The standard pooling problem is a NP-hard subclass of non-convex quadratically-constrained optimization problems that commonly arises in process systems engineering applications. We take a parametric approach to uncovering topological structure and sparsity, focusing on the single quality standard pooling problem in its p-formulation. The structure uncovered in this approach validates Professor Christodoulos A. Floudas' intuition that pooling problems are rooted in piecewise-defined functions. We introduce dominant active topologies under relaxed flow availability to explicitly identify pooling problem sparsity and show that the sparse patterns of active topological structure are associated with a piecewise objective function. Finally, the paper explains the conditions under which sparsity vanishes and where the combinatorial complexity emerges to cross over the P / NP boundary. We formally present the results obtained and their derivations for various specialized single quality pooling problem subclasses.

中文翻译：

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合并问题中的分段参数结构：从稀疏的强多项式解到NP硬度。

标准合并问题是过程系统工程应用程序中常见的非凸二次约束优化问题的NP-hard子类。我们采用参数化方法来揭示拓扑结构和稀疏性，重点是p公式中的单个质量标准池问题。这种方法揭示的结构验证了Christodoulos A. Floudas教授的直觉，即合并问题源于分段定义的函数。我们在宽松的流量可用性下引入主要的主动拓扑，以明确识别池问题的稀疏性，并表明主动拓扑结构的稀疏模式与分段目标函数相关。最后，论文解释了稀疏性消失的条件以及组合复杂性越过P / NP边界的条件。我们正式介绍了获得的结果及其对各种专门的单一质量合并问题子类的推导。