A universality theorem for allowable sequences with applications

Abstract

Order types are a well known abstraction of combinatorial properties of a point set. By Mn\"ev's universality theorem for each semi-algebraic set $V$ there is an order type with a realization space that is \emph{stably equivalent} to $V$. We consider realization spaces of \emph{allowable sequences}, a refinement of order types. We show that the realization spaces of allowable sequences are \emph{universal} and consequently deciding the realizability is complete in the \emph{existential theory of the reals} (\ER). This result holds even if the realization space of the order type induced by the allowable sequence is non-empty. Furthermore, we argue that our result is a useful tool for further geometric reductions. We support this by giving \ER-hardness proofs for the realizability of abstract convex geometries and for the recognition problem of visibility graphs of polygons with holes using the hardness result for allowable sequences. This solves two longstanding open problems.