On the Hardness of Counting and Sampling Center Strings

Given a set S of n strings, each of length \ell , and a nonnegative value d , we define a center string as a string of length \ell that has Hamming distance at most d from each string in S . The \#{\rm CLOSEST STRING} problem aims to determine the number of center strings for a given set of strings S and input parameters n , \ell , and d . We show \#{\rm CLOSEST STRING} is impossible to solve exactly or even approximately in polynomial time, and that restricting \#{\rm CLOSEST STRING} so that any one of the parameters n , \ell , or d is fixed leads to a fully polynomial-time randomized approximation scheme (FPRAS). We show equivalent results for the problem of efficiently sampling center strings uniformly at random (u.a.r.).

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