On the general chain pair simplification problem

The Chain Pair Simplification problem (CPS) was posed by Bereg et al. who were motivated by the problem of efficiently computing and visualizing the structural resemblance between a pair of protein backbones. In this problem, given two polygonal chains of lengths n and m, the goal is to simplify both of them simultaneously, so that the lengths of the resulting simplifications as well as the discrete Fréchet distance between them are bounded. When the vertices of the simplifications are arbitrary (i.e., not necessarily from the original chains), the problem is called General CPS (GCPS). In this paper we consider for the first time the complexity of GCPS under both the discrete Fréchet distance (GCPS-3F) and the Hausdorff distance (GCPS-2H). (In the former version, the quality of the two simplifications is measured by the discrete Fréchet distance, and in the latter version it is measured by the Hausdorff distance.) We prove that GCPS-3F is polynomially solvable, by presenting an eO((n+m)⁶ min{n,m}) time algorithm for the corresponding minimization problem. We also present an O((n + m)⁴) 2-approximation algorithm for the problem. On the other hand, we show that GCPS-2H is NP-complete, and present an approximation algorithm for the problem.