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

Abstract

Given a set S ={s ₁,s ₂,...,s _n } of strings each of length m, and an integer L, we study the following two problems.

k -Closest Substring problem: find k center strings c ₁,c ₂ ,...,c _k of length L minimizing d such that for each s _j ∈ S, there is a length-L substring t _j (closest substring) of s _j with min _{1 ≤ i ≤ k} d(c _i ,t _j ) ≤ d. We give a PTAS for this problem, for k=O(1).

k -Consensus Pattern problem: find k median strings c ₁,c ₂ ,...,c _k of length L and a substring t _j (consensus pattern) of length L from each s _j minimizing the total cost $w= \sum \limits_{j=1}^n \min \limits _{1 \le i \le k} d(c_i,t_j)$. 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-ε)-approximation of the Hamming radius k -clustering problem, a special case of the k -Closest Substring problem restricted to L=m.