Improved clustering algorithms with limited extra centers and strict capacities
Uniform Capacitated Clustering with Few Extra Centers
Computational GeometryData Structures and Algorithms
Summary
The paper deals with a common problem in organizing clients by assigning them to facilities without exceeding the maximum capacity per facility. The authors develop new algorithms that allow opening slightly more than the original number of centers, while ensuring no center is overloaded and the overall client-to-center distance is kept low. These algorithms work well in various metric spaces, including Euclidean spaces and others with a notion of doubling dimension. Their main technical advance is a smart way to handle extra centers to reduce the cost of assigning clients, improving previous guarantees.
What this means in practice
- •For logistics planners: Optimize delivery or service locations with strict capacity limits while allowing few extra distribution centers to reduce travel costs.
- •For network designers: Design server clusters that strictly limit user load per server but can provision a small number of additional servers to improve latency efficiently.
Authors
Kangke Cheng, Guanlin Mo, Hu Ding
Abstract
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.