Uniform Capacitated Clustering with Few Extra Centers

In uniform capacitated $(k,z)$-clustering, we open $k$ centers and assign every client to a center so that no center serves more than $U$ clients, minimizing the sum of the $z$-th powers of the assignment distances; $k$-median and $k$-means are the cases $z=1$ and $z=2$. We study algorithms that may open $k+\lfloor\varepsilon k\rfloor$ centers for a fixed $\varepsilon\in(0,1]$ and must respect the capacities exactly. In low dimensions, we give deterministic algorithms that are, except for $k$-median in the plane, the first polynomial-time algorithms with $(1+\varepsilon)k$ centers whose ratio is bounded independently of $\varepsilon$. In Euclidean space of any dimension, with clients of bounded doubling dimension, the ratio is $\mathcal{B}_z-(\mathcal{B}_z-1)\varepsilon+δ$ for every fixed $δ>0$, where $\mathcal{B}_z<3^z$ for $z>1$; this is $3-2\varepsilon+δ$ for $k$-median and $5-4\varepsilon+δ$ for $k$-means. In doubling metrics, the ratio is $3^z-(3^z-1)\varepsilon+δ$ when centers may share a location, and a constant otherwise, such as $\min\{7-4\varepsilon,\,3+2\ln(1/\varepsilon)\}+δ$ for $k$-median. In general metrics, deleting centers greedily gives ratio $3+2\ln(1/\varepsilon)+δ$ for $k$-median when centers may share a location and $7+2\ln(1/\varepsilon)+δ$ otherwise, improving the previous $O(\varepsilon^{-2}\log(1/\varepsilon))$. Our main tool is a rescue lemma. Splitting a near-optimal uncapacitated solution gives at most $2k-1$ blocks of at most $U$ clients, and giving a block a private center saves its share of the extra cost of following an optimal solution. Since a center of the optimal solution can be closed once all its blocks are rescued, $\lfloor\varepsilon k\rfloor$ extra centers save almost an $\varepsilon$-fraction of this extra cost instead of an $\varepsilon/2$-fraction.