Area-Optimal Simple Polygonalizations: The CG Challenge 2019

Demaine, Erik D.; Fekete, Sándor P.; Keldenich, Phillip; Krupke, Dominik; Mitchell, Joseph S. B.

doi:10.1145/3504000

ACM J. Exp. Algorithmics

2022

DOI: 10.1145/3504000

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Area-Optimal Simple Polygonalizations: The CG Challenge 2019

Erik D. Demaine

Sándor P. Fekete

Phillip Keldenich

et al.

Abstract: We give an overview of theoretical and practical aspects of finding a simple polygon of minimum ( Min-Area ) or maximum ( Max-Area ) possible area for a given set of n points in the plane. Both problems are known to be NP -hard and were the subject of the 2019 Computational Geometry Challenge, which presented the quest of finding good solutions to more than 200 instances, ranging from n = 10 … Show more

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Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time

Eppstein

2024

Algorithmica

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We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in the number of surrounding cycles. As a consequence, we can list the non-crossing Hamiltonian paths or the polygonalizations, in time polynomial in the output size, by filtering the output of simple backtracking algorithms for non-crossing paths or surrounding cycles respectively. We do not assume that the points are in general position. To prove these results we relate the numbers of non-crossing structures to two easily-computed parameters of the point set: the minimum number of points whose removal results in a collinear set, and the number of points interior to the convex hull. These relations also lead to polynomial-time approximation algorithms for the numbers of structures of all four types, accurate to within a constant factor of the logarithm of these numbers.

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Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time

Eppstein

2024

Algorithmica

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Computing Area-Optimal Simple Polygonizations

Fekete¹,

Haas²,

Keldenich³

et al. 2022

ACM J. Exp. Algorithmics

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We consider methods for finding a simple polygon of minimum ( Min-Area ) or maximum ( Max-Area ) possible area for a given set of points in the plane. Both problems are known to be NP-hard; at the center of the recent CG Challenge, practical methods have received considerable attention. However, previous methods focused on heuristic methods, with no proof of optimality. We develop exact methods, based on a combination of geometry and integer programming. As a result, we are able to solve instances of up to \( n=25 \) points to provable optimality. While this extends the range of solvable instances by a considerable amount, it also illustrates the practical difficulty of both problem variants.

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Area-Optimal Simple Polygonalizations: The CG Challenge 2019

Cited by 2 publications

References 28 publications

Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time

Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time

Computing Area-Optimal Simple Polygonizations

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