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 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.
| Original language | English |
|---|---|
| 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
Graph orientation to maximize the minimum weighted outdegree. / Asahiro, Yuichi; Jansson, Jesper; Miyano, Eiji; Ono, Hirotaka.
In: International Journal of Foundations of Computer Science, Vol. 22, No. 3, 01.04.2011, p. 583-601.Research output: Contribution to journal › Article
}
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.
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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 -