Towards a theoretical understanding of why local search works for clustering with fair-center representation

The representative k-median problem generalizes the classical clustering formulations in that it partitions the data points into ℓ disjoint demographic groups and imposes a lower-bound constraint on the number of opened facilities from each group, such that all the groups are fairly represented by the opened facilities. Due to its simplicity, the local-search heuristic, which iteratively swaps a bounded number of closed facilities for the same number of opened ones to improve the solution, has been frequently used in the representative k-median problem. It is known that the local-search heuristic, when restricted to constant-size swaps, yields a constant-factor approximation if $\ell =2$, and has an unbounded approximation ratio if ℓ is super-constant. However, for any constant $\ell >2$, the existence of a constant-factor approximation under constant-size swaps remained an open question for a long time. In response to this question, we demonstrate that the local-search heuristic guarantees a $(4\ell +5)$-approximation when up to $\ell (\ell +1)$ facilities are allowed to be swapped in each iteration, thus providing an affirmative answer to the question.

Our main technical contribution is a novel approach for theoretically analyzing the local-search heuristic, which bounds its approximation ratio by linearly combining the clustering cost increases induced by a set of hierarchically organized swaps. Our techniques also generalize to the k-means clustering formulation and reveal similar approximation guarantees for the local-search heuristic.