AI 中文总结
研究多选拟阵秘书问题，介绍多轨和并集两种实现方式。给出无容量限制横截拟阵的最优算法及概率竞争比，分析有容量限制横截拟阵的单阈值路由算法，还对$k$列稀疏和层状拟阵实例化相关方法。
AI 中文摘要
我们引入并研究了多选拟阵秘书问题，记为$(J,\kappa)$-MSP。对于秩为一的拟阵且$\kappa=\infty$，它可简化为具有$J$种选择的经典秘书问题。元素按均匀随机顺序到达。算法可在满足$|\mathrm{AUX}|\leq\kappa\cdot\mathrm{rank}(\mathcal{M})$的$J$重并拟阵$\mathcal{M}^{(J)}$中维护一个可行候选池$\mathrm{AUX}$。最后，在$\mathcal{M}$中提取$\mathrm{AUX}$的最大权重独立子集。该模型将在线存储与最终可行解分离。我们研究了两种多选实现：多轨算法（维护$\mathcal{M}$的$J$个独立集）和基于并集的算法（直接在$\mathcal{M}^{(J)}$中维护池）。我们的主要结果是在无容量限制的$(J,\infty)$设置下为横截拟阵提供了一种精确的最优算法。对于固定的$J$，其概率竞争比等于经典$J$选秘书问题的最优成功概率。因此，秩为一的实例是整个横截类的最坏情况，并且随着$J$的增长，最优保证以指数速度快速收敛到$1$。我们还分析了一种用于具有局部容量$b$和全局容量$\kappa\cdot\mathrm{rank}(\mathcal{M})$的有容量限制横截拟阵的简单单阈值路由算法。其分析提供了明确的有限参数界和渐近公式，展示了全局容量导致的有限秩损失如何衰减，以及$b$、$J$和$\kappa$如何相互作用。最后，我们为$k$列稀疏拟阵实例化了多轨方法（保证$1 - O(e^{-J/(ke)})$）和为层状拟阵实例化了基于并集的方法（保证$1 - O(e^{-J/e})$）。
英文摘要
We introduce and study the multiple-choice matroid secretary problem, denoted $(J,κ)$-MSP. For rank-one matroids and $κ=\infty$, it reduces to the classical secretary problem with $J$ choices. Elements arrive in uniformly random order. Algorithms may keep a candidate pool $\mathrm{AUX}$ feasible in the $J$-fold union matroid $\mathcal{M}^{(J)}$ satisfying $|\mathrm{AUX}|\le κ\cdot\mathrm{rank}(\mathcal{M})$. Finally, one extracts the maximum-weight independent subset of $\mathrm{AUX}$ in $\mathcal{M}$. This model separates online storage from the final feasible solution. We study two multiple-choice implementations: multi-track algorithms (maintaining $J$ independent sets of $\mathcal{M}$) and union-based algorithms (maintaining the pool directly in $\mathcal{M}^{(J)}$). Our main result is an exact optimal algorithm for transversal matroids in the uncapacitated $(J,\infty)$ setting. For fixed $J$, its probability-competitive ratio equals the optimal success probability of the classical $J$-choice secretary problem. Thus, rank-one instances are the worst case for the whole transversal class, and the optimal guarantee converges exponentially fast to $1$ as $J$ grows. We also analyze a simple single-threshold routing algorithm for capacitated transversal matroids with local capacities $b$ and global capacity $κ\cdot\mathrm{rank}(\mathcal{M})$. Its analysis provides explicit finite-parameter bounds and asymptotic formulas, showing how finite-rank loss caused by global capacity decays, and how $b$, $J$, and $κ$ interact. Finally, we instantiate the multi-track approach for $k$-column-sparse matroids (guarantee $1-O(e^{-J/(ke)})$) and the union-based approach for laminar matroids (guarantee $1-O(e^{-J/e})$).
