We show that the feasibility of a booking in the European entry-exit gas market can be decided in polynomial time on single-cycle networks that are passive, i.e., do not contain controllable elements. The feasibility of a booking can be characterized by solving polynomially many nonlinear potential-based flow models for computing so-called potential-difference maximizing load flow scenarios. We thus analyze the structure of these models and exploit both the cyclic graph structure as well as specific properties of potential-based flows. This enables us to solve the decision variant of the nonlinear potential-difference maximization by reducing it to a system of polynomials of constant dimension that is independent of the cycle's size. This system of fixed dimension can be handled with tools from real algebraic geometry to derive a polynomial-time algorithm. The characterization in terms of potential-difference maximizing load flow scenarios then leads to a polynomial-time algorithm for deciding the feasibility of a booking. Our theoretical results extend the existing knowledge about the complexity of deciding the feasibility of bookings from trees to single-cycle networks.

中文翻译：

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在无源网络的情况下，决定在欧洲天然气市场上预订的可行性是在 P 中

我们表明，在欧洲进出天然气市场预订的可行性可以在多项式时间内在被动的单周期网络上决定，即不包含可控元素。预订的可行性可以通过多项式求解许多非线性基于势的流模型来表征，以计算所谓的势差最大化潮流场景。因此，我们分析了这些模型的结构，并利用了循环图结构以及基于势的流的特定属性。这使我们能够通过将非线性势差最大化的决策变体简化为与循环大小无关的常数维多项式系统来求解。这个固定维度的系统可以用来自实代数几何的工具来处理，以推导出多项式时间算法。根据潜在差异最大化潮流场景的表征然后导致用于决定预订可行性的多项式时间算法。我们的理论结果扩展了关于确定从树到单周期网络的预订可行性的复杂性的现有知识。