A lower bound on approximation algorithms for the closest substring problem

The Closest Substring problem (CSP), where a short string is sought that minimizes the number of mismatches between it and each of a given set of strings, is a minimization problem with polynomial time approximation schemes. In this paper, a lower bound on approximation algorithms for the CSP problem is developed. We prove that unless the Exponential Time Hypothesis (ETH Hypothesis, i.e., not all search problems in SNP are solvable in subexponential time) fails, the CSP problem has no polynomial time approximation schemes of running time f(1/epsilon)|x|(O(1/epsilon)) for any function f, where |x| is the size of input instance. This essentially excludes the possibility that the CSP problem has a practical polynomial time approximation scheme even for moderate values of the error bound epsilon. As a consequence, it is unlikely that the study of approximation schemes for the CSP problem in the literature would lead to practical approximation algorithms for the CSP problem for small error bound E.