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Sometimes $$\\exists \\mathbb {R}$$<\/jats:tex-math>\n \n \u2203<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is referred to as the \u2018real analog\u2019 of\u00a0NP<\/jats:sc>. While NP<\/jats:sc> is a class of computational problems that deals with existentially quantified boolean<\/jats:italic> variables, $$\\exists \\mathbb {R}$$<\/jats:tex-math>\n \n \u2203<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> deals with existentially quantified real<\/jats:italic> variables. In analogy to $$\\Pi _2^p$$<\/jats:tex-math>\n \n \u03a0<\/mml:mi>\n 2<\/mml:mn>\n p<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\Sigma _2^p$$<\/jats:tex-math>\n \n \u03a3<\/mml:mi>\n 2<\/mml:mn>\n p<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in the famous polynomial hierarchy, we study the complexity classes $$\\forall \\exists \\mathbb {R}$$<\/jats:tex-math>\n \n \u2200<\/mml:mo>\n \u2203<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$ \\exists \\forall \\mathbb {R}$$<\/jats:tex-math>\n \n \u2203<\/mml:mo>\n \u2200<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with real<\/jats:italic> variables. Our main interest is the Area<\/jats:sc>Universality<\/jats:sc> problem, where we are given a plane graph\u00a0G<\/jats:italic>, and ask if for each assignment of areas to the inner faces of G<\/jats:italic>, there exists a straight-line drawing of G<\/jats:italic> realizing the assigned areas. We conjecture that Area<\/jats:sc>Universality<\/jats:sc> is $$\\forall \\exists \\mathbb {R}$$<\/jats:tex-math>\n \n \u2200<\/mml:mo>\n \u2203<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-complete and support this conjecture by proving $$\\exists \\mathbb {R}$$<\/jats:tex-math>\n \n \u2203<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>- and $$\\forall \\exists \\mathbb {R}$$<\/jats:tex-math>\n \n \u2200<\/mml:mo>\n \u2203<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-completeness of two variants of Area<\/jats:sc>Universality<\/jats:sc>. To this end, we introduce tools to prove $$\\forall \\exists \\mathbb {R}$$<\/jats:tex-math>\n \n \u2200<\/mml:mo>\n \u2203<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-hardness and membership. Finally, we present geometric problems as candidates for $$\\forall \\exists \\mathbb {R}$$<\/jats:tex-math>\n \n \u2200<\/mml:mo>\n \u2203<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.<\/jats:p>","DOI":"10.1007\/s00454-022-00381-0","type":"journal-article","created":{"date-parts":[[2022,5,18]],"date-time":"2022-05-18T15:06:58Z","timestamp":1652886418000},"page":"154-188","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Completeness for the Complexity Class $$\\forall \\exists \\mathbb {R}$$ and Area-Universality"],"prefix":"10.1007","volume":"70","author":[{"given":"Michael Gene","family":"Dobbins","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3786-916X","authenticated-orcid":false,"given":"Linda","family":"Kleist","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tillmann","family":"Miltzow","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pawe\u0142","family":"Rz\u0105\u017cewski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,5,18]]},"reference":[{"key":"381_CR1","doi-asserted-by":"crossref","unstructured":"Abrahamsen, M., Adamaszek, A., Miltzow, T.: The art gallery problem is $$\\exists \\mathbb{R}$$-complete. 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