A Unified Approach to Online Allocation Algorithms via Randomized Dual Fitting

We present a unified framework for designing and analyzing algorithms for online budgeted allocation problems (including online matching) and their generalization, the Online Generalized Assignment Problem (OnGAP). These problems have been intensively studied as models of how to allocate impressions for online advertising. In contrast to previous analyses of online budgeted allocation algorithms (the so-called "balance" or "water-filling" family of algorithms) our analysis is based on the method of randomized dual fitting, analogous to the recent analysis of the RANKING algorithm for online matching due to Devanur et al. Our main contribution is thus to provide a unified method of proof that simultaneously derives the optimal competitive ratio bounds for online matching and online fractional budgeted allocation. The same method of proof also supplies $(1-1/e)$ competitive ratio bounds for greedy algorithms for both problems, in the random order arrival model; this simplifies existing analyses of greedy online allocation algorithms with random order of arrivals, while also strengthening them to apply to a larger family of greedy algorithms. Finally, for the more general OnGAP problem, we show that no algorithm can be constant-competitive; instead we present an algorithm whose competitive ratio depends logarithmically on a certain parameter of the problem instance, and we show that this dependence cannot be improved.