舍入公平最大最小多样化问题的球线性规划

中文摘要 AI 辅助

给定度量空间中的 $n$ 个点，这些点被划分为若干组 $X_1,\dots,X_m$，并给定整数配额 $k_1,\dots,k_m$，其总和为 $k$，公平最大最小多样化问题要求选择 $k$ 个点，其中从每组 $X_i$ 中恰好选取 $k_i$ 个点，以最大化最小成对距离。Addanki 等人（ICDT 2022）针对该问题描述了一个球线性规划（ball LP）以及相应的舍入算法，这些算法可产生因子为 2 的近似解，但其公平性仅在期望意义下成立；还可产生因子为 6 的近似解，但公平性保证较为宽松；此外，他们提出了一种具有精确公平性的 $(m+1)$-近似算法。我们引入了两种新算法。第一种方法改进了 Addanki 等人的舍入过程，得到因子为 2 的近似解，并且该解以高概率是 $\nvarepsilon$-公平的，即从每个组 $X_i$ 中至少选取 $(1-\nvarepsilon) k_i$ 个点。第二种方法在多项式时间内返回精确公平性版本最优值的 4-近似解。在时间 $n^{O(1)} 2^{O(k)}$ 内（该复杂度关于 $k$ 是固定参数可处理的），我们实现了精确公平的 4-近似。我们的方法改编了 Haxell 定理（Graphs Combin., 1995）背后的增广过程。该近似因子不依赖于 $m$。此外，我们证明了对于精确公平性，任何对球线性规划的舍入都无法实现更小的因子。

英文摘要

Given $n$ points in a metric space, partitioned into groups, $X_1,\dots,X_m$, and integer quotas, $k_1,\dots,k_m$, summing to $k$, the Fair Max-Min Diversification problem asks for a set of $k$ points, exactly $k_i$ from each group $X_i$, maximizing the minimum pairwise distance. Addanki et al. (ICDT 2022) described a ball LP for this problem and rounding algorithms that yield a factor 2 approximation whose fairness holds only in expectation and a factor 6 approximation with relaxed fairness guarantees, as well as an $(m+1)$-approximation with exact fairness. We introduce two new algorithms. The first method refines the rounding of Addanki et al., yielding a 2-approximate solution that is $\varepsilon$-fair with high probability, meaning that from every group $X_i$, at least $(1-\varepsilon) k_i$ points are chosen. The second method in polynomial time returns a 4-approximation to the optimal value of the exact fairness version. In time $n^{O(1)} 2^{O(k)}$, which is fixed-parameter tractable in $k$, we achieve an exactly fair 4-approximation. Our method adapts the augmenting procedure behind Haxell's theorem (Graphs Combin., 1995). This approximation factor does not depend on $m$. Moreover, we show that no rounding of the ball LP achieves a smaller factor with exact fairness.

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15 pages, 1 figure