On the Chain Pair Simplification Problem

Abstract

The problem of efficiently computing and visualizing the structural resemblance between a pair of protein backbones in 3D has led Bereg et al. [4] to pose the Chain Pair Simplification problem (CPS). In this problem, given two polygonal chains A and B of lengths m and n, respectively, one needs to simplify them simultaneously, such that each of the resulting simplified chains, $A'$ and $B'$, is of length at most k and the discrete Fréchet distance between $A'$ and $B'$ is at most $\delta$, where k and $\delta$ are given parameters. In this paper we study the complexity of CPS under the discrete Fréchet distance (CPS-3F), i.e., where the quality of the simplifications is also measured by the discrete Fréchet distance. Since CPS-3F was posed in 2008, its complexity has remained open. In this paper, we prove that CPS-3F is actually polynomially solvable, by presenting an $O(m^2n^2\min \{m,n\})$ time algorithm for the corresponding minimization problem. On the other hand, we prove that if the vertices of the chains have integral weights then the problem is weakly NP-complete.

A complete version including the one-sided cases and empirical results can be found at http://arxiv.org/abs/1409.2457.

Work by O. Filtser has been partially supported by the Lynn and William Frankel Center for Computer Sciences. Work by M. Katz and O. Filtser has been partially supported by grant 1045/10 from the Israel Science Foundation.