Improved Approximation Guarantees for Shortest Superstrings using Cycle Classification by Overlap to Length Ratios

TL;DR

This paper improves the approximation guarantees for the Shortest Superstring problem, showing that the GREEDY algorithm has a better upper bound and providing a new approximation ratio for the problem.

Contribution

It presents the first improvement on the GREEDY algorithm's approximation ratio since 2005 and offers a new, tighter approximation factor for the Shortest Superstring problem.

Findings

GREEDY algorithm's approximation ratio is at most approximately 3.425.

Shortest Superstring problem can be approximated within a factor of approximately 2.475.

Provides the first progress on GREEDY's approximation guarantee since 2005.

Abstract

In the Shortest Superstring problem, we are given a set of strings and we are asking for a common superstring, which has the minimum number of characters. The Shortest Superstring problem is NP-hard and several constant-factor approximation algorithms are known for it. Of particular interest is the GREEDY algorithm, which repeatedly merges two strings of maximum overlap until a single string remains. The GREEDY algorithm, being simpler than other well-performing approximation algorithms for this problem, has attracted attention since the 1980s and is commonly used in practical applications. Tarhio and Ukkonen (TCS 1988) conjectured that GREEDY gives a 2-approximation. In a seminal work, Blum, Jiang, Li, Tromp, and Yannakakis (STOC 1991) proved that the superstring computed by GREEDY is a 4-approximation, and this upper bound was improved to 3.5 by Kaplan and Shafrir (IPL 2005). We show that the approximation guarantee of GREEDY is at most $(13+\sqrt{57})/6 \approx 3.425$, making the first progress on this question since 2005. Furthermore, we prove that the Shortest Superstring can be approximated within a factor of $(37+\sqrt{57})/18\approx 2.475$, improving slightly upon the currently best $2\frac{11}{23}$-approximation algorithm by Mucha (SODA 2013).