A Performance Bound for the Greedy Algorithm in a Generalized Class of String Optimization Problems
For researchers in combinatorial optimization, this provides a more general and tighter performance guarantee for greedy algorithms on string problems, though the improvement is incremental.
The paper generalizes greedy curvature bounds for submodular set functions to string optimization problems, proving a simpler and superior bound, and correcting a prior bound. Applications to sensor coverage and social welfare maximization demonstrate the bound's effectiveness.
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.