From Electrical Power Flows to Unsplittabe Flows: A QPTAS for OPF with Discrete Demands in Line Distribution Networks

The {\it AC Optimal Power Flow} (OPF) problem is a fundamental problem in power systems engineering which has been known for decades. It is a notoriously hard problem due mainly to two reasons: (1) non-convexity of the power flow constraints and (2) the (possible) existence of discrete power injection constraints. Recently, sufficient conditions were provided for certain convex relaxations of OPF to be exact in the continuous case, thus allowing one to partially address the issue of non-convexity. In this paper we make a first step towards addressing the combinatorial issue. Namely, by establishing a connection to the well-known {\it unsplittable flow problem} (UFP), we are able to generalize known techniques for the latter problem to provide approximation algorithms for OPF with discrete demands. As an application, we give a quasi-polynomial time approximation scheme for OPF in line networks under some mild assumptions and a single generation source. We believe that this connection can be further leveraged to obtain approximation algorithms for more general settings, such as multiple generation sources and tree networks