Abstract We investigate the problem of deciding whether a given preference profile is close to a nicely structured preference profile of a certain type, as for instance single-peaked, single-caved, singlecrossing, value-restricted, best-restricted, worstrestricted, medium-restricted, or group-separable profiles. We measure this distance by the number of voters or alternatives that have to be deleted so as to reach a nicely structured profile. Our results classify all considered problem variants with respect to their computational complexity, and draw a clear line between computationally tractable (polynomial time solvable) and computationally intractable (NP-hard) questions.