Iterative Rounding for the Closest String Problem

The closest string problem is an NP-hard problem, which arises in computational molecular biology and coding theory. Its task is to find a string that minimizes maximum Hamming distance to a given set of strings. This problem can be reduced to an integer program (IP). However, to date, there exists no known polynomial-time algorithm for integer programs. In 2004, Meneses et al. introduced a branch-and-bound (B&B) method for solving the IP problem. Their algorithm is not always efficient and has the exponential time complexity. In the paper, we attempt to solve efficiently the IP problem by a greedy iterative rounding technique. The proposed algorithm is polynomial time and much faster than the existing B&B IP for the CSP. If the string count is limited to 3, the algorithm is provably at most 1 away from the optimum. The empirical results show that in many cases we can find an exact solution. Even though we fails to find an exact solution, the solution found is very close to exact solution. We think that the approach is useful in solving other IP problems in domains beyond the CSP.