A $2^{O(n^{1-{1\over d}}\log n)}$ Time Algorithm for d-Dimensional Protein Folding in the HP-Model

Abstract

The protein folding problem in the HP-model is NP-hard in both 2D and 3D [4,6]. The problem is to put a sequence, consisting of two characters H and P, on a d-dimensional grid to have the maximal number of HH contacts. We design a $2^{O(n^{1-{1\over d}}\log n)}$ time algorithm for d-dimensional protein folding in the HP-model. In particular, our algorithm has $O(2^{6.145\sqrt{n}\log n})$ and $O(2^{4.306n^{2\over 3}\log n})$ computational time in 2D and 3D respectively. The algorithm is derived via our separator theorem for points on a d-dimensional grid. For example, for a set of n points P on a 2-dimensional grid, there is a separator with at most $1.129\sqrt{n}$ points that partitions P into two sides with at most $({2\over 3})n$ points on each side. Our separator theorem for grid points has a greatly reduced upper bound than that for the general planar graph [2].