On protein structure alignment under distance constraint

In this paper we study the protein structure comparison problem where each protein is modeled as a sequence of 3D points, and a contact edge is placed between every two of these points that are sufficiently close. Given two proteins represented this way, our problem is to find a subset of points from each protein, and a bijective matching of points between these two subsets, with the objective of maximizing either (A) the size of the subsets (the LCP problem), or (B) the number of edges that exist simultaneously in both subsets (the CMO problem), under the requirement that only points within a specified proximity can be matched. It is known that the general CMO problem (without the proximity requirement) is hard to approximate. However, with the proximity requirement, it is known that if a minimum inter-residue distance is imposed on the input, approximate solutions can be efficiently obtained. In this paper we mainly show that the CMO problem under these conditions: (1) is NP-hard, but (2) allows a PTAS. The rest of this paper shows algorithms for the LCP problem which improve on known results.