Why Large Closest String Instances Are Easy to Solve in Practice

We initiate the study of the smoothed complexity of the Closest String problem by proposing a semi-random model of Hamming distance. We restrict interest to the optimization version of the Closest String problem, and give a 2-approximation algorithm, which we refer to as CSP-Greedy, that runs in O(n` + `)-time, where ` is the string length and n is the number of strings. Using smoothed analysis, we prove CSP-Greedy achieves a (1 + 2 )approximation guarantee, where > 0 is a small value. This approximation and runtime guarantee demonstrates that Closest String instances with a relatively large number of input strings are efficiently solved in practice. We also give experimental results demonstrating that CSP-Greedy runs efficiently on instances with a large number of strings. The counter-intuitive fact that large Closest String instances are relatively easy and efficient to solve gives new insight into this well-investigated problem.