Batching and Optimal Multi-stage Bipartite Allocations

In several applications of real-time matching of demand to supply in online marketplaces, the platform allows for some latency to batch the demand and improve the efficiency of the resulting matching. Motivated by these applications, we study the optimal trade-off between batching and inefficiency in the context of designing robust online allocations. In particular, we consider K-stage variants of the classic vertex weighted bipartite b-matching and AdWords problems in the adversarial setting, where online vertices arrive stage-wise and in K batches---in contrast to online arrival. Our main result for both problems is an optimal (1-(1-1/K)^K)-competitive (fractional) matching algorithm, improving the classic (1-1/e) competitive ratio bound known for the online variants of these problems (Mehta et al., 2007; Aggarwal et al., 2011).Our main technique at high-level is developing algorithmic tools to vary the trade-off between "greedy-ness'' and `"hedging" of the matching algorithm across stages. We rely on a particular family of convex-programming based matchings that distribute the demand in a specifically balanced way among supply in different stages, while carefully modifying the balancedness of the resulting matching across stages. More precisely, we identify a sequence of polynomials with decreasing degrees to be used as strictly concave regularizers of the maximum weight matching linear program to form these convex programs. At each stage, our fractional multi-stage algorithm returns the corresponding regularized optimal solution as the matching of this stage (by solving the convex program). By providing structural decomposition of the underlying graph using the optimal solutions of these convex programs and recursively connecting the regularizers together, we develop a new multi-stage primal-dual framework to analyze the competitive ratio of this algorithm. We further show this algorithm is optimal competitive, even in the unweighted case, by providing an upper-bound instance in which no online algorithm obtains a competitive ratio better than (1-(1-1/K)^K). We extend our results to integral allocations in the vertex weighted b-matching problem with large budgets, and in the AdWords problem with small bid over budget ratios