakrishnan},
  journal={2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)},
  year={2019},
  pages={876-895}
}
We construct a family of planar graphs {G_n}_n≥4, where G_n has n vertices including a source vertex s, a sink vertex t, and edge weights that change linearly with a parameter λ such that, as λ varies in (-∞,+∞), the piece-wise linear cost of the shortest path from s to t has n^Ω(logn) pieces. This shows that lower bounds obtained by Carstensen (1983) and Mulmuley & Shah (2001) for general graphs also hold for planar graphs, refuting a conjecture of Nikolova (2009). Gusfield (1980) and Dean… Expand
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