On the Closest String via Rank Distance

Abstract

Given a set S of k strings of maximum length n, the goal of the closest substring problem (CSSP) is to find the smallest integer d (and a corresponding string t of length l≤n) such that each string s∈S has a substring of length l of "distance" at most d to t. The closest string problem (CSP) is a special case of CSSP where l=n. CSP and CSSP arise in many applications in bioinformatics and are extensively studied in the context of Hamming and edit distance. In this paper we consider a recently introduced distance measure, namely the rank distance. First, we show that the CSP and CSSP via rank distance are NP-hard. Then, we present a polynomial time k-approximation algorithm for the CSP problem. Finally, we give a parametrized algorithm for the CSP (the parameter is the number of input strings) if the alphabet is binary and each string has the same number of 0's and 1's.