Orienting Ordered Scaffolds: Complexity and Algorithms

TL;DR

This paper studies the problem of orienting ordered genomic scaffolds, formalizes it as an optimization problem, proves NP-hardness, and proposes algorithms for special and general cases to improve genome assembly accuracy.

Contribution

It formalizes the scaffold orientation problem, proves its NP-hardness, and introduces polynomial and fixed-parameter tractable algorithms for solving it.

Findings

The scaffold orientation problem is NP-hard.

A polynomial-time algorithm solves a special case with limited constraints.

An FPT algorithm addresses the general case effectively.

Abstract

Despite the recent progress in genome sequencing and assembly, many of the currently available assembled genomes come in a draft form. Such draft genomes consist of a large number of genomic fragments (scaffolds), whose order and/or orientation (i.e., strand) in the genome are unknown. There exist various scaffold assembly methods, which attempt to determine the order and orientation of scaffolds along the genome chromosomes. Some of these methods (e.g., based on FISH physical mapping, chromatin conformation capture, etc.) can infer the order of scaffolds, but not necessarily their orientation. This leads to a special case of the scaffold orientation problem (i.e., deducing the orientation of each scaffold) with a known order of the scaffolds. We address the problem of orientating ordered scaffolds as an optimization problem based on given weighted orientations of scaffolds and their pairs (e.g., coming from pair-end sequencing reads, long reads, or homologous relations). We formalize this problem using notion of a scaffold graph (i.e., a graph, where vertices correspond to the assembled contigs or scaffolds and edges represent connections between them). We prove that this problem is NP-hard, and present a polynomial-time algorithm for solving its special case, where orientation of each scaffold is imposed relatively to at most two other scaffolds. We further develop an FPT algorithm for the general case of the OOS problem.