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 -a -augmentable functions and prove that it encompasses several important subclasses, such as functions of bounded submodularity ratio, a -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 \gamma\alpha\cdote\alpha e\alpha - 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 [A. Bernstein et al., Math. Program., 191 (2022), pp. 953--979] by obtaining a tight lower bound for a -augmentable functions for all a \geq 1. For weighted rank functions of independence systems, our tight bound becomes \alpha least q. \gamma , which recovers the known bound of 1/q for independence systems of rank quotient at