Maximizing Submodular plus Supermodular Functions Subject to a Fairness Constraint

We investigate the problem of maximizing the sum of submodular and supermodular functions under a fairness constraint. This sum function is non-submodular in general. For an offline model, we introduce two approximation algorithms: A greedy algorithm and a threshold greedy algorithm. For a streaming model, we propose a one-pass streaming algorithm. We also analyze the approximation ratios of these algorithms, which all depend on the total curvature of the supermodular function. The total curvature is computable in polynomial time and widely utilized in the literature.