text: We study the average performance of online greedy matching algorithms on G(n,n,p), the random bipartite graph with n vertices on each side and edges occurring independently with probability p = p(n). In the online model, vertices on one side of the graph are given up front while vertices on the other side arrive sequentially; when a vertex arrives its edges are revealed and it must be immediately matched or dropped. We begin by analyzing the oblivious algorithm, which tries to match each arriving vertex to a random neighbor, even if the neighbor has already been matched. The algorithm is shown to have a performance ratio of at least 1 − 1 e for all monotonic functions p(n), where the performance ratio is defined asymptotically as the ratio of the expected matching size given by the algorithm to the expected maximum matching size. Next we show that the conventional greedy algorithm, which assigns each vertex to a random unmatched neighbor, has a performance ratio of at least 0.837 for all monotonic functions p(n). Under the G(n,n,p) model, the performance of greedy is equivalent to the performance of the well known ranking algorithm, so our results show that ranking has a performance ratio of at least 0.837. We finally consider vertex-weighted bipartite matching. Our proofs are based on simple differential equations that describe the evolution of the matching process.
  @cite_0: There has been a great deal of interest recently in the relative power of on-line and off-line algorithms. An on-line algorithm receives a sequence of requests and must respond to each request as soon as it is receiveD. An off-line algorithm may wait until all requests have been received before determining its responses. One approach to evaluating an on-line algorithm is to compare its performance with that of the best possible off-line algorithm for the same problem. Thus, given a measure of "profit", the performance of an on-line algorithm can be measured by the worst-case ratio of its profit to that of the optimal off-line algorithm. This general approach has been applied in a number of contexts, including data structures [SITa], bin packing [CoGaJo], graph coloring [GyLe] and the k-server problem [MaMcSI]. Here we apply it to bipartite matching and show that a simple randomized on-line algorithm achieves the best possible performance.
  @cite_15: 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.
  @cite_16: We examine the classic on-line bipartite matching problem studied by Karp, Vazirani, and Vazirani [8] and provide a simple proof of their result that the Ranking algorithm for this problem achieves a competitive ratio of 1 -- 1 e.
  @cite_11: We give a simple proof that the ranking algorithm of Karp, Vazirani and Vazirani [KVV90] is 1-1 e competitive for the online bipartite matching problem. The proof is via a randomized primal-dual argument. Primal-dual algorithms have been successfully used for many online algorithm problems, but the dual constraints are always satisfied deterministically. This is the first instance of a non-trivial randomized primal-dual algorithm in which the dual constraints only hold in expectation. The approach also generalizes easily to the vertex-weighted version considered by [AGKM11]. Further we show that the proof is very similar to the deterministic primal-dual argument for the online budgeted allocation problem with small bids (also called the AdWords problem) of [MSVV05].
summary: The online matching problem was originally analyzed by Karp, Vazirani and Vazirani, who introduced the algorithm and showed that it obtains a competitive ratio of @math @cite_0 . Simpler proofs of the algorithm have since been found @cite_11 @cite_16 . A @math competitive algorithm is also known for vertex-weighted online bipartite matching, which was given by Aggarwal, Goel, Karande, and Mehta @cite_15 .