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 Cornu\'{e}jols (1984). We consider three constants, $\alpha_G$, $\alpha_G'$, and $\alpha_G''$ introduced by Conforti and Cornu\'{e}jols (1984), that are used in performance bounds of greedy schemes in submodular set optimization. We first generalize both of the $\alpha_G$ and $\alpha_G''$ 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 $\alpha_G$ and $\alpha_G''$ bounds and provide a counterexample to show that the $\alpha_G'$ bound is incorrect under the assumptions in Conforti and Cornu\'{e}jols (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：我们为在字符串优化问题中取得强结果的贪心方案提出了一个简单的性能界。我们的方法极大地推广了Conforti和Cornu\'{e}jols (1984) 先前建立的贪心曲率界所涵盖的类别。我们考虑了Conforti和Cornu\'{e}jols (1984) 引入的三个常数 $\alpha_G$、$\alpha_G'$ 和 $\alpha_G''$，这些常数用于子模集优化中贪心方案的性能界。我们首先将 $\alpha_G$ 和 $\alpha_G''$ 界推广到字符串优化问题，其方式包含了在拟阵上最大化子模集函数作为特例。然后，我们推导出一个更简单且可计算的界，该界允许应用于具有字符串定义域的更广泛的函数类。我们证明了我们的界优于 $\alpha_G$ 和 $\alpha_G''$ 界，并提供了一个反例来表明在Conforti和Cornu\'{e}jols (1984) 的假设下 $\alpha_G'$ 界是不正确的。最后我们给出两个应用。第一个是将我们的结果应用于传感器覆盖问题。我们展示了当目标函数是集合子模和字符串子模时我们的性能界。第二个是应用于具有黑盒效用函数的社会福利最大化问题。