Summary: We propose an effective polynomial-time preprocessing strategy for intractable median problems. Developing a new methodological framework, we show that if the input objects of generally intractable problems exhibit a sufficiently high degree of similarity between each other on average, then there are efficient exact solving algorithms. In other words, we show that the median problems Swap Median Permutation, Consensus Clustering, Kemeny Score, and Kemeny Tie Score all are fixed-parameter tractable with respect to the parameter “average distance between input objects”. To this end, we develop the novel concept of “partial kernelization” and, furthermore, identify polynomial-time solvable special cases for the considered problems.