On Covering Problems of Codes

On the String Consensus Problem and the Manhattan Sequence Consensus Problem

“…The problem is known to be NPcomplete even for binary alphabet [8]. The algorithmic study of Hamming string consensus (HSC) problem started in 1999 with the first approximation algorithms [15].…”

Closest String and Substring Problems

“…, 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.…”

“…These problems find applications in locating binding sites and finding conserved regions in unaligned sequences [20,9,8,19], genetic drug target identification [11], designing genetic probes [11], universal PCR primer design [10,3,17,11], and, outside computational biology, in coding theory [4,5]. Such problems may be considered to be various generalizations of the common substring problem, allowing errors.…”

“…In the context of coding theory it was shown to be NP-hard [4]. In DNA sequence related topics, [2] gave an exact algorithm when the distance d is a constant.…”

“…Note that the small d is key in applications such as genetic drug target search where we look for similar regions to which a complementary drug sequence would bind. It is a major open problem [4,2,1,5,11] to achieve the best approximation ratio for this problem. (Justifications for using Hamming distance can also be found in these references, especially [11].)…”