Prompt mechanism for ad placement over time

Consider video ad placement into commercial breaks in a television channel. The ads arrive online over time and each has an expiration date. The commercial breaks are typically of some uniform duration; however, the video ads may have an arbitrary size. Each ad has a private value and should be posted into some break at most once by its expiration date. The player who own the ad gets her value if her ad had been broadcasted by the ad's expiration date (obviously, after ad's arrival date), and zero value otherwise. Arranging the ads into the commercial breaks while maximizing the players' profit is a classical problem of ad placement subject to the capacity constraint that should be solved truthfully. However, we are interested not only in truthfulness but also in a prompt mechanism where the payment is determined for an agent at the very moment of the broadcast. The promptness of the mechanism is a crucial requirement for our algorithm, since it allows a payment process without any redundant relation between an auctioneer and players. An inability to resolve this problem could even prevent the application of such mechanisms in a real marketing process. We design a 6-approximation prompt mechanism for the problem. Previously Cole et al considered a special case where all ads have the same size which is equal to the break duration. For this particular case they achieved a 2-approximation prompt mechanism. The general case of ads with arbitrary size is considerably more involved and requires designing a new algorithm, which we call the Gravity Algorithm.