Improved Approximation Guarantees for Shortest Superstrings using Cycle Classification by Overlap to Length Ratios
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+ root 57)/6 approximate to 3.425. Furthermore, we prove that the Shortest Superstring can be approximated within a factor of (37+root 57)/18 approximate to 2.475, improving slightly upon the currently best 2 11 23 -approximation algorithm by Mucha (SODA 2013).