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Abstract

We deal with the problem of finding such an orientation of a given graph that the largest number of edges leaving a vertex (called the outdegree of the orientation) is small. For any $\varepsilon\in(0,1)$ we show an $\tilde{O}(|E(G)|/\varepsilon)$ time algorithm which finds an orientation of an input graph $G$ with outdegree at most $\lceil(1+\varepsilon)d^*\rceil$, where $d^*$ is the maximum density of a subgraph of $G$. It is known that the optimal value of orientation outdegree is $\lceil d^* \rceil$. Our algorithm has applications in constructing labeling schemes, introduced by Kannan et al. and in approximating such graph density measures as arboricity, pseudoarboricity and maximum density. Our results improve over the previous, 2-approximation algorithms by Aichholzer et al. (for orientation / pseudoarboricity), by Arikati et al. (for arboricity) and by Charikar (for maximum density).