On Maximizing the Difference Between an Approximately Submodular Function and a Linear Function Subject to a Matroid Constraint

In this paper, we investigate the problem of maximizing the difference between an approximately submodular function and a non-negative linear function subject to a matroid constraint. This model has widespread applications in real life, such as the team formation problem in labor market and the assortment optimization in sales market. We provide a bicriteria approximation algorithm with bifactor ratio (gamma/1+gamma, 1), where gamma is an element of (0, 1] is a parameter to characterize the approximate submodularity. Our result extends Ene's recent work on maximizing the difference between a monotone submodular function and a linear function. Also, a generalized version of the proposed algorithm is capable to deal with huge volume data set.