Computational Geometry Column 62

Abstract

In this column, we consider natural problems in computational geometry that are polynomialtime equivalent to finding a real solution to a system of polynomial inequalities. Such problems are called ⇿R-complete, and typically involve geometric graphs. We describe the foundations of those completeness proofs, in particular Mnëv's Universality Theorem, as well as some known ⇿R-completeness results, and recent additions to the list. The results shed light on the complex structure of those problems, beyond mere NP-hardness.