In this paper, a bound on the “error subspace” projection is established, which is then used to show that any two clusterings with small distortion are close. Which immediately follows that, if a good clustering, $C$, is discovered for a given dataset (say manually, or by whatever ssuitable means), then another clustering, $C_1$, is going to be close to the optimal clustering if it minimizes the distortion between the two clusterings.
The paper was presented as optimality for k-means. But I could not find any specific detail that stops it from being applicable to general clustering algorithms. That's why I want to confirm here by asking whether this method of determining existence of an optimal algorithmic clustering applicable to other algorithms beyond k-means.
Meilă, Marina. "The uniqueness of a good optimum for k-means." In Proceedings of the 23rd international conference on Machine learning, pp. 625-632. ACM, 2006.