On the k-Closest Substring and k-Consensus Pattern Problems

Given a set S = {s1, s2, . . . , sn} of strings each of length m, and an integer L, we study the following two problems. k-Closest Substring problem: find k center strings c1, c2, . . . , ck of length L minimizing d such that for each sj ∈ S, there is a length-L substring tj (closest substring) of sj with min1≤i≤k d(ci, tj) ≤ d. We give a PTAS for this problem, for k = O(1). k-Consensus Pattern problem: find k median strings c1, c2, . . . , ck of length L and a substring tj (consensus pattern) of length L from each sj minimizing the total cost w = n ∑ j=1 min 1≤i≤k d(ci, tj). We give a PTAS for this problem, for k = O(1). Our results improve recent results of [10] and [16] both of which depended on the random linear transformation technique in [16]. As for general k case, we give an alternative and direct proof of the NP-hardness of (2)-approximation of the Hamming radius k-clustering problem, a special case of the k-Closest Substring problem restricted to L = m.