We study the problem of online prophet-inequality matching in bipartite graphs. There is a static set of bidders and an online stream of items. We represent the interest of bidders in items by a weighted bipartite graph. Each bidder has a capacity, i.e., an upper bound on the number of items that can be allocated to her. The weight of a matching is the total weight of edges matched to the bidders. Upon the arrival of an item, the online algorithm should either allocate it to a bidder or discard it. The objective is to maximize the weight of the resulting matching. We consider this model in a stochastic setting where we know the distribution of the incoming items in advance. Furthermore, we allow the items to be drawn from different distributions, i.e., we may assume that the t th item is drawn from distribution Dt. In contrast to i.i.d. model, this allows us to model the change in the distribution of items throughout the time. We call this setting the Prophet-Inequality Matching because of the possibility of having a different distribution for each time. We generalize the classic prophet inequality by presenting an algorithm with the approximation ratio of 1 − 1 √ k+3 where k is the minimum capacity. In case of k = 2, the algorithm gives a tight ratio of 1 2 which is a different proof of the prophet inequality. We also consider a model in which the bidders do not have a capacity, instead each bidder has a budget. The weight of a matching is the minimum of the budget of each vertex and the total weight of edges matched to it, when summed over all bidders. We show that if the bid to the budget ratio of every bidder is at most 1 k then a natural randomized online algorithm has an approximation ratio of 1 − k k e k k! ≈ 1 − 1 √ 2πk compared to the optimal offline (in which the ratio goes to 1 as k becomes large).We also present the applications of our model in Adword Allocation, Display Ad Allocation, and AdCell Model.

We consider the problem of estimating the size of a maximum matching when the edges are revealed in a streaming fashion. When the input graph is planar, we present a simple and elegant streaming algorithm that with high probability estimates the size of a maximum matching within a constant factor usingÕ(n 2/3 ) space, where n is the number of vertices. The approach generalizes to the family of graphs that have bounded arboricity, which include graphs with an excluded constant-size minor. To the best of our knowledge, this is the first result for estimating the size of a maximum matching in the adversarial-order streaming model (as opposed to the random-order streaming model) in o(n) space. We circumvent the barriers inherent in the adversarial-order model by exploiting several structural properties of planar graphs, and more generally, graphs with bounded arboricity. We further reduce the required memory size toÕ( √ n) for three restricted settings: (i) when the input graph is a forest; (ii) when we have 2-passes and the input graph has bounded arboricity; and (iii) when the edges arrive in random order and the input graph has bounded arboricity.Finally, we design a reduction from the Boolean Hidden Matching Problem to show that there is no randomized streaming algorithm that estimates the size of the maximum matching to within a factor better than 3/2 and uses only o(n 1/2 ) bits of space. Using the same reduction, we show that there is no deterministic algorithm that computes this kind of estimate in o(n) bits of space. The lower bounds hold even for graphs that are collections of paths of constant length.

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Abstract. We present a 1 − 1 √ k -competitive algorithm for the online stochastic generalized assignment problem under the assumption that no item takes up more than 1 k fraction of the capacity of any bin. Items arrive online; each item has a value and a size; upon arrival, an item can be placed in a bin or discarded; the objective is to maximize the total value of the placement. Both value and size of an item may depend on the bin in which the item is placed; the size of an item is revealed only after it has been placed in a bin; distribution information is available about the value and size of each item in advance (not necessarily i.i.d), however items arrive in adversarial order (non-adaptive adversary). We also present an application of our result to subscription-based advertising where each advertiser, if served, requires a given minimum number of impressions (i.e., the "all or nothing" model).

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