We present a simple performance bound for the greedy scheme in string optimization problems that obtains strong results. Our approach vastly generalizes the group of previously established greedy curvature bounds by Conforti and Cornuéjols (1984). We consider three constants, $α_G$, $α_G'$, and $α_G''$ introduced by Conforti and Cornuéjols (1984), that are used in performance bounds of greedy schemes in submodular set optimization. We first generalize both of the $α_G$ and $α_G''$ bounds to string optimization problems in a manner that includes maximizing submodular set functions over matroids as a special case. We then derive a much simpler and computable bound that allows for applications to a far more general class of functions with string domains. We prove that our bound is superior to both the $α_G$ and $α_G''$ bounds and provide a counterexample to show that the $α_G'$ bound is incorrect under the assumptions in Conforti and Cornuéjols (1984). We conclude with two applications. The first is an application of our result to sensor coverage problems. We demonstrate our performance bound in cases where the objective function is set submodular and string submodular. The second is an application to a social welfare maximization problem with black-box utility functions.

翻译：我们提出了字符串优化问题中贪婪方案的简单性能界，该界获得了强结果。我们的方法极大地推广了Conforti与Cornuéjols（1984）先前建立的贪婪曲率界集合。我们考虑了Conforti与Cornuéjols（1984）引入的三个常数$α_G$、$α_G'$和$α_G''$，这些常数用于子模集优化中贪婪方案的性能界。我们首先将$α_G$和$α_G''$两个界推广到字符串优化问题，其方式包括在拟阵上最大化子模集函数作为特例。接着，我们推导出一个更简单且可计算的界，该界适用于定义域为字符串的、更广泛的一般函数类。我们证明我们的界优于$α_G$和$α_G''$两个界，并提供一个反例表明，在Conforti与Cornuéjols（1984）的假设下，$α_G'$界是不正确的。最后，我们给出两个应用。第一个应用是将我们的结果应用于传感器覆盖问题，我们在目标函数为集子模和字符串子模的情况下展示了我们的性能界。第二个应用是将其应用于具有黑箱效用函数的社会福利最大化问题。