Minimum decomposition of a digital surface into digital plane segments is NP-hard

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 λ 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.