The Santa Claus problem is a fundamental problem in {\em fair division}: the goal is to partition a set of {\em heterogeneous} items among {\em heterogeneous} agents so as to maximize the minimum value of items received by any agent. In this paper, we study the online version of this problem where the items are not known in advance and have to be assigned to agents as they arrive over time. If the arrival order of items is arbitrary, then no good assignment rule exists in the worst case. However, we show that, if the arrival order is random, then for $n$ agents and any $\varepsilon > 0$, we can obtain a competitive ratio of $1-\varepsilon$ when the optimal assignment gives value at least $\Omega(\log n / \varepsilon^2)$ to every agent (assuming each item has at most unit value). We also show that this result is almost tight: namely, if the optimal solution has value at most $C \ln n / \varepsilon$ for some constant $C$, then there is no $(1-\varepsilon)$-competitive algorithm even for random arrival order.

We study mechanism design for procurement auctions in which the goal is to buy a subset of items or hire a team of providers. In order to measure the efficiency of a mechanism, one defines an appropriate benchmark which denotes a reasonable expectation of the payments and defines the overpayment of a mechanism based on the benchmark. This ratio is called the \em frugality ratio of the mechanism. Procurement auctions are well-studied and benchmarks proposed for these auctions have evolved over a sequence of papers ~\citearcher2007frugal,chen2010frugal,elkind2007frugality,karlinbeyond,kempe2010frugal. In this work, we introduce a newer benchmark, and based on that, study classic procurement auctions. Our benchmark addresses critical issues raised by the unintuitive behavior of the previous benchmarks. We show two attractive properties for our benchmark which have been lacking in the previous proposals: \em monotonicity and \em smoothness. Based on our benchmark, we provide positive results for vertex cover and knapsack auctions. Prior to this work, \kempefrugal\citekempe2010frugal propose a constant approximation mechanism for vertex cover auctions. However, their analysis suffers from an error. We give a correct analysis to the mechanism of \kempefrugal\citekempe2010frugal with respect to our benchmark. In particular, we prove their mechanism is optimal up to a constant factor. Our analysis is different from what \kempefrugal\citekempe2010frugal propose. We also study the knapsack auctions and give a truthful mechanism for such auctions with a bounded frugality ratio. We show that this is almost tight by presenting a lower bound on the frugality ratio of any truthful mechanism for such auctions. All our results depend on both properties of the benchmark.

As online ad offerings become increasingly complex, with multiple size configurations and layouts available to advertisers, the sale of web advertising space increasingly resembles a combinatorial auction with complementarities. Standard ad auction formats do not immediately extend to these settings, and truthful combinatorial auctions, such as the Vickrey-Clarke-Groves auction, can yield unacceptably low revenue. Core selecting auctions, which apply to combinatorial markets, boost revenue by setting prices so that no group of agents, including the auctioneer, can jointly improve their utilities by switching to a different allocation and payments. Among outcomes in the core, bidder-optimal core points have been the most widely studied due to their incentive properties, such as being implementable at natural equilibria.

The Generalized Second Price (GSP) auction has appealing properties when ads are simple (text based and identical in size), but does not generalize to richer ad settings, whereas truthful mechanisms such as VCG do. However, a straight switch from GSP to VCG incurs significant revenue loss for the search engine. We introduce a transitional mechanism which encourages advertisers to update their bids to their valuations, while mitigating revenue loss. In this setting, it is easier to propose first a payment function rather than an allocation function, so we give a general framework which guarantees incentive compatibility by requiring that the payment functions satisfy two specific properties. Finally, we analyze the revenue impacts of our mechanism on a sample of Bing data.

In the Movement Repairmen (MR) problem, we are given a metric space ( V , d ) along with a set R of k repairmen r 1 , r 2 , …, r k with their start depots s 1 , s 2 , …, s k ∈ V and speeds v 1 , v 2 , …, v k ⩾ 0, respectively, and a set C of m clients c 1 , c 2 , …, c m having start locations s ′ 1 , s ′ 2 , …, s ′ m ∈ V and speeds v ′ 1 , v ′ 2 , …, v ′ m ⩾ 0, respectively. If t is the earliest time a client c j is collocated with any repairman (say, r i ) at a node u , we say that the client is served by r i at u and that its latency is t . The objective in the (S um -MR) problem is to plan the movements for all repairmen and clients to minimize the sum (average) of the clients’ latencies. The motivation for this problem comes, for example, from Amazon Locker Delivery [Amazon 2010] and USPS gopost [Service 2010]. We give the first O (log n )-approximation algorithm for the S um -MR problem. In order to approximate S um -MR, we formulate an LP for the problem and bound its integrality gap. Our LP has exponentially many variables; therefore, we need a separation oracle for the dual LP. This separation oracle is an instance of the Neighborhood Prize Collecting Steiner Tree (NPCST) problem in which we want to find a tree with weight at most L collecting the maximum profit from the clients by visiting at least one node from their neighborhoods. The NPCST problem, even with the possibility to violate both the tree weight and neighborhood radii, is still very hard to approximate. We deal with this difficulty by using LP with geometrically increasing segments of the timeline, and by giving a tricriteria approximation for the problem. The rounding needs a relatively involved analysis. We give a constant approximation algorithm for S um -MR in Euclidean Space where the speed of the clients differs by a constant factor. We also give a constant approximation for the makespan variant.

One of the major drawbacks of the celebrated VCG auction is its low (or zero) revenue even when the agents have high value for the goods and a competitive outcome would have generated a significant revenue. A competitive outcome is one for which it is impossible for the seller and a subset of buyers to 'block' the auction by defecting and negotiating an outcome with higher payoffs for themselves. This corresponds to the well-known concept of core in cooperative game theory.

One of the important optimization questions in highly parallel systems is the problem of assigning computational resources to communicating tasks. While scheduling tasks/operators, tasks assigned to nearby resources (e.g. on the same CPU core) have low communication costs, whereas tasks assigned to distant resources (e.g. on different server racks) have high communication costs. An optimal solution of task to resource assignment minimizes the communication cost of the task ensemble while satisfying the load balancing requirements. We model such an optimization question of minimizing communication cost as a new class of graph partitioning problems called hierarchical graph partitioning.

Online advertising is an essential part of the Internet and the main source of revenue for many web-centric firms such as search engines, social networks, and online publishers. A key component of online advertising is the auction mechanism which selects and prices the set of winning ads. This work is inspired by one of the biggest practical drawbacks of the widely popular Vickrey-Clarke-Groves (VCG) mechanism, which is the unique incentive-compatible mechanism that maximizes social welfare. It is known that VCG lacks a desired property of revenue monotonicity - a natural notion which states that the revenue of a mechanism shouldn't go down as the number of bidders increase or if the bidders increase their bids. Most firms which depend on online advertising revenue have a large sales team to attract more bidders on their inventory as the general belief is that more bidders will increase competition, and hence revenue. However, the lack of revenue monotonicity of VCG conflicts with this general belief and can be strategically confusing for the firm's business.