公平最大最小多样化的局部搜索
Local Search for Fair Max-Min Diversification
- Microsoft Research（微软研究院）
- Citadel Securities（城堡证券）
- Princeton University（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对公平最大最小多样化问题，提出首个常数因子局部搜索算法，精确满足所有划分约束，并推广至任意颜色子集及上下界约束。
AI中文摘要：
给定度量空间中的 $n$ 个点，最大最小多样化问题要求选择 $k$ 个点的子集，使得所选点之间的最小成对距离最大化。这可以说是最基本的多樣性概念，在广泛的应用领域中都有应用。我们在划分约束下考虑该问题，此前被称为公平最大最小多样化（FMMD）。这里，每个点具有 $[m]$ 中的一种颜色，可行解必须恰好包含 $k_i$ 个颜色为 $i$ 的点，其中 $k_1,\ldots,k_m$ 是满足 $\sum_i k_i=k$ 的给定参数。我们首次给出了使用局部搜索的常数因子近似算法，运行时间为 $f(m)\cdot \operatorname{poly}(n)$，并且所有约束都被精确满足。所有先前已知的算法要么提供 $\widetilde \Theta(m)$ 的近似因子，要么运行时间随解的大小 $k$ 呈指数增长，要么仅近似或期望地满足公平约束。我们进一步将结果推广到每个点可以属于任意颜色子集的问题。给定每种颜色 $i$ 的下界和上界 $\ell_i$ 和 $u_i$，目标是找到 $k$ 个点，其颜色计数满足所有这些界，同时最大化其多样性。
英文摘要：
Given $n$ points in a metric space, Max-Min diversification asks for a subset of $k$ points maximizing the minimum pairwise distance between the selected points. This is arguably the most fundamental notion of diversity with applications across a wide range of domains. We consider this problem under partition constraints, previously studied as Fair Max-Min Diversification (FMMD). Here, each point has a color in $[m]$, and a feasible solution must contain exactly $k_i$ points of color $i$, where $k_1,\ldots,k_m$ are prescribed parameters satisfying $\sum_i k_i=k$. We give the first constant factor approximation for the problem using local search, that runs in time $f(m)\cdot \operatorname{poly}(n)$, in which all constraints are satisfied exactly. All previously known algorithms either provided an $\widetilde Θ(m)$ approximation factor, had running times exponential in the solution size $k$, or satisfied the fairness constraints only approximately or in expectation. We further generalize our result to the problem where each point may belong to an arbitrary subset of colors. Given lower and upper bounds $\ell_i$ and $u_i$ for every color $i$, the goal is to find $k$ points whose color counts satisfy all these bounds while maximizing their diversity.
