Optimal Groupon Allocations

Group-buying websites represented by Groupon.com are very popular in electronic commerce and online shopping nowadays. They have multiple slots to provide deals with significant discounts to their visitors every day. The current user traffic allocation mostly relies on human decisions. We study the problem of automatically allocating the user traffic of a group-buying website to different deals to maximize the total revenue and refer to it as the Group-buying Allocation Problem (\GAP). The key challenge of \GAP\ is how to handle the tipping point (lower bound) and the purchase limit (upper bound) of each deal. We formulate \GAP\ as a knapsack-like problem with variable-sized items and majorization constraints. Our main results for \GAP\ can be summarized as follows. (1) We first show that for a special case of \GAP, in which the lower bound equals the upper bound for each deal, there is a simple dynamic programming-based algorithm that can find an optimal allocation in pseudo-polynomial time. (2) The general case of \GAP\ 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 leveraging those properties. This algorithm can find an optimal allocation in pseudo-polynomial time. (3) We convert the two-layer dynamic programming based algorithm to a fully polynomial time approximation scheme (FPTAS), using the technique developed in \cite{ibarra1975fast}, combined with some careful modifications of the dynamic programs. Besides these results, we further investigate some natural generalizations of \GAP, and propose effective algorithms.