Parametric path problems arise independently in diverse domains, ranging from transportation to finance, where they are studied under various assumptions. We formulate a general path problem with relaxed assumptions, and describe how this formulation is applicable in these domains. We study the complexity of the general problem, and a variant of it where preprocessing is allowed. We show that when the parametric weights are linear functions, algorithms remain tractable even under our relaxed assumptions. Furthermore, we show that if the weights are allowed to be non-linear, the problem becomes NP-hard. We also study the mutli-dimensional version of the problem where the weight functions are parameterized by multiple parameters. We show that even with two parameters, the problem is NP-hard.

中文翻译：

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广义参数路径问题

参数路径问题在从运输到金融的不同领域中独立出现，需要在各种假设下进行研究。我们用宽松的假设来表述一般路径问题，并描述这种表述如何在这些领域中适用。我们研究了一般问题的复杂性，以及允许预处理的情况的变体。我们表明，当参数权重为线性函数时，即使在我们宽松的假设下，算法仍然易于处理。此外，我们表明，如果权重被允许是非线性的，那么问题就变成了NP难题。我们还研究了权重函数由多个参数进行参数化的问题的多维度版本。我们证明，即使有两个参数，问题仍然是NP困难的。