Improved Approximation Algorithms and Hardness Results for Shortest Common Superstring with Reverse Complements

TL;DR

This paper introduces an improved approximation algorithm for the Shortest Common Superstring with Reverse Complements problem, achieving a ratio of 8/3, and establishes its NP-hardness to approximate within a factor of 333/332.

Contribution

It presents a novel approximation algorithm with a better ratio and proves the problem's hardness of approximation, extending known NP-hardness results to SCS-RC.

Findings

New approximation algorithm with ratio 8/3

Reduction to maximum-weight perfect matching

NP-hardness of approximation within 333/332

Abstract

The Shortest Common Superstring (SCS) problem is a fundamental task in sequence analysis. In genome assembly, however, the double-stranded nature of DNA implies that each fragment may occur either in its original orientation or as its reverse complement. This motivates the Shortest Common Superstring with Reverse Complements (SCS-RC) problem, which asks for a shortest string that contains, for each input string, either the string itself or its reverse complement as a substring. The previously best-known approximation ratio for SCS-RC was $\frac{23}{8}$. In this paper, we present a new approximation algorithm achieving an improved ratio of $\frac{8}{3}$. Our approach computes an optimal constrained cycle cover by reducing the problem, via a novel gadget construction, to a maximum-weight perfect matching in a general graph. We also investigate the computational hardness of SCS-RC. While the decision version is known to be NP-complete, no explicit inapproximability results were previously established. We show that the hardness of SCS carries over to SCS-RC through a polynomial-time reduction, implying that it is NP-hard to approximate SCS-RC within a factor better than $\frac{333}{332}$. Notably, this hardness result holds even for the DNA alphabet.