On the Complexity of Protein Local Structure Alignment Under the Discrete Fr\'echet Distance

TL;DR

This paper proves that aligning local protein structures under the discrete Fréchet distance is computationally very hard in general, but feasible in polynomial time when the number of proteins is fixed.

Contribution

It establishes the computational complexity of protein local structure alignment under the discrete Fréchet distance, showing hardness in the general case and polynomial solvability for fixed numbers of proteins.

Findings

Alignment problem is as hard as Independent Set.

No approximation within factor n^{1-ε} unless P=NP.

Polynomial time solution when the number of proteins is constant.

Abstract

We show that given $m$ proteins (or protein backbones, which are modeled as 3D polygonal chains each of length O(n)) the problem of protein local structure alignment under the discrete Fr\'{e}chet distance is as hard as Independent Set. So the problem does not admit any approximation of factor $n^{1-\epsilon}$. This is the strongest negative result regarding the protein local structure alignment problem. On the other hand, if $m$ is a constant, then the problem can be solved in polygnomial time.