, Volume 65, Issue 2, pp 317–338 | Cite as

Route-Enabling Graph Orientation Problems

  • Takehiro Ito
  • Yuichiro Miyamoto
  • Hirotaka Ono
  • Hisao Tamaki
  • Ryuhei Uehara


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.


Approximation algorithm Cactus Dynamic programming Fully polynomial-time approximation scheme Graph orientation Planar graph Reachability 



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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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Takehiro Ito
    • 1
  • Yuichiro Miyamoto
    • 2
  • Hirotaka Ono
    • 3
  • Hisao Tamaki
    • 4
  • Ryuhei Uehara
    • 5
  1. 1.Graduate School of Information SciencesTohoku UniversitySendaiJapan
  2. 2.Faculty of Science and TechnologySophia UniversityTokyoJapan
  3. 3.Faculty of EconomicsKyushu UniversityFukuokaJapan
  4. 4.School of Science and TechnologyMeiji UniversityKanagawaJapan
  5. 5.School of Information ScienceJAISTIshikawaJapan

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