### 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 language | English |
---|---|

Pages (from-to) | 317-338 |

Number of pages | 22 |

Journal | Algorithmica |

Volume | 65 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 1 2013 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Algorithmica*,

*65*(2), 317-338. https://doi.org/10.1007/s00453-011-9589-z

**Route-enabling graph orientation problems.** / Ito, Takehiro; Miyamoto, Yuichiro; Ono, Hirotaka; Tamaki, Hisao; Uehara, Ryuhei.

Research output: Contribution to journal › Article

*Algorithmica*, vol. 65, no. 2, pp. 317-338. https://doi.org/10.1007/s00453-011-9589-z

}

TY - JOUR

T1 - Route-enabling graph orientation problems

AU - Ito, Takehiro

AU - Miyamoto, Yuichiro

AU - Ono, Hirotaka

AU - Tamaki, Hisao

AU - Uehara, Ryuhei

PY - 2013/2/1

Y1 - 2013/2/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84892902998&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84892902998&partnerID=8YFLogxK

U2 - 10.1007/s00453-011-9589-z

DO - 10.1007/s00453-011-9589-z

M3 - Article

AN - SCOPUS:84892902998

VL - 65

SP - 317

EP - 338

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 2

ER -