Abstract
A convex partition of a point set P in the plane is a planar subdivision of the convex hull of P whose edges are segments with both endpoints in P and such that all internal faces are empty convex polygons. In the Minimum Convex Partition Problem (mcpp) one seeks to find a convex partition with the least number of faces. The complexity of the problem is still open and so far no computational tests have been reported. In this paper, we formulate the mcpp as an integer program that is used both to solve the problem exactly and to design heuristics. Thorough experiments are conducted to compare these algorithms in terms of solution quality and runtime, showing that the duality gap is decidedly small and grows quite slowly with the instance size.
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Notes
- 1.
Here, \((i+\overrightarrow{u})\) denotes any point obtained by a translation of i in the direction \(\overrightarrow{u}\).
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Acknowledgments
This research was financed in part by grants from Conselho Nacional de Desenvolvimento CientÃfico e Tecnológico (CNPq) #311140/2014-9, #304727/2014-8, Coordenação de Aperfeiçoamento do Pessoal do Ensino Superior - Brasil (Capes) #001 and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) #2014/12236-1, #2017/12523-9, #2018/14883-5.
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Barboza, A.S., de Souza, C.C., de Rezende, P.J. (2019). Minimum Convex Partition of Point Sets. In: Heggernes, P. (eds) Algorithms and Complexity. CIAC 2019. Lecture Notes in Computer Science(), vol 11485. Springer, Cham. https://doi.org/10.1007/978-3-030-17402-6_3
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DOI: https://doi.org/10.1007/978-3-030-17402-6_3
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