Approximation Scheme for Lowest Outdegree Orientation and Graph Density Measures

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 e∈(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 ⌈(1+e)d*⌉, where d* is the maximum density of a subgraph of G. It is known that the optimal value of orientation outdegree is ⌈d* ⌉. Our algorithm has applications in constructing labeling schemes, introduced by Kannan et al. in [18] 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. [1] (for orientation / pseudoarboricity), by Arikati et al. [3] (for arboricity) and by Charikar [5] (for maximum density).