A Deterministic Separation Lemma

2026-05-27

AI Summary

This work addresses the limitation in hardness proofs for path-packing problems that rely on randomized weight assignments by introducing the first deterministic variant of the isolation lemma. Combining combinatorial constructions with algebraic techniques, the authors explicitly design deterministic weights and employ formal verification to guarantee their correctness. This approach successfully eliminates probabilistic assumptions from several known hardness results, replacing randomized assignments with fully deterministic ones. Consequently, it achieves complete derandomization of the corresponding complexity lower-bound proofs and significantly broadens the applicability of the isolation lemma within theoretical computer science.

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.

Problem

• Separation Lemma
• deterministic construction
• derandomization
• hardness results
• path-packing problems

Innovation

• Deterministic Construction
• Separation Lemma
• Derandomization
• Path-Packing Problems
• Hardness Results