Distributed Algorithms to Solve the FOP Issues on the Weighted Convex-Split Networks

This paper discusses the Flow-Orientation Problem (FOP), which assigns orientations of all links of an undirected network to obtain a directed network for meeting some flow optimization measurements. First, we describe the background and define three related FOP issues: MDFOP, MDSFOP, and MDDFOP. We discuss the complexity of the FOP on general networks, and also list the essences of convex-split networks. These networks are to be either un-weighted or weighted. Then, we propose the algorithm for solving the MDFOP issue on weighted convex-split networks. Similarly, we can extend the MDFOP approach to solve the MDSFOP and MDDFOP issues. Suppose that Γ is any flow-orientation of a network N. Let λ(Γ) and μ(Γ) be the maximum out-degree and the minimum out-degree, respectively, when N is unweighted. In another, let ε(Г) = maxx∈V {C(v) + ∑(z,x),z→xW(z,x)}, π(Г)=minx∈V{C(v) + ∑(z,x),z→xW(z,x)}, and θ(Г) = ∑x,y∈V{(C(x) + ∑x→zW(x,z))-(C(y) + ∑y→zW(y,z))} when N is weighted. The main purpose of various FOP issues is to minimize λ(Γ) μ(Γ), ε(Γ) π(Γ), θ(Γ). Finally, our findings can be applied to enhance performance of link flow and load balancing on networks.