AI 中文总结
该研究探讨划分约束下的A/E-设计问题，证明其不存在合理近似，与有良好近似保证的D-设计形成对比，解答了相关学者提出的问题。
AI 中文摘要
我们研究划分约束下的A/E-设计问题：给定向量$v_1,\ldots,v_N\in\mathbb{R}^d$和集合$[N]$上的一个划分拟阵，求该拟阵的一个基$S$，使得$\tr(M(S)^{-1})$或$\lambda_{\max}(M(S)^{-1})$最小，其中$M(S)=\sum_{i \in S} v_i v_i^\top$。与D-设计不同，已知D-设计的良好估计和近似保证是$d$的函数，我们证明A/E-设计不存在合理的近似，这回答了Brown、Laddha和Singh提出的问题。该证明基于从三维匹配出发的基本归约。
英文摘要
We consider the A/E-design problem under partition constraints: Given vectors $v_1,\ldots,v_N\in \R^d$ and a partition matroid on $[N]$, find a base $S$ of the matroid that minimizes $\tr(M(S)^{-1})$ or $λ_{\max}(M(S)^{-1})$ where $M(S)=\sum_{i \in S} v_i v_i^\top$. In contrast to D-design, where good estimation and approximation guarantees are known as a function of $d$, we show that no reasonable approximation exists for A/E-design. This answers a question of Brown, Laddha and Singh. The proof is based on an elementary reduction from three-dimensional matching.
