Minimum Decomposition of a Digital Surface into Digital Plane Segments is NP-Hard

Isabelle Sivignon 1 David Coeurjolly 1
1 M2DisCo - Geometry Processing and Constrained Optimization
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
Abstract : This paper deals with the complexity of the decomposition of a digital surface into digital plane segments (DPS for short). We prove that the decision problem (does there exist a decomposition with less than $\lambda$ DPS ?) is NP-complete, and thus that the optimisation problem (finding the minimum number of DPS) is NP-hard. The proof is based on a polynomial reduction of any instance of the well-known 3-SAT problem to an instance of the digital surface decomposition problem. A geometric model for the 3-SAT problem is proposed.
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Isabelle Sivignon, David Coeurjolly. Minimum Decomposition of a Digital Surface into Digital Plane Segments is NP-Hard. Discrete Applied Mathematics, Elsevier, 2008, 157 (3), pp.558--570. ⟨10.1016/j.dam.2008.05.028⟩. ⟨hal-00350145⟩

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