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 ǫ > 0, there exists a polynomial-time algorithm with an approximation ratio 1 − c/e − ǫ, where c ∈ [0, 1] is the (total) curvature of the input function. This approximation ratio is tight up to ǫ for any c ∈ [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. ∗Supported by JSPS Grant-in-Aid for Young Scientists (B) (26730009), MEXT Grant-in-Aid for Scientific Research on Innovative Areas (24106003), and JST, ERATO, Kawarabayashi Large Graph Project.