A heuristic algorithm based on Lagrangian relaxation for the closest string problem

The closest string problem that arises in both computational biology and coding theory is to find a string that minimizes the maximum Hamming distance from a given set of strings. This study proposes an efficient heuristic algorithm for this NP-hard problem. The key idea is to apply the Lagrangian relaxation technique to the problem formulated as an integer programming problem. This enables us to decompose the problem into trivial subproblems corresponding to each position of the strings. Furthermore, a feasible solution can be easily obtained from a solution of the relaxation. Based on this, a heuristic algorithm is constructed by combining a Lagrangian multiplier adjustment procedure and a tabu search. Numerical experiments will show the effectiveness of the proposed algorithm.