Computational Geometry Column 62

@article{Cardinal2015ComputationalGC,
  title={Computational Geometry Column 62},
  author={Jean Cardinal},
  journal={ACM SIGACT News},
  year={2015},
  volume={46},
  pages={69 - 78}
}
  • J. Cardinal
  • Published 1 December 2015
  • Mathematics
  • ACM SIGACT News
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… 

Figures from this paper

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.

References

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