Route-enabling graph orientation problems

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

Research output: Contribution to journalArticle

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 polynomialtime 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.

Original languageEnglish
Pages (from-to)317-338
Number of pages22
JournalAlgorithmica
Volume65
Issue number2
DOIs
Publication statusPublished - 2013 Feb 1

Keywords

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

ASJC Scopus subject areas

  • Computer Science(all)
  • Computer Science Applications
  • Applied Mathematics

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  • Cite this

    Ito, T., Miyamoto, Y., Ono, H., Tamaki, H., & Uehara, R. (2013). Route-enabling graph orientation problems. Algorithmica, 65(2), 317-338. https://doi.org/10.1007/s00453-011-9589-z