We consider the problem of maximizing a submodular function with access to a noisy value oracle for the function instead of an exact value oracle. Similar to prior work, we assume that the noisy oracle is persistent in that multiple calls to the oracle for a specific set always return the same value. In this model, Hassidim and Singer (2017) design a $(1-1/e)$-approximation algorithm for monotone submodular maximization subject to a cardinality constraint, and Huang et al (2022) design a $(1-1/e)/2$-approximation algorithm for monotone submodular maximization subject to any arbitrary matroid constraint. In this paper, we design a meta-algorithm that allows us to take any "robust" algorithm for exact submodular maximization as a black box and transform it into an algorithm for the noisy setting while retaining the approximation guarantee. By using the meta-algorithm with the measured continuous greedy algorithm, we obtain a $(1-1/e)$-approximation (resp. $1/e$-approximation) for monotone (resp. non-monotone) submodular maximization subject to a matroid constraint under noise. Furthermore, by using the meta-algorithm with the double greedy algorithm, we obtain a $1/2$-approximation for unconstrained (non-monotone) submodular maximization under noise.

翻译：我们研究在只能通过噪声值预言机而非精确值预言机访问函数值的情况下，最大化子模函数的问题。与先前工作类似，我们假设噪声预言机具有持久性，即对特定集合的多次调用总是返回相同的值。在此模型中，Hassidim 和 Singer (2017) 针对基数约束下的单调子模最大化问题设计了 $(1-1/e)$-近似算法，而 Huang 等人 (2022) 针对任意拟阵约束下的单调子模最大化问题设计了 $(1-1/e)/2$-近似算法。本文中，我们设计了一种元算法，能够将任何精确子模最大化问题的"鲁棒"算法作为黑盒，并将其转换为适用于噪声环境的算法，同时保持近似保证。通过将元算法与测度连续贪心算法结合，我们获得了噪声下拟阵约束单调（相应非单调）子模最大化的 $(1-1/e)$-近似（相应 $1/e$-近似）解。此外，通过将元算法与双重贪心算法结合，我们获得了噪声下无约束（非单调）子模最大化的 $1/2$-近似解。