Cycle basis distributed ADMM solution for optimal network flow problem over biconnected graphs

Minimum cost network flow problems appear in a wide range of applications including general network optimization problems, transportation systems, electricity networks and information networks. In these applications, the underlying physical layer of the systems can generate a very large graph resulting in an optimization problem with a large decision variable space. Various arc and node based cyber layer layouts have been proposed to solve this problem in a distributed manner. In this paper, for a physical layer network of n nodes and m arcs with biconnected graph topology, we take advantage of the cycle basis concept in the graph theory to reduce the search variables of an optimal network flow from m to m - n + 1 variables. We show that our proposed new formulation of the optimal network flow problem is amenable to a distributed solution using alternating direction method of multipliers (ADMM) method via a cyber layer whose nodes are defined based on the fundamental cycles of a cycle basis of the physical layer graph. We demonstrate the performance of our algorithm through a numerical example.