A Deterministic Separation Lemma

Abstract

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.