The Complexity of Drawing a Graph in a Polygonal Region

We prove that the following problem is complete for the existential theory of the reals: Given a planar graph and a polygonal region, with some vertices of the graph assigned to points on the boundary of the region, place the remaining vertices to create a planar straight-line drawing of the graph inside the region. This establishes a wider context for the NP-hardness result by Patrignani on extending partial planar graph drawings. Our result is one of the first showing that a problem of drawing planar graphs with straight-line edges is hard for the existential theory of the reals. The complexity of the problem is open in the case of a simply connected region. We also show that, even for integer input coordinates, it is possible that drawing a graph in a polygonal region requires some vertices to be placed at irrational coordinates. By contrast, the coordinates are known to have bounded bit complexity for the special case of a convex region, or for drawing a path in any polygonal region. In addition, we prove a Mnëv-type universality result—loosely speaking, that the solution spaces of instances of our graph drawing problem are equivalent, in a topological and algebraic sense, to bounded algebraic varieties

Completeness for the Complexity Class ∀ ∃ R and Area-Universality

Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class ∃R plays a crucial role in the study of geometric problems. Sometimes ∃R is referred to as the ‘real analog’ of NP. While NP is a class of computational problems that deals with existentially quantified boolean variables, ∃R deals with existentially quantified real variables. In analogy to Πp2 and Σp2 in the famous polynomial hierarchy, we study the complexity classes ∀∃R and ∃∀R with real variables. Our main interest is the AREA UNIVERSALITY problem, where we are given a plane graph G, and ask if for each assignment of areas to the inner faces of G, there exists a straight-line drawing of G realizing the assigned areas. We conjecture that AREA UNIVERSALITY is ∀∃R -complete and support this conjecture by proving ∃R - and ∀∃R -completeness of two variants of AREA UNIVERSALITY. To this end, we introduce tools to prove ∀∃R -hardness and membership. Finally, we present geometric problems as candidates for ∀∃R -complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability

Dynamic Toolbox for ETRINV

Recently, various natural algorithmic problems have been shown to be $\exists \mathbb{R}$-complete. The reduction relied in many cases on the $\exists \mathbb{R}$-completeness of the problem ETR-INV, which served as a useful intermediate problem. Often some strengthening and modification of ETR-INV was required. This lead to a cluttered situation where no paper included all the previous details. Here, we give a streamlined exposition in a self-contained manner. We also explain and prove various universality results regarding ETR-INV. These notes should not be seen as a research paper with new results. However, we do describe some refinements of earlier results which might be useful for future research. We plan to extend and update this exposition as seems fit

Dynamic Toolbox for ETRINV

Recently, various natural algorithmic problems have been shown to be $\exists \mathbb{R}$-complete. The reduction relied in many cases on the $\exists \mathbb{R}$-completeness of the problem ETR-INV, which served as a useful intermediate problem. Often some strengthening and modification of ETR-INV was required. This lead to a cluttered situation where no paper included all the previous details. Here, we give a streamlined exposition in a self-contained manner. We also explain and prove various universality results regarding ETR-INV. These notes should not be seen as a research paper with new results. However, we do describe some refinements of earlier results which might be useful for future research. We plan to extend and update this exposition as seems fit