Smoothing the gap between NP and ER

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.Comment: 31 pages, 11 figures, FOCS 2020, SICOMP 202

Practical Approaches to Partially Guarding a Polyhedral Terrain

We study the problem of placing guard towers on a terrain such that the terrain can be seen from at least one tower. This problem is important in many applications, and has an extensive history in the literature (known as, e.g., multiple observer siting). In this paper, we consider the problem on polyhedral terrains, and we allow the guards to see only a fixed fraction of the terrain, rather than everything. We experimentally evaluate how the number of required guards relates to the fraction of the terrain that can be covered. In addition, we introduce the concept of dominated guards, which can be used to preprocess the potential guard locations and speed up the subsequent computations. » » Look Inside MyCopy Softcover Edition 24.99 EUR/USD/GBP/CHF Buy Now Other actions Reprints and Permissions Export citatio