Robust Reductions

TL;DR

This paper re-establishes a key theorem on the power of robust reductions, analyzes their restrictions, and introduces a new strong form of the Karp-Lipton Theorem.

Contribution

It corrects and completes the proof of an earlier theorem on robust reductions and explores the impact of restrictions on their computational power.

Findings

Re-established the optimal separation between robust and nondeterministic strong reductions.

Identified how restrictions like underproductivity and overproductivity affect reduction power.

Derived a new, strong form of the Karp-Lipton Theorem from robust reductions.

Abstract

We continue the study of robust reductions initiated by Gavalda and Balcazar. In particular, a 1991 paper of Gavalda and Balcazar claimed an optimal separation between the power of robust and nondeterministic strong reductions. Unfortunately, their proof is invalid. We re-establish their theorem. Generalizing robust reductions, we note that robustly strong reductions are built from two restrictions, robust underproductivity and robust overproductivity, both of which have been separately studied before in other contexts. By systematically analyzing the power of these reductions, we explore the extent to which each restriction weakens the power of reductions. We show that one of these reductions yields a new, strong form of the Karp-Lipton Theorem.