We study the online budgeted allocation (also called ADWORDS) problem, where a set of impressions arriving online are allocated to a set of budget-constrained advertisers to maximize revenue. Motivated by connections to Internet advertising, several variants of this problem have been studied since the seminal work of Mehta, Saberi, Vazirani, and Vazirani (FOCS 2005). However, this entire body of work focuses on a single budget for every advertising campaign, whereas in order to fully represent the actual agenda of an advertiser, an advertising budget should be expressible over multiple tiers of user-attribute granularity. A simple example is an advertising campaign that is constrained by an overall budget but is also accompanied by a set of sub-budgets for each target demographic. In such a contract scheme, an advertiser can specify their true user-targeting goals, allowing the publisher to fulfill them through relevant allocations. In this paper, we give a complete characterization of the ADWORDS problem for general advertising budgets. In the most general setting, we show that, unlike in the single-budget ADWORDS problem, obtaining a constant competitive ratio is impossible and give asymptotically tight upper and lower bounds. However for our main result, we observe that in many real-world scenarios (as in the above example), multi-tier budgets have a laminar structure, since most relevant consumer or product classifications are hierarchical. For laminar budgets, we obtain a competitive ratio of e/(e − 1) in the small bids case, which matches the best known ADWORDS result for single budgets. Our algorithm has a primal-dual structure and generalizes the primaldual analysis for single-budget ADWORDS first given by Buchbinder, Jain, and Naor (ESA 2007). However many new ideas are required to overcome the barriers introduced by laminar budgets-our algorithm uses a novel formulation that overcomes non-monotonicity in the syntactically defined dual variables, as well as a dynamically maintained labeling scheme that tracks the "most-limiting" budgets in the hierarchy.
REF considered Online Budgeted Allocation with General Budgets, where offline agents have multiple tiers of budget constraints forming a laminar structure.
Online Budgeted Allocation with General Budgets
EC '16
Mathematics Computer Science