Computational Geometry Column 62

  title={Computational Geometry Column 62},
  author={Jean Cardinal},
  journal={ACM SIGACT News},
  pages={69 - 78}
  • J. Cardinal
  • Published 1 December 2015
  • Mathematics
In this column, we consider natural problems in computational geometry that are polynomialtime equivalent to finding a real solution to a system of polynomial inequalities. Such problems are called ⇿R-complete, and typically involve geometric graphs. We describe the foundations of those completeness proofs, in particular Mnëv's Universality Theorem, as well as some known ⇿R-completeness results, and recent additions to the list. The results shed light on the complex structure of those problems… 

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Geometric Embeddability of Complexes is $\exists \mathbb R$-complete

We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in R is complete for the Existential Theory of the Reals for all d ≥ 3 and k

Optimal Curve Straightening is ∃R-Complete

We prove that the following problem has the same computational complexity as the existential theory of the reals: Given a generic self-intersecting closed curve γ in the plane and an integer m, is

Fine-Grained Complexity of Coloring Unit Disks and Balls

It is proved that fatness is crucial to  obtain subexponential algorithms for coloring and shows that existence of an algorithm coloring an intersection graph of segments using a constant number of colors in time $2^{o(n)}$ already refutes the ETH.

Framework for $\exists \mathbb{R}$-Completeness of Two-Dimensional Packing Problems

It is shown that many natural two-dimensional packing problems are algorithmically equivalent to finding real roots of multivariate polynomials and to deciding whether a given system of polynomial equations and inequalities with integer coefficients has a real solution.

A Framework for Robust Realistic Geometric Computations

It is proved that suitable algorithms can (under smoothed analysis) be robustly executed with expected logarithmic bit-precision and concluded with a real RAM analogue to the Cook-Levin Theorem, which gives an easy proof of ER-membership.

The art gallery problem is ∃ ℝ-complete

It is shown that for every compact semi-algebraic set S⊂ [0,1]2 there exists a polygon with rational coordinates that enforces one of the guards to be at any position p∈ S, in any optimal guarding, which rules out many natural geometric approaches to the art gallery problem.

A universality theorem for allowable sequences with applications

This work shows that the realization spaces of allowable sequences are universal and consequently deciding the realizability is complete in the existential theory of the reals (\ER), and argues that this result is a useful tool for further geometric reductions.

Optimal Curve Straightening is $\exists\mathbb{R}$-Complete.

We prove that the following problem has the same computational complexity as the existential theory of the reals: Given a generic self-intersecting closed curve $\gamma$ in the plane and an integer

The Art Gallery Problem is ∃ℝ-complete

The AGP is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the AGP, and (2) theAGP is not in the complexity class NP unless NP = ∃ℝ.

The Complexity of the Hausdorff Distance

It is shown that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for the complexity class ∀∃<R.hard, which implies that the problem is NP-, co-NP-, ∃Rand ∀R-hard.



Coordinate representation of order types requires exponential storage

We give doubly exponential upper and lower bounds on the size of the smallest grid on which we can embed every planar configuration of n points in general position up to order type. The lower bound

Some algebraic and geometric computations in PSPACE

A PSPACE algorithm for determining the signs of multivariate polynomials at the common zeros of a system of polynomial equations is given and it is shown that the existential theory of the real numbers can be decided in PSPACE.

Complexity of Some Geometric and Topological Problems

It is shown that recognizing intersection graphs of convex sets has the same complexity as deciding truth in the existential theory of the reals, and it is argued that there is a need to recognize this level of complexity as its own class.

The Complexity of Simultaneous Geometric Graph Embedding

Given a collection of planar graphs G1;:::;Gk on the same set V of n vertices, the simultaneous geometric embedding (with mapping) problem, or simply k-SGE, is to nd a set P of n points in the plane

A characterization of ideal polyhedra in hyperbolic $3$-space

The goals of this paper are to provide a characterization of dihedral angles of convex ideal (those with all vertices on the sphere at infinity) polyhedra in H3, and also of those convex polyhedra

Some provably hard crossing number problems

It is shown that any given arrangement can be forced to occur in every minimum-crossing drawing of an appropriate graph, and that there exists no polynomial-time algorithm for producing a straight-line drawing of a graph, with minimum number of crossings from among all such drawings.

Universality Theorems for Inscribed Polytopes and Delaunay Triangulations

It is proved that every primary basic semi-algebraic set is homotopy equivalent to the set of inscribed realizations (up to Möbius transformation) of a polytope, and that all algebraic extensions of Q are needed to coordinatize inscribed polytopes.

Visibility Graphs and Oriented Matroids

For every graph satisfying these conditions it is shown that a uniform rank 3 oriented matroid can be constructed in polynomial time, which if affinely co- ordinatizable would yield a simple polygon whose visibility graph is isomorphic to the given graph.

On Simultaneous Planar Graph Embeddings

Positive and negative results for the problem of simultaneous embedding of planar graphs are presented, and it is shown that any number of outerplanar graphs can be embedded simultaneously on an O(n)xO( n) grid.

Intersection Graphs of Segments

It is proved that the recognition of SEG-graphs is of the same complexity as the decision of solvability of a system of strict polynomial inequalities in the reals, i.e., as the decisions of a special existentially quantified sentence in the theory of real closed fields, and thus it belongs to PSPACE.