Optimal Algorithms for Online b-Matching with Variable Vertex Capacities

We study the b-matching problem, which generalizes classical online matching introduced by Karp, Vazirani and Vazirani (STOC 1990). Consider a bipartite graph G = (S ?? R,E). Every vertex s ? S is a server with a capacity b_s, indicating the number of possible matching partners. The vertices r ? R are requests that arrive online and must be matched immediately to an eligible server. The goal is to maximize the cardinality of the constructed matching. In contrast to earlier work, we study the general setting where servers may have arbitrary, individual capacities. We prove that the most natural and simple online algorithms achieve optimal competitive ratios. As for deterministic algorithms, we give a greedy algorithm RelativeBalance and analyze it by extending the primal-dual framework of Devanur, Jain and Kleinberg (SODA 2013). In the area of randomized algorithms we study the celebrated Ranking algorithm by Karp, Vazirani and Vazirani. We prove that the original Ranking strategy, simply picking a random permutation of the servers, achieves an optimal competitiveness of 1-1/e, independently of the server capacities. Hence it is not necessary to resort to a reduction, replacing every server s by b_s vertices of unit capacity and to then run Ranking on this graph with ?_{s ? S} b_s vertices on the left-hand side. From a theoretical point of view our result explores the power of randomization and strictly limits the amount of required randomness. From a practical point of view it leads to more efficient allocation algorithms. Technically, we show that the primal-dual framework of Devanur, Jain and Kleinberg cannot establish a competitiveness better than 1/2 for the original Ranking algorithm, choosing a permutation of the servers. Therefore, we formulate a new configuration LP for the b-matching problem and then conduct a primal-dual analysis. We extend this analysis approach to the vertex-weighted b-matching problem. Specifically, we show that the algorithm PerturbedGreedy by Aggarwal, Goel, Karande and Mehta (SODA 2011), again with a sole randomization over the set of servers, is (1-1/e)-competitive. Together with recent work by Huang and Zhang (STOC 2020), our results demonstrate that configuration LPs can be strictly stronger than standard LPs in the analysis of more complex matching problems

Matroid Online Bipartite Matching and Vertex Cover

The Adwords and Online Bipartite Matching problems have enjoyed a renewed attention over the past decade due to their connection to Internet advertising. Our community has contributed, among other things, new models (notably stochastic) and extensions to the classical formulations to address the issues that arise from practical needs. In this paper, we propose a new generalization based on matroids and show that many of the previous results extend to this more general setting. Because of the rich structures and expressive power of matroids, our new setting is potentially of interest both in theory and in practice. In the classical version of the problem, the offline side of a bipartite graph is known initially while vertices from the online side arrive one at a time along with their incident edges. The objective is to maintain a decent approximate matching from which no edge can be removed. Our generalization, called Matroid Online Bipartite Matching, additionally requires that the set of matched offline vertices be independent in a given matroid. In particular, the case of partition matroids corresponds to the natural scenario where each advertiser manages multiple ads with a fixed total budget. Our algorithms attain the same performance as the classical version of the problems considered, which are often provably the best possible. We present $1-1/e$-competitive algorithms for Matroid Online Bipartite Matching under the small bid assumption, as well as a $1-1/e$-competitive algorithm for Matroid Online Bipartite Matching in the random arrival model. A key technical ingredient of our results is a carefully designed primal-dual waterfilling procedure that accommodates for matroid constraints. This is inspired by the extension of our recent charging scheme for Online Bipartite Vertex Cover.Comment: 19 pages, to appear in EC'1

Online Matching with Stochastic Rewards: Optimal Competitive Ratio via Path Based Formulation

The problem of online matching with stochastic rewards is a generalization of the online bipartite matching problem where each edge has a probability of success. When a match is made it succeeds with the probability of the corresponding edge. Introducing this model, Mehta and Panigrahi (FOCS 2012) focused on the special case of identical edge probabilities. Comparing against a deterministic offline LP, they showed that the Ranking algorithm of Karp et al. (STOC 1990) is 0.534 competitive and proposed a new online algorithm with an improved guarantee of $0.567$ for vanishingly small probabilities. For the case of vanishingly small but heterogeneous probabilities Mehta et al. (SODA 2015), gave a 0.534 competitive algorithm against the same LP benchmark. For the more general vertex-weighted version of the problem, to the best of our knowledge, no results being $1/2$ were previously known even for identical probabilities. We focus on the vertex-weighted version and give two improvements. First, we show that a natural generalization of the Perturbed-Greedy algorithm of Aggarwal et al. (SODA 2011), is $(1-1/e)$ competitive when probabilities decompose as a product of two factors, one corresponding to each vertex of the edge. This is the best achievable guarantee as it includes the case of identical probabilities and in particular, the classical online bipartite matching problem. Second, we give a deterministic $0.596$ competitive algorithm for the previously well studied case of fully heterogeneous but vanishingly small edge probabilities. A key contribution of our approach is the use of novel path-based analysis. This allows us to compare against the natural benchmarks of adaptive offline algorithms that know the sequence of arrivals and the edge probabilities in advance, but not the outcomes of potential matches.Comment: Preliminary version in EC 202

Online Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals

We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time goes by, and each online vertex matches the neighbor with the highest offer at its arrival