Why Large Closest String Instances Are Easy to Solve in Practice

Abstract

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 randomized algorithm, we refer to as CSP-Greedy, that computes the closest string on smoothed instances up to a constant factor approximation in time O(ℓ³), where ℓ is the string length. Using smoothed analysis, we prove CSP-Greedy achieves a $\left( ( 1 + \frac{\epsilon e}{2^n})\right)^{\ell}$-approximation guarantee, where ε> 0 is any small value and n is the number of strings. These approximation and runtime guarantees demonstrate 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 extremely efficiently on instances with a large number of strings. This counter-intuitive fact that “large” Closest String instances are easier and more efficient to solve gives new insight into this well-investigated problem.