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.
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 .
@cite_0 @cite_15 @cite_16 @cite_11
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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. 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. 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]. We study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of @math , a very familiar bound that holds for many online problems; further, the bound is tight in this case. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the @math bound.Our main result is a @math -approximation online algorithm for stochastic bipartite matching, breaking this @math barrier. Furthermore, we show that no online algorithm can produce a @math approximation for an arbitrarily small @math for this problem. Our algorithms are based on computing an optimal offline solution to the expected instance, and using this solution as a guideline in the process of online allocation. We employ a novel application of the idea of the power of two choices from load balancing: we compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order. To identify these two disjoint solutions, we solve a max flow problem in a boosted flow graph, and then carefully decompose this maximum flow to two edge-disjoint (near-) matchings. In addition to guiding the online decision making, these two offline solutions are used to characterize an upper bound for the optimum in any scenario. This is done by identifying a cut whose value we can bound under the arrival distribution. At the end, we discuss extensions of our results to more general bipartite allocations that are important in a display ad application.