Maximizing a Monotone Submodular Function with a Bounded Curvature under a Knapsack Constraint

We consider the problem of maximizing a monotone submodular function under a knapsack constraint. We show that, for any fixed $ε> 0$, there exists a polynomial-time algorithm with an approximation ratio $1-c/e-ε$, where $c \in [0,1]$ is the (total) curvature of the input function. This approximation ratio is tight up to $ε$ for any $c \in [0,1]$. To the best of our knowledge, this is the first result for a knapsack constraint that incorporates the curvature to obtain an approximation ratio better than $1-1/e$, which is tight for general submodular functions. As an application of our result, we present a polynomial-time algorithm for the budget allocation problem with an improved approximation ratio.