Optimal Allocation for Chunked-Reward Advertising

Chunked-reward advertising is commonly used in the industry, such as the guaranteed delivery in display advertising and the dailydeal services (e.g., Groupon) in online shopping. In chunked-reward advertising, the publisher promises to deliver at least a certain volume (a.k.a. tipping point or lower bound) of user traffic to an advertiser according to their mutual contract. At the same time, the advertiser may specify a maximum volume (upper bound) of traffic that he/she would like to pay for according to his/her budget constraint. The objective of the publisher is to design an appropriate mechanism to allocate the user traffic so as to maximize the overall revenue obtained from all such advertisers. In this paper, we perform a formal study on this problem, which we call Chunked-reward Allocation Problem (CAP). In particular, we formulate CAP as a knapsack-like problem with variable-sized items and majorization constraints. Our main results regarding CAP are as follows. (1) We first show that for a special case of CAP, in which the lower bound equals the upper bound for each contract, there is a simple dynamic programming-based algorithm that can find an optimal allocation in pseudo-polynomial time. (2) The general case of CAP is much more difficult than the special case. To solve the problem, we first discover several structural properties of the optimal allocation, and then design a two-layer dynamic programming-based algorithm that can find an optimal allocation in pseudo-polynomial time by leveraging these properties. (3) We convert the two-layer dynamic programming based algorithm to a fully polynomial time approximation scheme (FPTAS), using the technique developed in [8], combined with some careful modifications of the dynamic programs. Besides these results, we also investigate some natural generalizations of CAP, and propose effective algorithms to solve them.