### Abstract

We study a new variant of the graph orientation problem called MAXMINO where the input is an undirected, edge-weighted graph and the objective is to assign a direction to each edge so that the minimum weighted outdegree (taken over all vertices in the resulting directed graph) is maximized. All edge weights are assumed to be positive integers. This problem is closely related to the job scheduling on parallel machines, called the machine covering problem, where its goal is to assign jobs to parallel machines such that each machine is covered as much as possible. First, we prove that MAXMINO is strongly NP-hard and cannot be approximated within a ratio of 2 ε for any constant ε > 0 in polynomial time unless P=NP, even if all edge weights belong to {1, 2}, every vertex has degree at most three, and the input graph is bipartite or planar. Next, we show how to solve MAXMINO exactly in polynomial time for the special case in which all edge weights are equal to 1. This technique gives us a simple polynomial-time w_{max}/w_{min}-approximation algorithm for MAXMINO where w_{max} and w_{min} denote the maximum and minimum weights among all the input edges. Furthermore, we also observe that this approach yields an exact algorithm for the general case of MAXMINO whose running time is polynomial whenever the number of edges having weight larger than w_{min} is at most logarithmic in the number of vertices. Finally, we show that MAXMINO is solvable in polynomial time if the input is a cactus graph.

Original language | English |
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Pages (from-to) | 583-601 |

Number of pages | 19 |

Journal | International Journal of Foundations of Computer Science |

Volume | 22 |

Issue number | 3 |

DOIs | |

Publication status | Published - Apr 1 2011 |

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### All Science Journal Classification (ASJC) codes

- Computer Science (miscellaneous)

### Cite this

*International Journal of Foundations of Computer Science*,

*22*(3), 583-601. https://doi.org/10.1142/S0129054111008246

**Graph orientation to maximize the minimum weighted outdegree.** / Asahiro, Yuichi; Jansson, Jesper; Miyano, Eiji; Ono, Hirotaka.

Research output: Contribution to journal › Article

*International Journal of Foundations of Computer Science*, vol. 22, no. 3, pp. 583-601. https://doi.org/10.1142/S0129054111008246

}

TY - JOUR

T1 - Graph orientation to maximize the minimum weighted outdegree

AU - Asahiro, Yuichi

AU - Jansson, Jesper

AU - Miyano, Eiji

AU - Ono, Hirotaka

PY - 2011/4/1

Y1 - 2011/4/1

N2 - We study a new variant of the graph orientation problem called MAXMINO where the input is an undirected, edge-weighted graph and the objective is to assign a direction to each edge so that the minimum weighted outdegree (taken over all vertices in the resulting directed graph) is maximized. All edge weights are assumed to be positive integers. This problem is closely related to the job scheduling on parallel machines, called the machine covering problem, where its goal is to assign jobs to parallel machines such that each machine is covered as much as possible. First, we prove that MAXMINO is strongly NP-hard and cannot be approximated within a ratio of 2 ε for any constant ε > 0 in polynomial time unless P=NP, even if all edge weights belong to {1, 2}, every vertex has degree at most three, and the input graph is bipartite or planar. Next, we show how to solve MAXMINO exactly in polynomial time for the special case in which all edge weights are equal to 1. This technique gives us a simple polynomial-time wmax/wmin-approximation algorithm for MAXMINO where wmax and wmin denote the maximum and minimum weights among all the input edges. Furthermore, we also observe that this approach yields an exact algorithm for the general case of MAXMINO whose running time is polynomial whenever the number of edges having weight larger than wmin is at most logarithmic in the number of vertices. Finally, we show that MAXMINO is solvable in polynomial time if the input is a cactus graph.

AB - We study a new variant of the graph orientation problem called MAXMINO where the input is an undirected, edge-weighted graph and the objective is to assign a direction to each edge so that the minimum weighted outdegree (taken over all vertices in the resulting directed graph) is maximized. All edge weights are assumed to be positive integers. This problem is closely related to the job scheduling on parallel machines, called the machine covering problem, where its goal is to assign jobs to parallel machines such that each machine is covered as much as possible. First, we prove that MAXMINO is strongly NP-hard and cannot be approximated within a ratio of 2 ε for any constant ε > 0 in polynomial time unless P=NP, even if all edge weights belong to {1, 2}, every vertex has degree at most three, and the input graph is bipartite or planar. Next, we show how to solve MAXMINO exactly in polynomial time for the special case in which all edge weights are equal to 1. This technique gives us a simple polynomial-time wmax/wmin-approximation algorithm for MAXMINO where wmax and wmin denote the maximum and minimum weights among all the input edges. Furthermore, we also observe that this approach yields an exact algorithm for the general case of MAXMINO whose running time is polynomial whenever the number of edges having weight larger than wmin is at most logarithmic in the number of vertices. Finally, we show that MAXMINO is solvable in polynomial time if the input is a cactus graph.

UR - http://www.scopus.com/inward/record.url?scp=79955055443&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79955055443&partnerID=8YFLogxK

U2 - 10.1142/S0129054111008246

DO - 10.1142/S0129054111008246

M3 - Article

AN - SCOPUS:79955055443

VL - 22

SP - 583

EP - 601

JO - International Journal of Foundations of Computer Science

JF - International Journal of Foundations of Computer Science

SN - 0129-0541

IS - 3

ER -