Counting Polygon Triangulations is Hard

Eppstein, David

doi:10.1007/s00454-020-00251-7

Cited by 6 publications

(2 citation statements)

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“…For those values of k , the existential question is therefore settled. On the other hand, as stated in [ 21 ], it is not even known whether counting the number of plane perfect matchings on a set of n points is hard, that is, #P-complete.…”

Section: Discussionmentioning

confidence: 99%

Perfect Matchings with Crossings

et al. 2023

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For sets of n points, n even, in general position in the plane, we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least $$C_{n/2}$$ C n / 2 different plane perfect matchings, where $$C_{n/2}$$ C n / 2 is the n/2-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings. We show the following results. (1) For every $$k\le \frac{1}{64}n^2-\frac{35}{32}n\sqrt{n}+\frac{1225}{64}n$$ k ≤ 1 64 n 2 - 35 32 n n + 1225 64 n , any set with n points, n sufficiently large, admits a perfect matching with exactly k crossings. (2) There exist sets of n points where every perfect matching has at most $$\frac{5}{72}n^2-\frac{n}{4}$$ 5 72 n 2 - n 4 crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n. (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for $$k=0,1,2$$ k = 0 , 1 , 2 , and maximize the number of perfect matchings with $$\left( {\begin{array}{c}n/2\\ 2\end{array}}\right) $$ n / 2 2 crossings and with $${\left( {\begin{array}{c}n/2\\ 2\end{array}}\right) }\!-\!1$$ n / 2 2 - 1 crossings.

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Section: Discussionmentioning

confidence: 99%

Perfect Matchings with Crossings

et al. 2023

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“…Given n points in the plane, is finding the number of polygons with vertices at the n points also #P-complete? See [10] for related results and further questions on hardness of counting problem in discrete geometry.…”

Section: Future Workmentioning

confidence: 99%

Existence and Hardness of Conveyor Belts

Baird

Billey

Demaine³

et al. 2020

Electron. J. Combin.

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An open problem of Manuel Abellanas asks whether every set of disjoint closed unit disks in the plane can be connected by a conveyor belt, which means a tight simple closed curve that touches the boundary of each disk, possibly multiple times. We prove three main results: For unit disks whose centers are both $x$-monotone and $y$-monotone, or whose centers have $x$-coordinates that differ by at least two units, a conveyor belt always exists and can be found efficiently. It is NP-complete to determine whether disks of arbitrary radii have a conveyor belt, and it remains NP-complete when we constrain the belt to touch disks exactly once. Any disjoint set of $n$ disks of arbitrary radii can be augmented by $O(n)$ "guide" disks so that the augmented system has a conveyor belt touching each disk exactly once, answering a conjecture of Demaine, Demaine, and Palop.

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Perfect Matchings with Crossings

Aichholzer

Fabila-Monroy

Kindermann

et al. 2022

Lecture Notes in Computer Science

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For sets of n = 2m points in general position in the plane we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least Cm different plane perfect matchings, where Cm is the m-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings.We show the following results. (1) For every k ≤ 1 64 n 2 − O(n √ n), * Some results of this work have been published in the proceedings of IWOCA 2022 [1]. Early results of this work have been presented at the "Computational Geometry: Young Researchers Forum" in 2021 [2]. Perfect Matchings with Crossingsany set with an even number of n points, n sufficiently large, admits a perfect matching with exactly k crossings. ( 2) There exist sets of n points (n even) where every perfect matching has fewer than 5 72 n 2 crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n. (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for k = 0, 1, 2, and maximize the number of perfect matchings with n/2 2 crossings and with n/2 2 − 1 crossings.

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Counting Polygon Triangulations is Hard

Abstract: We prove that it is #P-complete to count the triangulations of a (non-simple) polygon.

Cited by 6 publications

References 40 publications

Perfect Matchings with Crossings

Perfect Matchings with Crossings

Existence and Hardness of Conveyor Belts

Perfect Matchings with Crossings

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