Unified Greedy Approximability Beyond Submodular Maximization

We consider classes of objective functions of cardinality-constrained maximization problems for which the greedy algorithm guarantees a constant approximation. We propose the new class of gamma-alpha-augmentable functions and prove that it encompasses several important subclasses, such as functions of bounded submodularity ratio, alpha-augmentable functions, and weighted rank functions of an independence system of bounded rank quotient - as well as additional objective functions for which the greedy algorithm yields an approximation. For this general class of functions, we show a tight bound of alpha/gamma . e(alpha)/e(alpha-1) ea-1 on the approximation ratio of the greedy algorithm that tightly interpolates between bounds from the literature for functions of bounded submodularity ratio and for a-augmentable functions. In particular, as a by-product, we close a gap in [Math.Prog., 2020] by obtaining a tight lower bound for alpha-augmentable functions for all a >= 1. For weighted rank functions of independence systems, our tight bound becomes alpha/gamma, which recovers the known bound of 1/q for independence systems of rank quotient at least q.