Smoothing the gap between NP and ER

TL;DR

This paper establishes a real RAM analogue of the Cook-Levin theorem for ER, enabling polynomial-time verification of ER problems and demonstrating that high-precision witnesses are typically due to contrived inputs, thus bridging NP and ER.

Contribution

It proves a real RAM version of the Cook-Levin theorem for ER and develops a framework for smoothed analysis of ER-complete problems, showing most witnesses require only logarithmic precision.

Findings

ER membership is equivalent to polynomial-time verification on a real RAM.

Most ER witnesses can be represented with logarithmic input-precision under smoothed analysis.

Contrived inputs are responsible for high-precision witnesses, bridging NP and ER.

Abstract

We study algorithmic problems that belong to the complexity class of the existential theory of the reals (ER). A problem is ER-complete if it is as hard as the problem ETR and if it can be written as an ETR formula. Traditionally, these problems are studied in the real RAM, a model of computation that assumes that the storage and comparison of real-valued numbers can be done in constant space and time, with infinite precision. The complexity class ER is often called a real RAM analogue of NP, since the problem ETR can be viewed as the real-valued variant of SAT. In this paper we prove a real RAM analogue to the Cook-Levin theorem which shows that ER membership is equivalent to having a verification algorithm that runs in polynomial-time on a real RAM. This gives an easy proof of ER-membership, as verification algorithms on a real RAM are much more versatile than ETR-formulas. We use this result to construct a framework to study ER-complete problems under smoothed analysis. We show that for a wide class of ER-complete problems, its witness can be represented with logarithmic input-precision by using smoothed analysis on its real RAM verification algorithm. This shows in a formal way that the boundary between NP and ER (formed by inputs whose solution witness needs high input-precision) consists of contrived input. We apply our framework to well-studied ER-complete recognition problems which have the exponential bit phenomenon such as the recognition of realizable order types or the Steinitz problem in fixed dimension.