Frequency capping in online advertising

We study the following online problem. There are n advertisers. Each advertiser $$a_i$$ai has a total demand $$d_i$$di and a value $$v_i$$vi for each supply unit. Supply units arrive one by one in an online fashion, and must be allocated to an agent immediately. Each unit is associated with a user, and each advertiser $$a_i$$ai is willing to accept no more than $$f_i$$fi units associated with any single user (the value $$f_i$$fi is called the frequency cap of advertiser $$a_i$$ai). The goal is to design an online allocation algorithm maximizing the total value. We first show a deterministic $$3/4$$3/4-competitiveness upper bound, which holds even when all frequency caps are $$1$$1, and all advertisers share identical values and demands. A competitive ratio approaching $$1-1/e$$1-1/e can be achieved via a reduction to a different model considered by Goel and Mehta (WINE ‘07: Workshop on Internet and Network, Economics: 335–340, 2007). Our main contribution is analyzing two $$3/4$$3/4-competitive greedy algorithms for the cases of equal values, and arbitrary valuations with equal integral demand to frequency cap ratios. Finally, we give a primal-dual algorithm which may serve as a good starting point for improving upon the ratio of $$1-1/e$$1-1/e.