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 epsilon > 0, there exists a polynomial-time algorithm with an approximation ratio 1- c/e - epsilon, where c is an element of [0, 1] is the (total) curvature of the input function. This approximation ratio is tight up to c for any c is an element of [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.