A Tight Linear Time ( 1 / 2 ) - Approximation for Unconstrained Submodular

Consider a manager hiring workers to staff a new factory. Moving from zero employees to a thousand employees exponentially increases the factory’s production. Recognizing a pattern, the manager again hires a thousand employees. This increase in workers also raises factory production because machines are now used more efficiently, but the effect is not quite as dramatic as opening the factory from scratch. Emboldened by his previous success, the manager decides to hire another thousand workers. However, this time the additional workers leads to overcrowding, and the factory’s production barely increases. This phenomenon, the principle of diminishing returns, is a foundational concept in economics. Intuitively, adding an additional unit of a good to a small set has a much larger impact compared to adding a single unit of a good to a larger set. This notion is rigorously captured by the theory of submodular functions, which we define as follows.