Unified greedy approximability beyond submodular maximization
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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 γ-α-augmentable functions and prove that it encompasses several important subclasses, such as functions of bounded submodularity ratio, α-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 α/γ·e^α/e^α-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 α-augmentable functions. In paritcular, as a by-product, we close a gap left in [Math.Prog., 2020] by obtaining a tight lower bound for α-augmentable functions for all α≥1. For weighted rank functions of independence systems, our tight bound becomes α/γ, which recovers the known bound of 1/q for independence systems of rank quotient at least q.
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