Geometric Separators and Their Applications to Protein Folding in the HP-Model

Abstract

We develop a new method for deriving a geometric separator for a set of grid points. Our separator has a linear structure, which can effectively partition a grid graph. For example, we prove that for a grid graph G with a set of n points P in a two-dimensional grid, there is a separator with at most $1.129\sqrt{n}$ points in P that partitions G into two disconnected grid graphs each with at most ${2n\over 3}$ points. Our separator theorem for grid graphs has a significantly smaller upper bound than that was obtained for the general planar graphs in [H. N. Djidjev and S. M. Venkatesan, Acta Inform., 34 (1997), pp. 231–234]. The protein folding problem in the HP-model is to put a sequence, consisting of two characters H and P, in a d-dimensional grid to have maximal number of HH-contacts, where an HH-contact is a pair of non-consecutive H letters that are put at two grid points of distance 1. Our separator is then applied to develop an exact algorithm for the protein-folding problem in the HP-model, which is NP-hard both in both two and three dimensions [B. Berger and T. Leighton, J. Comput. Biol., 5 (1998), pp. 27–40; P. Crescenzi et al., J. Comput. Biol., 5 (1998), pp. 423–465]. We design a $2^{O(n^{1-{1\over d}}\log n)}$ time algorithm for the d-dimensional protein folding problem in the HP-model. In particular, our algorithm has $O(2^{6.145\sqrt{n}\log n})$ and $O(2^{6.913n^{2\over 3}\log n})$ computational time in two and three dimensions, respectively.