Smoothing the gap between 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 ERcomplete 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 realvalued 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 realvalued variant of SAT. In this paper we prove a real RAM analogue to the CookLevin theorem which shows that ER membership is equivalent to having a verification algorithm that runs in polynomialtime on a real RAM. This gives an easy proof of ERmembership, as verification algorithms on a real RAM are much more versatile than ETRformulas. We use this result to construct a framework to study ERcomplete problems under smoothed analysis. We show that for a wide class of ERcomplete problems, its witness can be represented with logarithmic inputprecision 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 inputprecision) consists of contrived input. We apply our framework to wellstudied ERcomplete recognition problems which have the exponential bit phenomenon such as the recognition of realizable order types or the Steinitz problem in fixed dimension.
 Publication:

arXiv eprints
 Pub Date:
 December 2019
 arXiv:
 arXiv:1912.02278
 Bibcode:
 2019arXiv191202278E
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Computational Complexity;
 Computer Science  Discrete Mathematics;
 Computer Science  Data Structures and Algorithms;
 Mathematics  Numerical Analysis
 EPrint:
 43 pages, 13 figures, accepted to FOCS 2020