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 −

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We consider the online bipartite matching problem in the unknown distribution input model. We show that the Ranking algorithm of [KVV90] achieves a competitive ratio of at least 0.653. This is the first analysis to show an algorithm which breaks the natural 1 − 1/e 'barrier' in the unknown distribution model (our analysis in fact works in the stricter, random order model) and answers an open question in [GM08]. We also describe a family of graphs on which Ranking does no better than 0.727 in the random order model. Finally, we show that for graphs which have k > 1 disjoint perfect matchings, Ranking achieves a competitive ratio of at least 1 − 1 k − 1 k 2 + 1 n -in particular Ranking achieves a factor of 1 − o(1) for graphs with ω(1) disjoint perfect matchings.

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Search engine ad auctions typically have a significant fraction of advertisers who are budget constrained, i.e., if allowed to participate in every auction that they bid on, they would spend more than their budget. This yields an important problem: selecting the ad auctions in which these advertisers participate, in order to optimize different system objectives such as the return on investment for advertisers, and the quality of ads shown to users. We present a system and algorithms for optimizing such budget constrained spend. The system is designed be deployed in a large search engine, with hundreds of thousands of advertisers, millions of searches per hour, and with the query stream being only partially predictable. We have validated the system design by implementing it in the Google ads serving system and running experiments on live traffic. We have also compared our algorithm to previous work that casts this problem as a large linear programming problem limited to popular queries, and show that our algorithms yield substantially better results.

A variety of lossless compression schemes have been proposed to reduce the storage requirements of web graphs. One successful approach is virtual node compression [7], in which often-used patterns of links are replaced by links to virtual nodes, creating a compressed graph that succinctly represents the original. In this paper, we show that several important classes of web graph algorithms can be extended to run directly on virtual node compressed graphs, such that their running times depend on the size of the compressed graph rather than the original. These include algorithms for link analysis, estimating the size of vertex neighborhoods, and a variety of algorithms based on matrix-vector products and random walks. Similar speed-ups have been obtained previously for classical graph algorithms like shortest paths and maximum bipartite matching. We measure the performance of our modified algorithms on several publicly available web graph datasets, and demonstrate significant empirical speedups that nearly match the compression ratios.

A variety of lossless compression schemes has been proposed to reduce the storage requirements of web graphs. One successful approach is virtual-node compression [Buehrer and Chellapilla 08], in which often-used patterns of links are replaced by links to virtual nodes, creating a compressed graph that succinctly represents the original. In this paper, we show that several important classes of web graph algorithms can be extended to run directly on virtual-node-compressed graphs, such that their running times depend on the size of the compressed graph rather than on that of the original. These include algorithms for link analysis, estimating the size of vertex neighborhoods, and a variety of algorithms based on matrix-vector products and random walks. Similar speedups have been obtained previously for classical graph algorithms such as shortest paths and maximum bipartite matching. We measure the performance of our modified algorithms on several publicly available web graph data sets, and demonstrate significant empirical speedups that nearly match the compression ratios.

Identical products being sold at different prices in different locations is a common phenomenon. Price differences might occur due to various reasons such as shipping costs, trade restrictions and price discrimination. To model such scenarios, we supplement the classical Fisher model of a market by introducing transaction costs. For every buyer i and every good j, there is a transaction cost of c ij ; if the price of good j is p j , then the cost to the buyer i per unit of j is p j + c ij . This allows the same good to be sold at different (effective) prices to different buyers.We provide a combinatorial algorithm that computes ǫ-approximate equilibrium prices and allocations in O 1 ǫ (n + log m)mn log(B/ǫ) operations -where m is the number goods, n is the number of buyers and B is the sum of the budgets of all the buyers.In order to capture all these scenarios, we supplement the classical Fisher model of a market (see below for a formal definition) by introducing transaction costs. For every buyer i and every good j, there is a transaction cost of c ij ; if the price of good j is p j , then the cost to the buyer i per unit of j is p j + c ij . This allows the same good to be sold at different (effective) prices to different buyers. Note that apart * Technion. sourav@cs.technion.ac.il † Microsoft Research.