### Abstract

Let P be a set of n points in the plane. A crossing-free structure on P is a straight-edge planar graph with vertex set in P. Examples of crossing-free structures include triangulations of P, and spanning cycles of P, also known as polygonalizations of P, among others. There has been a large amount of research trying to bound the number of such structures. In particular, bounding the number of triangulations spanned by P has received considerable attention. It is currently known that every set of n points has at most O(30n) and at least Ω(2.43n) triangulations. However, much less is known about the algorithmic problem of counting crossing-free structures of a given set P. For example, no algorithm for counting triangulations is known that, on all instances, performs faster than enumerating all triangulations. In this paper we develop a general technique for computing the number of crossing-free structures of an input set P. We apply the technique to obtain algorithms for computing the number of triangulations and spanning cycles of P. The running time of our algorithms is upper bounded by n ^{O(k)}, where k is the number of onion layers of P. In particular, we show that our algorithm for counting triangulations is not slower than O(3.1414n). Given that there are several well-studied configurations of points with at least Ω(3.464n) triangulations, and some even with Ω(8n) triangulations, our algorithm is the first to asymptotically outperform any enumeration algorithm for such instances. In fact, it is widely believed that any set of n points must have at least Ω(3.464n) triangulations. If this is true, then our algorithm is strictly sub-linear in the number of triangulations counted. We also show that our techniques are general enough to solve the restricted triangulation counting problem, which we prove to be W [2]-hard in the parameter k. This implies a "no free lunch" result: In order to be fixed-parameter tractable, our general algorithm must rely on additional properties that are specific to the considered class of structures.

Original language | English (US) |
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Title of host publication | Proceedings of the 28th Annual Symposuim on Computational Geometry, SCG 2012 |

Pages | 61-68 |

Number of pages | 8 |

DOIs | |

State | Published - Jul 23 2012 |

Event | 28th Annual Symposuim on Computational Geometry, SCG 2012 - Chapel Hill, NC, United States Duration: Jun 17 2012 → Jun 20 2012 |

### Other

Other | 28th Annual Symposuim on Computational Geometry, SCG 2012 |
---|---|

Country | United States |

City | Chapel Hill, NC |

Period | 6/17/12 → 6/20/12 |

### Fingerprint

### Keywords

- Counting
- Onion layers
- Parameterized complexity
- Triangulations

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

### Cite this

*Proceedings of the 28th Annual Symposuim on Computational Geometry, SCG 2012*(pp. 61-68) https://doi.org/10.1145/2261250.2261259

**Counting crossing-free structures.** / Alvarez, Victor; Bringmann, Karl; Curticapean, Radu; Ray, Saurabh.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 28th Annual Symposuim on Computational Geometry, SCG 2012.*pp. 61-68, 28th Annual Symposuim on Computational Geometry, SCG 2012, Chapel Hill, NC, United States, 6/17/12. https://doi.org/10.1145/2261250.2261259

}

TY - GEN

T1 - Counting crossing-free structures

AU - Alvarez, Victor

AU - Bringmann, Karl

AU - Curticapean, Radu

AU - Ray, Saurabh

PY - 2012/7/23

Y1 - 2012/7/23

N2 - Let P be a set of n points in the plane. A crossing-free structure on P is a straight-edge planar graph with vertex set in P. Examples of crossing-free structures include triangulations of P, and spanning cycles of P, also known as polygonalizations of P, among others. There has been a large amount of research trying to bound the number of such structures. In particular, bounding the number of triangulations spanned by P has received considerable attention. It is currently known that every set of n points has at most O(30n) and at least Ω(2.43n) triangulations. However, much less is known about the algorithmic problem of counting crossing-free structures of a given set P. For example, no algorithm for counting triangulations is known that, on all instances, performs faster than enumerating all triangulations. In this paper we develop a general technique for computing the number of crossing-free structures of an input set P. We apply the technique to obtain algorithms for computing the number of triangulations and spanning cycles of P. The running time of our algorithms is upper bounded by n O(k), where k is the number of onion layers of P. In particular, we show that our algorithm for counting triangulations is not slower than O(3.1414n). Given that there are several well-studied configurations of points with at least Ω(3.464n) triangulations, and some even with Ω(8n) triangulations, our algorithm is the first to asymptotically outperform any enumeration algorithm for such instances. In fact, it is widely believed that any set of n points must have at least Ω(3.464n) triangulations. If this is true, then our algorithm is strictly sub-linear in the number of triangulations counted. We also show that our techniques are general enough to solve the restricted triangulation counting problem, which we prove to be W [2]-hard in the parameter k. This implies a "no free lunch" result: In order to be fixed-parameter tractable, our general algorithm must rely on additional properties that are specific to the considered class of structures.

AB - Let P be a set of n points in the plane. A crossing-free structure on P is a straight-edge planar graph with vertex set in P. Examples of crossing-free structures include triangulations of P, and spanning cycles of P, also known as polygonalizations of P, among others. There has been a large amount of research trying to bound the number of such structures. In particular, bounding the number of triangulations spanned by P has received considerable attention. It is currently known that every set of n points has at most O(30n) and at least Ω(2.43n) triangulations. However, much less is known about the algorithmic problem of counting crossing-free structures of a given set P. For example, no algorithm for counting triangulations is known that, on all instances, performs faster than enumerating all triangulations. In this paper we develop a general technique for computing the number of crossing-free structures of an input set P. We apply the technique to obtain algorithms for computing the number of triangulations and spanning cycles of P. The running time of our algorithms is upper bounded by n O(k), where k is the number of onion layers of P. In particular, we show that our algorithm for counting triangulations is not slower than O(3.1414n). Given that there are several well-studied configurations of points with at least Ω(3.464n) triangulations, and some even with Ω(8n) triangulations, our algorithm is the first to asymptotically outperform any enumeration algorithm for such instances. In fact, it is widely believed that any set of n points must have at least Ω(3.464n) triangulations. If this is true, then our algorithm is strictly sub-linear in the number of triangulations counted. We also show that our techniques are general enough to solve the restricted triangulation counting problem, which we prove to be W [2]-hard in the parameter k. This implies a "no free lunch" result: In order to be fixed-parameter tractable, our general algorithm must rely on additional properties that are specific to the considered class of structures.

KW - Counting

KW - Onion layers

KW - Parameterized complexity

KW - Triangulations

UR - http://www.scopus.com/inward/record.url?scp=84863889850&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84863889850&partnerID=8YFLogxK

U2 - 10.1145/2261250.2261259

DO - 10.1145/2261250.2261259

M3 - Conference contribution

AN - SCOPUS:84863889850

SN - 9781450312998

SP - 61

EP - 68

BT - Proceedings of the 28th Annual Symposuim on Computational Geometry, SCG 2012

ER -