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 −

In this paper, we present the first approximation algorithms for the problem of designing revenue optimal Bayesian incentive compatible auctions when there are multiple (heterogeneous) items and when bidders have arbitrary demand and budget constraints (and additive valuations). Our mechanisms are surprisingly simple: We show that a sequential all-pay mechanism is a 4 approximation to the revenue of the optimal ex-interim truthful mechanism with a discrete type space for each bidder, where her valuations for different items can be correlated. We also show that a sequential posted price mechanism is a O(1) approximation to the revenue of the optimal ex-post truthful mechanism when the type space of each bidder is a product distribution that satisfies the standard hazard rate condition. We further show a logarithmic approximation when the hazard rate condition is removed, and complete the picture by showing that achieving a sub-logarithmic approximation, even for regular distributions and one bidder, requires pricing bundles of items. Our results are based on formulating novel LP relaxations for these problems, and developing generic rounding schemes from first principles.

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In this paper we consider a mechanism design problem in the context of large-scale crowdsourcing markets such as Amazon's Mechanical Turk (MTrk), ClickWorker (ClkWrkr), CrowdFlower (CrdFlwr). In these markets, there is a requester who wants to hire workers to accomplish some tasks. Each worker is assumed to give some utility to the requester on getting hired. Moreover each worker has a minimum cost that he wants to get paid for getting hired. This minimum cost is assumed to be private information of the workers. The question then is -if the requester has a limited budget, how to design a direct revelation mechanism that picks the right set of workers to hire in order to maximize the requester's utility?We note that although the previous work (Singer (2010); Chen et al. (2011)) has studied this problem, a crucial difference in which we deviate from earlier work is the notion of large-scale markets that we introduce in our model. The notion of a large-scale market that we consider is a natural one which states that the (private) cost of each worker is small compared to the budget of the requester. Without the large market assumption, it is known that no mechanism can achieve a competitive ratio better than 0.414 and 0.5 for deterministic and randomized mechanisms respectively (while the best known deterministic and randomized mechanisms achieve an approximation ratio of 0.292 and 0.33 respectively). In this paper, we design a budget-feasible mechanism for large markets that achieves a competitive ratio of 1−1/e 0.63. Our mechanism can be seen as a generalization of an alternate way to look at the proportional share mechanism, which is used in all the previous works so far on this problem. Interestingly, we can also show that our mechanism is optimal by showing that no truthful mechanism can achieve a factor better than 1 − 1/e; thus, fully resolving this setting. Finally we consider the more general case of submodular utility functions and give new and improved mechanisms for the case when the market is large.

A central issue in applying auction theory in practice is the problem of dealing with budget-constrained agents. A desirable goal in practice is to design incentive compatible, individually rational, and Pareto optimal auctions while respecting the budget constraints. Achieving this goal is particularly challenging in the presence of nontrivial combinatorial constraints over the set of feasible allocations.Toward this goal and motivated by AdWords auctions, we present an auction for polymatroidal environments satisfying these properties. Our auction employs a novel clinching technique with a clean geometric description and only needs an oracle access to the submodular function defining the polymatroid. As a result, this auction not only simplifies and generalizes all previous results, it applies to several new applications including AdWords Auctions, bandwidth markets, and video on demand. In particular, our characterization of the AdWords auction as polymatroidal constraints might be of independent interest. This allows us to design the first mechanism for Ad Auctions taking into account simultaneously budgets, multiple keywords and multiple slots.We show that it is impossible to extend this result to generic polyhedral constraints. This also implies an impossibility result for multiunit auctions with decreasing marginal utilities in the presence of budget constraints.

Abstract-Motivated by an application in kidney exchange, we study the following query-commit problem: we are given the set of vertices of a non-bipartite graph G. The set of edges in this graph are not known ahead of time. We can query any pair of vertices to determine if they are adjacent. If the queried edge exists, we are committed to match the two endpoints. Our objective is to maximize the size of the matching.This restriction in the amount of information available to the algorithm constraints us to implement myopic, greedy-like algorithms. A simple deterministic greedy algorithm achieves a factor 1/2 which is tight for deterministic algorithms. An important open question in this direction is to give a randomized greedy algorithm that has a significantly better approximation factor. This question was first asked almost 20 years ago by Dyer and Frieze [9] where they showed that a natural randomized strategy of picking edges uniformly at random doesn't help and has an approximation factor of 1/2 + o(1). They left it as an open question to devise a better randomized greedy algorithm. In subsequent work, Aronson, Dyer, Frieze, and Suen [2] gave a different randomized greedy algorithm and showed that it attains a factor 0.5 + ǫ where ǫ is 0.0000025.In this paper we propose and analyze a new randomized greedy algorithm for finding a large matching in a general graph and use it to solve the query commit problem mentioned above. We show that our algorithm attains a factor of at least 0.56, a significant improvement over 0.50000025. We also show that no randomized algorithm can have an approximation factor better than 0.7916 for the query commit problem. For another large and interesting class of randomized algorithms that we call vertex-iterative algorithms, we show that no vertexiterative algorithm can have an approximation factor better than 0.75.