A Three-String Approach to the Closest String Problem

Abstract

Given a set of n strings of length L and a radius d, the closest string problem asks for a new string t _sol that is within a Hamming distance of d to each of the given strings. It is known that the problem is NP-hard and its optimization version admits a polynomial time approximation scheme (PTAS). Parameterized algorithms have been then developed to solve the problem when d is small. In this paper, with a new approach (called the 3-string approach), we first design a parameterized algorithm for binary strings that runs in $O\left(nL + nd^3 6.731^d\right)$ time, while the previous best runs in $O\left(nL + nd 8^d\right)$ time. We then extend the algorithm to arbitrary alphabet sizes, obtaining an algorithm that runs in $O \left( nL + nd 1.612^d \left( \alpha^2 +1 - 2 \alpha^{-1} + \alpha^{-2} \right) ^{3d}\right)$ time, where $\alpha = \sqrt[3]{\mathstrut \sqrt{|\Sigma|-1}+1 }$. This new time bound is better than the previous best for small alphabets, including the very important case where |Σ| = 4 (i.e., the case of DNA strings).