0 00 20 12 v 1 1 7 Fe b 20 00 On The Closest String and Substring Problems ∗

The problem of finding a center string that is ‘close’ to every given string arises and has many applications in computational molecular biology and coding theory. This problem has two versions: the Closest String problem and the Closest Substring problem. Assume that we are given a set of strings S = {s1, s2, . . . , sn} of strings, say, each of length m. The Closest String problem [1, 2, 4, 5, 11] asks for the smallest d and a string s of length m which is within Hamming distance d to each si ∈ S. This problem comes from coding theory when we are looking for a code not too far away from a given set of codes [4]. The problem is NP-hard [4, 11]. Berman et al [2] give a polynomial time algorithm for constant d. For super-logarithmic d, Ben-Dor et al [1] give an efficient approximation algorithm using linear program relaxation technique. The best polynomial time approximation has ratio 4 3 for all d, given by [11] and [5]. The Closest Substring problem looks for a string t which is within Hamming distance d away from a substring of each si. This problem only has a 2− 2 2|Σ|+1 approximation algorithm previously [11] and is much more elusive than the Closest String problem, but it has many applications in finding conserved regions, genetic drug target identification, and genetic probes in molecular biology [8, 9, 10, 16, 17, 19, 20, 21, 22, 23, 11]. Whether there are efficient approximation algorithms for both problems are major open questions in this area. We present two polynomial time approxmation algorithms with approximation ratio 1 + ǫ for any small ǫ to settle both questions. ∗Some of the results in this paper have been presented in Proc. 31st ACM Symp. Theory of Computing, May, 1999 [12], and in Proc. 11th Symp. Combinatorial Pattern Matching, June, 2000, [14].