Closest String and Substring Problems

Abstract

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 = {s 1 , s 2 , . . . , s n } 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 s i ∈ 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 s i . 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.