Combinatorial acyclicity models for potential‐based flows

Abstract

Potential-based flows constitute a basic model to represent physical behavior in networks. Under natural assumptions, the flow in such networks must be acyclic. The goal of this article is to exploit this property for the solution of corresponding optimization problems. To this end, we introduce several combinatorial models for acyclic flows, based on binary variables for flow directions. We compare these models and introduce a particular model that tries to capture acyclicity together with the supply/demand behavior. We analyze properties of this model, including variable fixing rules. Our computational results show that the usage of the corresponding constraints speeds up solution times by about a factor of 3 on average and a speed-up of a factor of almost 5 for the time to prove optimality.

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“…One of the main differences between the physically more accurate potential-based flows and the capacitated linear flows is that in a passive potential-based flow model, no cyclic flows are possible, which is not necessarily the case for capacitated linear flows. This structural property of potential-based flows can be algorithmically exploited; see e.g., Habeck and Pfetsch (2021). However, the link between node potentials and arc flows is usually given by nonlinear constraints.…”