Tight Approximation for the Minimum Bottleneck Generalized Matching Problem

Abstract

We study a problem arising in statistical analysis called the minimum bottleneck generalized matching problem that involves breaking up a population into blocks in order to carry out generalizable statistical analyses of randomized experiments. At a high level the problem is to find a clustering of the population such that each part is at least a given size and has at least a given number of elements from each treatment class (so that the experiments are statistically significant), and that all elements within a block are as similar as possible (to improve the accuracy of the analysis).

More formally, given a metric space $(V, d)$, a treatment partition $\mathcal {T} = \{T_1, \ldots , T_k\}$ of $V$, and a target cardinality vector $(b_0, b_1, \ldots , b_k) \in Z_+^{k+1}$ such that $b_0 \ge \sum _{j=1}^k b_j$. The objective is to find a partition $M_1, \ldots , M_\ell$ of V minimizing the maximum diameter of any part such that for each part we have $|M_i| \ge b_0$ and $|M_i \cap T_j| \ge b_j$ for all $j=1, \ldots , k$.

Our main contribution is to provide a tight 2-approximation for the problem. We also show how to modify the algorithm to get the same approximation ratio for the more general problem of finding a partition where each part spans a given matroid.