Abstract
We study the problem of orienting the edges of a graph such that the minimum over all the vertices of the absolute difference between the outdegree and the indegree of a vertex is maximized. We call this minimum the imbalance of the orientation, i.e. the higher it gets, the more imbalanced the orientation is. We study this problem denoted by MaxIm. We first present different characterizations of the graphs for which the optimal objective value of MaxIm is zero. Next we show that it is generally NP-complete and cannot be approximated within a ratio of \(\frac{1}{2}+\varepsilon \) for any constant \(\varepsilon >0\) in polynomial time unless \(\mathtt P =\mathtt NP \) even if the minimum degree of the graph \(\delta \) equals 2. Finally we describe a polynomial-time approximation algorithm whose ratio is equal to \(\frac{1}{2}\) for graphs where \(\delta \equiv 0[4]\) or \(\delta \equiv 1[4]\) and \((\frac{1}{2}-\frac{1}{\delta })\) for general graphs.
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Ben-Ameur, W., Glorieux, A., Neto, J. (2015). On the Most Imbalanced Orientation of a Graph. In: Xu, D., Du, D., Du, D. (eds) Computing and Combinatorics. COCOON 2015. Lecture Notes in Computer Science(), vol 9198. Springer, Cham. https://doi.org/10.1007/978-3-319-21398-9_2
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DOI: https://doi.org/10.1007/978-3-319-21398-9_2
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