A polynomial time approximation scheme for the closest substring problem

In this paper we study the following problem: Given n strings s(1),s(2),...,s(n), each of length n, find a substring t(i) of length L for each s(i), and a string s of length L, such that max(i=1)(n) d(s, t(i)) is minimized, where d(.,.) is the Hamming distance. The problem was raised in [6] in an application of genetic drug target search and is a key open problem in many applications [7]. The authors of [6] showed that it is NP-hard and can be trivially approximated within ratio 2. A non-trivial approximation algorithm with ratio better than 2 was found in [7]. A major open question in this area is whether there exists a polynomial time approximation scheme (PTAS) for this problem. In this paper, we answer this question positively. We also apply our method to two related problems.