Designing and Implementing Algorithms for the Closest String Problem

Abstract

Given a set of n strings of length L and a radius d, the closest string problem (CSP for short) asks for a 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). A number of parameterized algorithms have been then developed to solve the problem when d is small. Among them, the relatively new ones have not been implemented before and their performance in practice was unknown. In this study, we implement all of them by careful engineering. For those that have been implemented before, our implementation is much faster. For some of those that have not been implemented before, our experimental results show that there exist huge gaps between their theoretical and practical performances. We also design a new parameterized algorithm for the binary case of CSP. The algorithm is deterministic and runs in \(O\left( nL + n^2d\cdot 6.16^d\right) \) time, while the previously best deterministic algorithm runs in \(O\left( nL + nd^3\cdot 6.731^d\right) \) time.