Existential theory of the reals
The decision problem for the existential theory of the reals is the problem of finding an algorithm that decides, for each such formula, whether it is true or false. This decision problem is NP-hard and lies in PSPACE. Thus, it has significantly lower complexity than Alfred Tarski's quantifier elimination procedure for deciding statements in the first-order theory of the reals without the restriction to existential quantifiers.
- recognition of intersection graphs of line segments in the plane (that is, given an undirected graph, determining whether there is a set of line segments that has an isomorphic intersection graph);
- recognition of unit disk graphs (again, given only the graph itself as input);
- recognition of unit distance graphs, and testing whether the dimension or Euclidean dimension of a graph is at most a given value.
- recognition of intersection graphs of convex sets in the plane;
- stretchability of pseudolines (that is, given a family of curves in the plane, determining whether they are homeomorphic to a line arrangement);
- determining the rectilinear crossing number of a graph (the minimum number of pairs of edges that cross in any drawing with edges drawn as straight line segments in the plane);
- the algorithmic Steinitz problem (given a lattice, determine whether it is the face lattice of a convex polytope);
- testing whether a given graph can be drawn in the plane with straight line edges and with a given set of edge pairs as its crossings; and
- testing whether a 4-regular graph whose edges are colored with four colors has a drawing with edges as straight line segments of four slopes, with the slopes representing the colors in the coloring.
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