## Complexity of Some Geometric Problems

### BibTeX

@MISC{Schaefer_complexityof,

author = {Marcus Schaefer},

title = {Complexity of Some Geometric Problems},

year = {}

}

### OpenURL

### Abstract

We show that recognizing intersection graphs of convex sets has the same complexity as deciding truth in the existential theory of the reals. Comparing this to similar results on the rectilinear crossing number and intersection graphs of line segments, we argue that there is a need to recognize this level of complexity as its own class. 1

### Citations

208 |
Crossing number is NP-complete
- Garey, Johnson
- 1983
(Show Context)
Citation Context ...y edge is represented by a straight-line segment and at most two edges intersect in a point. The problem is NP-hard by Garey and Johnson’s original proof that computing the crossing number is NP-hard =-=[6]-=- and it remains NP-hard even if the graph is cubic and a rotation system is given [8, 14]. Bienstock gave an easy and elegant reduction that shows that SIMPLE STRETCHABILITY reduces to deciding whethe... |

129 | Some algebraic and geometric computations in PSPACE - Canny - 1988 |

64 |
The universality theorems on the classification problem of configuration varieties and convex polytopes varieties, in “Topology and geometry - Rohlin seminar
- Mnëv
- 1988
(Show Context)
Citation Context ...e existential theory of the reals. The first combinatorial problem shown complete for ∃R was stretchability of pseudoline arrangements, a result due to Mnëv as a byproduct of his universality theorem =-=[12, 16, 15]-=-. There have been several other problems classified ∗ Some of this work was done in the beautiful library at Oberwolfach during the seminar on Discrete Geometry in September 2008. 1 The complexity cla... |

35 |
Oriented matroids, volume 46 of Encyclopedia of Mathematics and its Applications
- Björner, Vergnas, et al.
- 1993
(Show Context)
Citation Context ...ls, which is BP(NP 0 R) [3]; however, it has not played a major role in that model (as reflected by the complexity of the notation). 1as ∃R-complete since, including the algorithmic Steinitz problem =-=[1]-=-, intersection graphs of line segments [9], and straight-line realizability of abstract topological graphs [11]. Very often, however, ∃R-completeness is not claimed explicitly; for example, in the cas... |

24 |
Constraint networks of topological relations and convexity
- Davis, Gotts, et al.
- 1999
(Show Context)
Citation Context ...her case, A is stretchable. The claim about exponential precision again follows because the reduction we gave is geometric. Remark 5.3. As a simple application, we can give a new proof of a result in =-=[5]-=- which showed that RCC8 (the region-connection calculus, that is, topological inference) together with a predicate for convexity of regions has a consistency problem that is ∃R-complete. We will inclu... |

22 |
Stretchability of pseudolines is NP–hard, Applied Geometry and Discrete Mathematics – The Victor Klee Festschrift
- Shor
- 1991
(Show Context)
Citation Context ...e existential theory of the reals. The first combinatorial problem shown complete for ∃R was stretchability of pseudoline arrangements, a result due to Mnëv as a byproduct of his universality theorem =-=[12, 16, 15]-=-. There have been several other problems classified ∗ Some of this work was done in the beautiful library at Oberwolfach during the seminar on Discrete Geometry in September 2008. 1 The complexity cla... |

19 |
Intersection graphs of segments
- Kratochvíl, Matouˇsek
- 1994
(Show Context)
Citation Context ...he plane has the same complexity as deciding the truth of existential first-order sentences over the real numbers. This connection between geometry and logic is not uncommon: Kratochvíl and Matouˇsek =-=[9]-=-, for example, showed that recognizing intersection graphs of line segments also has the same complexity as the existential theory of the reals (we include a slightly simplified proof of that result),... |

17 |
Computing Sums of Radicals in Polynomial Time
- Blömer
- 1993
(Show Context)
Citation Context ...d a rotation system is given [8, 14]. Bienstock gave an easy and elegant reduction that shows that SIMPLE STRETCHABILITY reduces to deciding whether lin-cr(G) ≤ k, even if G is restricted to be cubic =-=[2]-=-. 4Theorem 3.1 (Bienstock [2]). Computing the rectilinear crossing number of a (cubic) graph is ∃R-complete. There are graphs for which the coordinates of the vertices in an lin-cr-optimal drawing of... |

17 | Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets
- Bürgisser, Cucker
(Show Context)
Citation Context ...fach during the seminar on Discrete Geometry in September 2008. 1 The complexity class is not entirely new, it has a name in the Blum-Shub-Smale model of computing over the reals, which is BP(NP 0 R) =-=[3]-=-; however, it has not played a major role in that model (as reflected by the complexity of the notation). 1as ∃R-complete since, including the algorithmic Steinitz problem [1], intersection graphs of... |

17 | Crossing number is hard for cubic graphs
- Hliněn´y
(Show Context)
Citation Context ... point. The problem is NP-hard by Garey and Johnson’s original proof that computing the crossing number is NP-hard [6] and it remains NP-hard even if the graph is cubic and a rotation system is given =-=[8, 14]-=-. Bienstock gave an easy and elegant reduction that shows that SIMPLE STRETCHABILITY reduces to deciding whether lin-cr(G) ≤ k, even if G is restricted to be cubic [2]. 4Theorem 3.1 (Bienstock [2]). ... |

16 |
Coordinate representation of order types requires exponential storage. Prec. 21st Annu
- Goodman, Pollack, et al.
- 1989
(Show Context)
Citation Context ...ABILITY is NP-hard, sine ∃R-hardness implies NPhardness as we saw above. (Shor [16] also gave a direct proof.) ∃R-hard problems typically require large representations; Goodman, Pollack and Sturmfels =-=[7]-=- showed that there are stretchable arrangements of n pseudolines whose coordinate representation requires 2 cn bits for some constant c > 0. (Equivalently, if we want to draw the endpoints on a grid, ... |

8 |
Tóth: Thirteen problems on crossing numbers, Geombinatorics 9
- Pach, G
- 2000
(Show Context)
Citation Context ...ber is ∃R-complete. So—in a sense— the complexity of the problem is known precisely, but it is not unusual to see the complexity question for the rectilinear crossing number listed as an open problem =-=[13]-=-. There is some good reason for that: we do not know how to capture ∃R well with respect to classical complexity classes: we know that it contains NP (this follows easily from the definition of ∃R; al... |

6 |
Daniel ˇ Stefankovič. Crossing number of graphs with rotation systems
- Pelsmajer, Schaefer
- 2005
(Show Context)
Citation Context ... point. The problem is NP-hard by Garey and Johnson’s original proof that computing the crossing number is NP-hard [6] and it remains NP-hard even if the graph is cubic and a rotation system is given =-=[8, 14]-=-. Bienstock gave an easy and elegant reduction that shows that SIMPLE STRETCHABILITY reduces to deciding whether lin-cr(G) ≤ k, even if G is restricted to be cubic [2]. 4Theorem 3.1 (Bienstock [2]). ... |

4 | Mnëv’s universality theorem revisited
- Richter-Gebert
- 1995
(Show Context)
Citation Context ...e existential theory of the reals. The first combinatorial problem shown complete for ∃R was stretchability of pseudoline arrangements, a result due to Mnëv as a byproduct of his universality theorem =-=[12, 16, 15]-=-. There have been several other problems classified ∗ Some of this work was done in the beautiful library at Oberwolfach during the seminar on Discrete Geometry in September 2008. 1 The complexity cla... |

1 |
Geometric intersection graphs: do short cycles help? (extended abstract
- Kratochvíl, Pergel
- 2007
(Show Context)
Citation Context ...n (in the size of the graph). Remark 4.2. Kratochvíl and Pergel showed that the recognition of intersection graphs of line segments remains NP-hard if the graphs have girth at least k for any fixed k =-=[10]-=-. Can this be extended to ∃R-completeness? We give a slightly simplified proof of Theorem 4.1; the argument will also be used in Theorem 5.1. Lemma 4.3. Suppose we have Jordan curves ℓ, (ℓi) i∈[n], (s... |