The \emph{Separation Lemma} is a simple yet powerful tool, akin to the well-known \emph{Isolation Lemma}, that guarantees the uniqueness of certain set sums. Bandopadhyay et al.\ introduced this lemma to establish lower bounds for the \ALP problem with respect to certain structural parameters, relying on random weight assignments in the process. The lemma's applicability extends well beyond that specific work, especially in proving hardness results. However, while effective, these hardness results inherently rely on probabilistic assumptions. In this work, we give a fully \emph{deterministic} construction for the weight assignment required by the Separation Lemma. We provide formal proofs of correctness, explicit examples, and show how deterministic weights can replace randomized ones, thereby derandomizing existing hardness results for path-packing problems. Our exposition highlights a clear progression from the original randomized foundations to deterministic constructions and their practical implications.

翻译：\emph{分离引理}是一个简单而强大的工具，类似于著名的\emph{隔离引理}，它保证了某些集合和的唯一性。Bandopadhyay 等人引入该引理，用于在特定结构参数下为 \ALP 问题建立下界，其过程依赖于随机权值分配。该引理的适用范围远超该具体工作，尤其体现在证明困难性结果方面。然而，尽管这些困难性结果行之有效，但本质上依赖于概率性假设。在本工作中，我们为分离引理所要求的权值分配提供了一种完全\emph{确定性}的构造。我们给出了正确性的形式化证明、显式示例，并展示了确定性权值如何替代随机化权值，从而为路径打包问题的现有困难性结果实现去随机化。我们的阐述清晰地展示了从原始随机化基础到确定性构造及其实际意义的演进过程。