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+ε k centers for a fixed ε∈(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+ε)k centers whose ratio is bounded independently of ε. In Euclidean space of any dimension, with clients of bounded doubling dimension, the ratio is Bz-(Bz-1)ε+δ for every fixed $δ>0$, where Bz<3ᶻ for $z>1$; this is 3-2ε+δ for k-median and 5-4ε+δ for k-means. In doubling metrics, the ratio is 3ᶻ-(3ᶻ-1)ε+δ when centers may share a location, and a constant otherwise, such as min\7-4ε,\,3+2ln(1/ε)\+δ for k-median. In general metrics, deleting centers greedily gives ratio 3+2ln(1/ε)+δ for k-median when centers may share a location and 7+2ln(1/ε)+δ otherwise, improving the previous O(ε⁻²log(1/ε)). 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, ε k extra centers save almost an ε-fraction of this extra cost instead of an ε/2-fraction.