{Approximation Algorithm for the Shortest Superstring Problem

Given a collection of strings S = fs1; : : : ; sng over an alphabet , a superstring of S is a string containing each si as a substring; that is, for each i, 1 i n, contains a block of jsij consecutive characters that match si exactly. The shortest superstring problem is the problem of nding a superstring of minimum length. The shortest superstring problem has applications in both data compression and computational biology. In data compression, the problem is a part of a general model of string compression proposed by Gallant, Maier and Storer (JCSS '80). Much of the recent interest in the problem is due to its application to DNA sequence assembly. The problem has been shown to be NP-hard; in fact, it was shown by Blum et al.(JACM '94) to be MAX SNP-hard. The rst O(1)-approximation was also due to Blum et al., who gave an algorithm that always returns a superstring no more than 3 times the length of an optimal solution. Several researchers have published results that improve on the approximation ratio; of these, the best previous result is our algorithmShortString, which achieves a 2 4 {approximation (WADS '95). We present our new algorithm, G-ShortString, which achieves a ratio of 2 3 . It generalizes the ShortString algorithm, but the analysis di ers substantially from that of ShortString. Our previous work identi ed classes of strings that have a nested periodic structure, and which must be present in the worst case for our algorithms. We introduced machinery to descibe these strings and proved strong structural properties about them. In this paper we extend this study to strings that exhibit a more relaxed form of the same structure, and we use this understanding to obtain our improved result.