Cycle flow formulation of optimal network flow problems for centralized and decentralized solvers

When the underlying physical network layer in optimal network flow problems is a large graph, the associated optimization problem has a large set of decision variables. In this paper, we discuss how the cycle basis from graph theory can be used to reduce the size of this decision variable space. The idea is to eliminate the aggregated flow conservation constraint of these problems by explicitly characterizing its solutions in terms of the span of the columns of the transpose of a fundamental cycle basis matrix of the network plus a particular solution. We show that for any given input/output flow vector, a particular solution can be efficiently constructed from tracing any path that connects a source node to a sink node. We demonstrate our results over a minimum cost flow problem as well as an optimal power flow problem with storage and generation at the nodes. We also show that the new formulation of the minimum cost flow problem based on the cycle basis variables is amenable to a distributed solution. In this regard, we apply our method over a distributed alternating direction method of multipliers (ADMM) solution and demonstrate it over a numerical example.