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

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+√(57))/6 ≈ 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+√(57))/18≈ 2.475, improving slightly upon the currently best 211/23-approximation algorithm by Mucha (SODA 2013).

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