### Abstract

Given a simple, undirected graph G = (V ,E) and a weight function w : E→ℤ^{+}, we consider the problem of orienting all edges in E so that the maximum weighted outdegree among all vertices is minimized. It has previously been shown that the unweighted version of the problem is solvable in polynomial time while the weighted version is (weakly) NP-hard. In this paper, we strengthen these results as follows: (1) We prove that the weighted version is strongly NP-hard even if all edge weights belong to the set {1, k}, where k is any fixed integer greater than or equal to 2, and that there exists no pseudo-polynomial time approximation algorithm for this problem whose approximation ratio is smaller than (1 + 1/k) unless P = NP; (2) we present a new polynomial-time algorithm that approximates the general version of the problem within a ratio of (2-1/k), where k is the maximum weight of an edge in G; (3) we show how to approximate the special case in which all edge weights belong to {1, k} within a ratio of 3/2 for k = 2 (note that this matches the inapproximability bound above), and (2 -2/(k +1)) for any k ≤ 3, respectively, in polynomial time.

Original language | English |
---|---|

Pages (from-to) | 78-96 |

Number of pages | 19 |

Journal | Journal of Combinatorial Optimization |

Volume | 22 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 1 2011 |

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### All Science Journal Classification (ASJC) codes

- Computer Science Applications
- Discrete Mathematics and Combinatorics
- Control and Optimization
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Journal of Combinatorial Optimization*,

*22*(1), 78-96. https://doi.org/10.1007/s10878-009-9276-z

**Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree.** / Asahiro, Yuichi; Jansson, Jesper; Miyano, Eiji; Ono, Hirotaka; Zenmyo, Kouhei.

Research output: Contribution to journal › Article

*Journal of Combinatorial Optimization*, vol. 22, no. 1, pp. 78-96. https://doi.org/10.1007/s10878-009-9276-z

}

TY - JOUR

T1 - Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree

AU - Asahiro, Yuichi

AU - Jansson, Jesper

AU - Miyano, Eiji

AU - Ono, Hirotaka

AU - Zenmyo, Kouhei

PY - 2011/7/1

Y1 - 2011/7/1

N2 - Given a simple, undirected graph G = (V ,E) and a weight function w : E→ℤ+, we consider the problem of orienting all edges in E so that the maximum weighted outdegree among all vertices is minimized. It has previously been shown that the unweighted version of the problem is solvable in polynomial time while the weighted version is (weakly) NP-hard. In this paper, we strengthen these results as follows: (1) We prove that the weighted version is strongly NP-hard even if all edge weights belong to the set {1, k}, where k is any fixed integer greater than or equal to 2, and that there exists no pseudo-polynomial time approximation algorithm for this problem whose approximation ratio is smaller than (1 + 1/k) unless P = NP; (2) we present a new polynomial-time algorithm that approximates the general version of the problem within a ratio of (2-1/k), where k is the maximum weight of an edge in G; (3) we show how to approximate the special case in which all edge weights belong to {1, k} within a ratio of 3/2 for k = 2 (note that this matches the inapproximability bound above), and (2 -2/(k +1)) for any k ≤ 3, respectively, in polynomial time.

AB - Given a simple, undirected graph G = (V ,E) and a weight function w : E→ℤ+, we consider the problem of orienting all edges in E so that the maximum weighted outdegree among all vertices is minimized. It has previously been shown that the unweighted version of the problem is solvable in polynomial time while the weighted version is (weakly) NP-hard. In this paper, we strengthen these results as follows: (1) We prove that the weighted version is strongly NP-hard even if all edge weights belong to the set {1, k}, where k is any fixed integer greater than or equal to 2, and that there exists no pseudo-polynomial time approximation algorithm for this problem whose approximation ratio is smaller than (1 + 1/k) unless P = NP; (2) we present a new polynomial-time algorithm that approximates the general version of the problem within a ratio of (2-1/k), where k is the maximum weight of an edge in G; (3) we show how to approximate the special case in which all edge weights belong to {1, k} within a ratio of 3/2 for k = 2 (note that this matches the inapproximability bound above), and (2 -2/(k +1)) for any k ≤ 3, respectively, in polynomial time.

UR - http://www.scopus.com/inward/record.url?scp=79955053720&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79955053720&partnerID=8YFLogxK

U2 - 10.1007/s10878-009-9276-z

DO - 10.1007/s10878-009-9276-z

M3 - Article

AN - SCOPUS:79955053720

VL - 22

SP - 78

EP - 96

JO - Journal of Combinatorial Optimization

JF - Journal of Combinatorial Optimization

SN - 1382-6905

IS - 1

ER -