Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class plays a crucial role in the study of geometric problems. Sometimes is referred to as the 'real analog' of NP. While NP is a class of computational problems that deals with existentially quantified variables, deals with existentially quantified variables. In analogy to and in the famous polynomial hierarchy, we study the complexity classes and with variables. Our main interest is the Area Universality problem, where we are given a plane graph , and ask if for each assignment of areas to the inner faces of , there exists a straight-line drawing of realizing the assigned areas. We conjecture that Area Universality is -complete and support this conjecture by proving - and -completeness of two variants of Area Universality. To this end, we introduce tools to prove -hardness and membership. Finally, we present geometric problems as candidates for -complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.
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| http://dx.doi.org/10.1007/s00454-022-00381-0 | DOI Listing |
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