Protein local structure alignment under the discrete Fréchet distance

Protein structure alignment is a fundamental problem in computational and structural biology. While there has been lots of experimental/heuristic methods and empirical results, very few results are known regarding the algorithmic/complexity aspects of the problem, especially on protein local structure alignment. A well-known measure to characterize the similarity of two polygonal chains is the famous Fréchet distance, and with the application of protein-related research, a related discrete Fréchet distance has been used recently. In this paper, following the recent work of Jiang et al. we investigate the protein local structural alignment problem using bounded discrete Fréchet distance. Given m proteins (or protein backbones, which are 3D polygonal chains), each of length O(n), our main results are summarized as follows: * If the number of proteins, m, is not part of the input, then the problem is NP-complete; moreover, under bounded discrete Fréchet distance it is NP-hard to approximate the maximum size common local structure within a factor of n(1-epsilon). These results hold both when all the proteins are static and when translation/rotation are allowed. * If the number of proteins, m, is a constant, then there is a polynomial time solution for the problem.