Abstract
Given an undirected and edge-weighted graph G together with a set of ordered vertex-pairs, called st-pairs, we consider two problems of finding an orientation of all edges in G: min-sum orientation is to minimize the sum of the shortest directed distances between all st-pairs; and min-max orientation is to minimize the maximum shortest directed distance among all st-pairs. Note that these shortest directed paths for st-pairs are not necessarily edge-disjoint. In this paper, we first show that both problems are strongly NP-hard for planar graphs even if all edge-weights are identical, and that both problems can be solved in polynomial time for cycles. We then consider the problems restricted to cacti, which form a graph class that contains trees and cycles but is a subclass of planar graphs. Then, min-sum orientation is solvable in polynomial time, whereas min-max orientation remains NP-hard even for two st-pairs. However, based on LP-relaxation, we present a polynomial-time 2-approximation algorithm for min-max orientation. Finally, we give a fully polynomial-time approximation scheme (FPTAS) for min-max orientation on cacti if the number of st-pairs is a fixed constant.
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Acknowledgements
We thank the referees for their fruitful comments, one of which leads us to improvements of the hardness analyses for min-sum orientation.
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An extended abstract of this paper has been presented at the 20th International Symposium on Algorithms and Computation (ISAAC 2009) [7].
This work is partially supported by Grant-in-Aid for Scientific Research.
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Ito, T., Miyamoto, Y., Ono, H. et al. Route-Enabling Graph Orientation Problems. Algorithmica 65, 317–338 (2013). https://doi.org/10.1007/s00453-011-9589-z
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DOI: https://doi.org/10.1007/s00453-011-9589-z