On the Chain Pair Simplification Problem

TL;DR

This paper investigates the computational complexity of the Chain Pair Simplification problem under the discrete Fréchet distance, proving it is polynomially solvable and providing efficient algorithms with practical biological applications.

Contribution

It resolves the open complexity question of CPS-3F, showing it is polynomially solvable and offering algorithms that outperform previous methods in biological data visualization.

Findings

CPS-3F is polynomially solvable with an $O(m^2n^2 ext{min}\{m,n\ ext{)}$ algorithm.

Weighted CPS-3F remains polynomial when weights are assigned to vertices of only one chain.

Experimental results indicate CPS-3F outperforms previous algorithms in biological applications.

Abstract

The problem of efficiently computing and visualizing the structural resemblance between a pair of protein backbones in 3D has led Bereg et al. 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 \frechet\ 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 \frechet\ distance (CPS-3F), i.e., where the quality of the simplifications is also measured by the discrete \frechet\ distance. Since CPS-3F was posed in 2008, its complexity has remained open. However, it was believed to be \npc, since CPS under the Hausdorff distance (CPS-2H) was shown to be \npc. We first prove that the weighted version of CPS-3F is indeed weakly \npc\, even on the line, based on a reduction from the set partition problem. Then, 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. In fact, we prove a stronger statement, implying, for example, that if weights are assigned to the vertices of only one of the chains, then the problem remains polynomially solvable. We also study a few less rigid variants of CPS and present efficient solutions for them. Finally, we present some experimental results that suggest that (the minimization version of) CPS-3F is significantly better than previous algorithms for the motivating biological application.