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Finding the optimal $k$-means clustering is NP-hard in general and manyheuristics have been designed for minimizing monotonically the $k$-meansobjective. We first show how to extend Lloyd's batched relocation heuristic andHartigan's single-point relocation heuristic to take into account empty-clusterand single-point cluster events, respectively. Those events tend toincreasingly occur when $k$ or $d$ increases, or when performing severalrestarts. First, we show that those special events are a blessing because theyallow to partially re-seed some cluster centers while further minimizing the$k$-means objective function. Second, we describe a novel heuristic,merge-and-split $k$-means, that consists in merging two clusters and splittingthis merged cluster again with two new centers provided it improves the$k$-means objective. This novel heuristic can improve Hartigan's $k$-means whenit has converged to a local minimum. We show empirically that thismerge-and-split $k$-means improves over the Hartigan's heuristic which is the{\em de facto} method of choice. Finally, we propose the $(k,l)$-meansobjective that generalizes the $k$-means objective by associating the datapoints to their $l$ closest cluster centers, and show how to either directlyconvert or iteratively relax the $(k,l)$-means into a $k$-means in order toreach better local minima.

Further heuristics for $k$-means: The merge-and-split heuristic and the $(k,l)$-means