We introduce a weighted version of the ranking algorithm by (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 (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.
Recently, Cohen and Wajc @cite_3 considered the (with arbitrary arrival order) on regular graphs, and provided a @math -competitive algorithm, where @math is the degree of vertices. Very recently, @cite_10 proposed a fully online matching model, in which all vertices of the graph arrive online (in an arbitrary order). Extending the randomized primal-dual technique, they obtained competitive ratios above @math for both bipartite graphs and general graphs.
@cite_10 @cite_3
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We introduce a fully online model of maximum cardinality matching in which all vertices arrive online. On the arrival of a vertex, its incident edges to previously-arrived vertices are revealed. Each vertex has a deadline that is after all its neighbors’ arrivals. If a vertex remains unmatched until its deadline, the algorithm must then irrevocably either match it to an unmatched neighbor, or leave it unmatched. The model generalizes the existing one-sided online model and is motivated by applications including ride-sharing platforms, real-estate agency, etc. We show that the Ranking algorithm by (STOC 1990) is 0.5211-competitive in our fully online model for general graphs. Our analysis brings a novel charging mechanic into the randomized primal dual technique by (SODA 2013), allowing a vertex other than the two endpoints of a matched edge to share the gain. To our knowledge, this is the first analysis of Ranking that beats 0.5 on general graphs in an online matching problem, a first step towards solving the open problem by (STOC 1990) about the optimality of Ranking on general graphs. If the graph is bipartite, we show that the competitive ratio of Ranking is between 0.5541 and 0.5671. Finally, we prove that the fully online model is strictly harder than the previous model as no online algorithm can be 0.6317 In this paper we study the classic online matching problem, introduced in the seminal work of Karp, Vazirani and Vazirani (STOC 1990), in regular graphs. For such graphs, an optimal deterministic algorithm as well as efficient algorithms under stochastic input assumptions were known. In this work, we present a novel randomized algorithm with competitive ratio tending to one on this family of graphs, under adversarial arrival order. Our main contribution is a novel algorithm which achieves competitive ratio [EQUATION] in expectation on d-regular graphs. In contrast, we show that all previously-studied online algorithms have competitive ratio strictly bounded away from one. Moreover, we show the convergence rate of our algorithm's competitive ratio to one is nearly tight, as no algorithm achieves competitive ratio better than [EQUATION]. Finally, we show that our algorithm yields a similar competitive ratio with high probability, as well as guaranteeing each vertex a probability of being matched tending to one. In a seminal paper, Karp, Vazirani, and Vazirani show that a simple ranking algorithm achieves a competitive ratio of 1-1 e for the online bipartite matching problem in the standard adversarial model, where the ratio of 1-1 e is also shown to be optimal. Their result also implies that in the random arrivals model defined by Goel and Mehta, where the online nodes arrive in a random order, a simple greedy algorithm achieves a competitive ratio of 1-1 e. In this paper, we study the ranking algorithm in the random arrivals model, and show that it has a competitive ratio of at least 0.696, beating the 1-1 e ≈ 0.632 barrier in the adversarial model. Our result also extends to the i.i.d. distribution model of , removing the assumption that the distribution is known. Our analysis has two main steps. First, we exploit certain dominance and monotonicity properties of the ranking algorithm to derive a family of factor-revealing linear programs (LPs). In particular, by symmetry of the ranking algorithm in the random arrivals model, we have the monotonicity property on both sides of the bipartite graph, giving good "strength" to the LPs. Second, to obtain a good lower bound on the optimal values of all these LPs and hence on the competitive ratio of the algorithm, we introduce the technique of strongly factor-revealing LPs. In particular, we derive a family of modified LPs with similar strength such that the optimal value of any single one of these new LPs is a lower bound on the competitive ratio of the algorithm. This enables us to leverage the power of computer LP solvers to solve for large instances of the new LPs to establish bounds that would otherwise be difficult to attain by human analysis. We study the following vertex-weighted online bipartite matching problem: G(U, V, E) is a bipartite graph. The vertices in U have weights and are known ahead of time, while the vertices in V arrive online in an arbitrary order and have to be matched upon arrival. The goal is to maximize the sum of weights of the matched vertices in U. When all the weights are equal, this reduces to the classic online bipartite matching problem for which Karp, Vazirani and Vazirani gave an optimal (1−1 e)-competitive algorithm in their seminal work [10]. Our main result is an optimal (1−1 e)-competitive randomized algorithm for general vertex weights. We use random perturbations of weights by appropriately chosen multiplicative factors. Our solution constitutes the first known generalization of the algorithm in [10] in this model and provides new insights into the role of randomization in online allocation problems. It also effectively solves the problem of online budgeted allocations [14] in the case when an agent makes the same bid for any desired item, even if the bid is comparable to his budget - complementing the results of [14, 3] which apply when the bids are much smaller than the budgets.