Welfare Maximization with Production Costs: A Primal Dual Approach

We study online combinatorial auctions with production costs proposed by Blum et al. [4] using the online primal dual framework. In this model, buyers arrive online, and the seller can produce multiple copies of each item subject to a nondecreasing marginal cost per copy. The goal is to allocate items to maximize social welfare less total production cost. For arbitrary (strictly convex and differentiable) production cost functions, we characterize the optimal competitive ratio achievable by online mechanisms/algorithms. We show that online posted pricing mechanisms, which are incentive compatible, can achieve competitive ratios arbitrarily close to the optimal, and construct lower bound instances on which no online algorithms, not necessarily incentive compatible, can do better. Our positive results improve or match the results in several previous work, e.g., Bartal et al. [3], Blum et al. [4], and Buchbinder and Gonen [6]. Our lower bounds apply to randomized algorithms and resolve an open problem by Buchbinder and Gonen [6].