The Edge-Orientation Problem and Its Variants on Some Classes of Graphs

Abstract

This paper considers the Edge-Orientation Problem (EOP) and its variants on weighted undirected graphs. Let G(V, E, C, W) be a graph in which V is the set of n vertices and E is the set of m edges. Each vertex v is associated a real cost C(v) and each edge e = (u, v) is associated with two real weights: W(u → v) and W(u → v). The problem involves assigning an orientation to each edge such that G becomes a directed graph. First, the significance and motivations of the EOP and some practical variants are addressed. Then, nontrivial efficient algorithms for solving its two variants, the Out-Degree EOP and the Vertex-Weighted EOP, on general graphs are designed, respectively. Finally, the EOP is proven to be NP-hard on bipartite graphs and then an O(nlgn) time algorithm on weighted trees is proposed. In general, the used techniques and the algorithmic results of this paper may have great help for implementing the Weighted Fair Queuing (WFQ) on real networks. The major goal of the WFQ is to assign effective weight for each queue or flow that can enhance link utilization. Consequently, the findings here can be easily extended to other classes of graphs, such as cactus graphs, block graphs, interval graphs, etc.