Online Matching with Cancellation Costs

Abstract

Motivated by applications in hotel upsell revenue management and selling banner ads on popular websites, we study the online resource allocation problem with overbooking and cancellation costs, also known as the "buyback" setting. To model this problem, we consider a variation of the classic edge-weighted online matching problem in which the decision maker can reclaim any fraction of an offline resource that is pre-allocated to an earlier online vertex; however, by doing so not only the decision maker loses the previously allocated edge-weight, it also has to pay a non-negative constant factor f of this edge-weight as an extra penalty.

Parameterizing the problem by the buyback factor f, our main result is obtaining optimal competitive algorithms for all possible values of f through a novel primal-dual family of algorithms. Interestingly, our result shows a phase transition: for small buyback regime, i.e., f<(e-2)/2, the optimal competitive ratio is e/(e-1-f), and for large buyback regime, i.e., f>(e-2)/2, the competitive ratio is W(-1/(e(1+f))), when W is the non-principal branch of the Lambert W function.

We establish the optimality of our results by obtaining separate lower-bounds for each of small and large buyback factor regimes, and showing how our primal-dual algorithm exactly matches this lower-bound by appropriately tuning a parameter as a function of f. We further study lower and upper bounds on the competitive ratio in variants of this model, e.g., single-resource with different demand sizes, or matching with deterministic integral allocations. We show how algorithms in the our family of primal-dual algorithms can obtain the exact optimal competitive ratio in all of these variants --- which in turn demonstrates the power of our algorithmic framework for online resource allocations with costly buyback.

Keywords: Buyback problem, overbooking, cancellations, recallable resources, online resource allocation, primal-dual