# Degree-Constrained Graph Orientation: Maximum Satisfaction and Minimum Violation

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## Abstract

A *degree-constrained graph orientation* of an undirected graph *G* is an assignment of a direction to each edge in *G* such that the outdegree of every vertex in the resulting directed graph satisfies a specified lower and/or upper bound. Such graph orientations have been studied for a long time and various characterizations of their existence are known. In this paper, we consider four related optimization problems introduced in reference (Asahiro et al. LNCS **7422**, 332–343 (2012)): For any fixed non-negative integer *W*, the problems MAX *W*-LIGHT, MIN *W*-LIGHT, MAX *W*-HEAVY, and MIN *W*-HEAVY take as input an undirected graph *G* and ask for an orientation of *G* that maximizes or minimizes the number of vertices with outdegree at most *W* or at least *W*. As shown in Asahiro et al. LNCS **7422**, 332–343 (2012)).

## Keywords

Graph orientation Degree constraint (In)approximability Submodular function Greedy algorithm## Notes

### Acknowledgments

This work was supported by KAKENHI grant numbers 23500020, 25104521, 25330018, 26330017, and 26540005 and The Hakubi Project at Kyoto University. The authors would like to thank the anonymous reviewers for their detailed comments and suggestions that helped to improve the presentation of the paper, and Peter Floderus for pointing out an error in one of the figures.

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