We study optimal policy learning under combined budget and minimum coverage constraints. We show that the problem admits a knapsack-type structure and that the optimal policy can be characterized by an affine threshold rule involving both budget and coverage shadow prices. We establish that the linear programming relaxation of the combinatorial solution has an O(1) integrality gap, implying asymptotic equivalence with the optimal discrete allocation. Building on this result, we analyze two implementable approaches: a Greedy-Lagrangian (GLC) and a rank-and-cut (RC) algorithm. We show that the GLC closely approximates the optimal solution and achieves near-optimal performance in finite samples. By contrast, RC is approximately optimal whenever the coverage constraint is slack or costs are homogeneous, while misallocation arises only when cost heterogeneity interacts with a binding coverage constraint. Monte Carlo evidence supports these findings.

翻译：我们研究了在组合预算与最低覆盖约束下的最优策略学习问题。我们证明该问题具有背包型结构，且最优策略可通过一个涉及预算影子价格和覆盖影子价格的仿射阈值规则进行刻画。我们建立了组合解对应的线性规划松弛具有O(1)整数间隙，这意味着其与最优离散分配渐近等价。基于这一结果，我们分析了两种可实施方法：贪婪-拉格朗日算法（GLC）与秩-割算法（RC）。我们证明GLC能够紧密逼近最优解，并在有限样本下实现近最优性能。相比之下，当覆盖约束松弛或成本同质时，RC近似最优；而仅当成本异质性与紧约束覆盖相互作用时才会出现错配。蒙特卡洛证据支持上述结论。