In this work, we study ad conversion measurement, a central functionality in digital advertising, where an advertiser seeks to estimate advertiser website (or mobile app) conversions attributed to ad impressions that users have interacted with on various publisher websites (or mobile apps). Using differential privacy (DP), a notion that has gained in popularity due to its strong mathematical guarantees, we develop a formal framework for private ad conversion measurement. In particular, we define the notion of an operationally valid configuration of the attribution rule, DP adjacency relation, contribution bounding scope and enforcement point. We then provide, for the set of configurations that most commonly arises in practice, a complete characterization, which uncovers a delicate interplay between attribution and privacy.

We introduce a new decomposition technique for random variables that maps a generic instance of the prophet inequalities problem to a new instance where all but a constant number of variables have a tractable structure that we refer to as (ε, δ)-smallness. Using this technique, we make progress on several outstanding problems in the area: We show that, even in the case of non-identical distributions, it is possible to achieve (arbitrarily close to) the optimal approximation ratio of β ~0.745 when the items arrive in a random order (this version is commonly known as prophet secretary) as long as we are allowed to remove a small constant number of distributions. We show that forfrequent instances (where each distribution reoccurs some number of times) and random arrival order, it is possible to achieve the optimal approximation ratio of β (improving over the previous best-known bound of 0.738). We give a new, simpler proof of Kertz's optimal approximation guarantee of β ~0.745 for prophet inequalities with i.i.d. distributions. The proof is primal-dual and simultaneously produces upper and lower bounds. Using this decomposition in combination with a novel convex programming formulation, we construct the first (in parallel work with[1]) an Efficient PTAS (EPTAS) for the Optimal Ordering problem.

We study the question of setting and testing reserve prices in single item auctions when the bidders are not identical. At a high level, there are two generalizations of the standard second price auction: in the lazy version we first determine the winner, and then apply reserve prices; in the eager version we first discard the bidders not meeting their reserves, and then determine the winner among the rest. We show that the two versions have dramatically different properties: lazy reserves are easy to optimize, and A/B test in production, whereas eager reserves always lead to higher welfare, but their optimization is NP-complete, and naive A/B testing will lead to incorrect conclusions. Despite their different characteristics, we show that the overall revenue for the two scenarios is always within a factor of 2 of each other, even in the presence of correlated bids. Moreover, we prove that the eager auction dominates the lazy auction on revenue whenever the bidders are independent or symmetric. We complement our theoretical results with simulations on real world data that show that even suboptimally set eager reserve prices are preferred from a revenue standpoint.

In this article, we consider the orienteering problem in undirected and directed graphs and obtain improved approximation algorithms. The point to point-orienteering problem is the following: Given an edge-weighted graph G =( V, E ) (directed or undirected), two nodes s, t ∈ V and a time limit B , find an s - t walk in G of total length at most B that maximizes the number of distinct nodes visited by the walk. This problem is closely related to tour problems such as TSP as well as network design problems such as k -MST. Orienteering with time-windows is the more general problem in which each node v has a specified time-window [ R ( v ), D ( v )] and a node v is counted as visited by the walk only if v is visited during its time-window. We design new and improved algorithms for the orienteering problem and orienteering with time-windows. Our main results are the following: — A (2+ϵ) approximation for orienteering in undirected graphs, improving upon the 3-approximation of Bansal et al. [2004]. — An O (log 2 OPT) approximation for orienteering in directed graphs, where OPT ≤ n is the number of vertices visited by an optimal solution. Previously, only a quasipolynomial-time algorithm due to Chekuri and Pál [2005] achieved a polylogarithmic approximation (a ratio of O (log OPT)). — Given an α approximation for orienteering, we show an O (α ċ max{log OPT, log l max / l min }) approximation for orienteering with time-windows, where l max and l min are the lengths of the longest and shortest time-windows respectively.

Let $f:2^X \rightarrow \cal R_+$ be a monotone submodular set function, and let $(X,\cal I)$ be a matroid. We consider the problem ${\rm max}_{S \in \cal I} f(S)$. It is known that the greedy algorithm yields a $1/2$-approximation [M. L. Fisher, G. L. Nemhauser, and L. A. Wolsey, Math. Programming Stud., no. 8 (1978), pp. 73–87] for this problem. For certain special cases, e.g., ${\rm max}_{|S| \leq k} f(S)$, the greedy algorithm yields a $(1-1/e)$-approximation. It is known that this is optimal both in the value oracle model (where the only access to f is through a black box returning $f(S)$ for a given set S) [G. L. Nemhauser and L. A. Wolsey, Math. Oper. Res., 3 (1978), pp. 177–188] and for explicitly posed instances assuming $P \neq NP$ [U. Feige, J. ACM, 45 (1998), pp. 634–652]. In this paper, we provide a randomized $(1-1/e)$-approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [J. Combin. Optim., 8 (2004), pp. 307–328], and a continuous greedy process that may be of independent interest. As a special case, our algorithm implies an optimal approximation for the submodular welfare problem in the value oracle model [J. Vondrák, Proceedings of the $38$th ACM Symposium on Theory of Computing, 2008, pp. 67–74]. As a second application, we show that the generalized assignment problem (GAP) is also a special case; although the reduction requires $|X|$ to be exponential in the original problem size, we are able to achieve a $(1-1/e-o(1))$-approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.

We consider two- and multistage versions of stochastic combinatorial optimization problems with recourse: in this framework, the instance for the combinatorial optimization problem is drawn from a known probability distribution $\pi$ and is only revealed to the algorithm over two (or multiple) stages. At each stage, on receiving some more information about the instance, the algorithm is allowed to build some partial solution. Since the costs of elements increase with each passing stage, there is a natural tension between waiting for later stages, to gain more information about the instance, and purchasing elements in earlier stages, to take advantages of lower costs. We provide approximation algorithms for stochastic combinatorial optimization problems (such as the Steiner tree problem, the Steiner network problem, and the vertex cover problem) by means of a simple sampling-based algorithm. In every stage, our algorithm samples the probability distribution of the requirements and constructs a partial solution to serve the resulting sample. We show that if one can construct cost-sharing functions associated with the algorithms used to construct these partial solutions, then this strategy results in provable approximation guarantees for the overall stochastic optimization problem. We also extend this approach to provide an approximation algorithm for the stochastic version of the uncapacitated facility location problem, a problem that does not fit into the simpler framework of our main model.

Many advertisers (bidders) use Internet systems to buy display advertisements on publishers’ webpages or on traditional media such as radio, TV and newsprint. They seek a simple, online mechanism to reserve ad slots in advance. On the other hand, media publishers (sellers) represent a vast and varying inventory, and they too seek automatic, online mechanisms for pricing and allocating such reservations. We propose and study a simple model for auctioning such ad slot reservations in advance. A seller will display a set of slots at some point T in the future. Until T, bidders arrive sequentially and place a bid on the slots they are interested in. The seller must decide immediately whether or not to grant a reservation. Our model allows the seller to cancel at any time any reservation made earlier, in which case the holder of the reservation incurs a utility loss amounting to a fraction of her value for the reservation and may also receive a cancellation fee from the seller. Our main result is an online mechanism for allocation and pricing in this model with many desirable game-theoretic properties. It is individually rational. Winners have an incentive to be honest and bidding one's true value dominates any lower bid. Further, it bounds the earnings of speculators who are in the game to obtain the cancellation fees. The mechanism in addition has optimization guarantees. Its revenue is within a constant fraction of the a posteriori revenue of the Vickrey-Clarke-Groves (VCG) mechanism which is known to be truthful (in the offline case). Our mechanism's efficiency is within a constant fraction of the a posteriori optimally efficient solution. If efficiency also takes into account the utility losses of bidders whose reservation was canceled, we show that our mechanism matches (for appropriate values of the parameters) an upper bound on the competitive ratio of any deterministic online algorithm. Our mechanism's technical core is a variant of the online weighted bipartite matching problem where unlike prior variants in which one randomizes edge arrivals or bounds edge weights, we may revoke previously committed edges. Our results make no assumptions about bidders’ arrival order or value distribution. They still hold if we replace items with elements of a matroid and matchings with independent sets, or if all bidders have additive value for a set of items.

In sponsored search, a number of advertising slots is available on a search results page, and have to be allocated among a set of advertisers competing to display an ad on the page. This gives rise to a bipartite matching market that is typically cleared by the way of an automated auction. Several auction mechanisms have been proposed, with variants of the Generalized Second Price (GSP) being widely used in practice. There is a rich body of work on bipartite matching markets that builds upon the stable marriage model of Gale and Shapley and the assignment model of Shapley and Shubik. This line of research offers deep insights into the structure of stable outcomes in such markets and their incentive properties. In this paper, we model advertising auctions in terms of an assignment model with linear utilities, extended with bidder and item specific maximum and minimum prices. Auction mechanisms like the commonly used GSP or the well-known Vickrey-Clarke-Groves (VCG) can be interpreted as simply computing a bidder-optimal stable matching in this model, for a suitably defined set of bidder preferences, but our model includes much richer bidders and preferences. We prove that in our model the existence of a stable matching is guaranteed, and under a non-degeneracy assumption a bidder-optimal stable matching exists as well. We give an algorithm to find such matching in polynomial time, and use it to design truthful mechanism that generalizes GSP, is truthful for profit-maximizing bidders, correctly implements features like bidder-specific minimum prices and position-specific bids, and works for rich mixtures of bidders and preferences. Our main technical contributions are the existence of bidder-optimal matchings and strategyproofness of the resulting mechanism, and are proved by induction on the progress of the matching algorithm.

In multi-rate wireless LANs, throughput-based fair bandwidth allocation can lead to drastically reduced aggregate throughput. To balance aggregate throughput while serving users in a fair manner, proportional fair or time-based fair scheduling has been proposed to apply at each access point (AP). However, since a realistic deployment of wireless LANs can consist of a network of APs, this paper considers proportional fairness in this much wider setting. Our technique is to intelligently associate users with APs to achieve optimal proportional fairness in a network of APs. We propose two approximation algorithms for periodical offline optimization. Our algorithms are the first approximation algorithms in the literature with a tight worst-case guarantee for the NP-hard problem. Our simulation results demonstrate that our algorithms can obtain an aggregate throughput which can be as much as 2.3 times more than that of the max-min fair allocation in 802.11b. While maintaining aggregate throughput, our approximation algorithms outperform the default user-AP association method in the 802.11b standard significantly in terms of fairness.

Internet search companies sell advertisement slots based on users' search queries via an auction. While there has been previous work onthe auction process and its game-theoretic aspects, most of it focuses on the Internet company. In this work, we focus on the advertisers, who must solve a complex optimization problem to decide how to place bids on keywords to maximize their return (the number of user clicks on their ads) for a given budget. We model the entire process and study this budget optimization problem. While most variants are NP-hard, we show, perhaps surprisingly, that simply randomizing between two uniform strategies that bid equally on all the keywordsworks well. More precisely, this strategy gets at least a 1-1/e fraction of the maximum clicks possible. As our preliminary experiments show, such uniform strategies are likely to be practical. We also present inapproximability results, and optimal algorithms for variants of the budget optimization problem.

We present constant-factor approximation algorithms for several widely-studied NP-hard optimization problems in network design, including the multicommodity rent-or-buy, virtual private network design, and single-sink buy-at-bulk problems. Our algorithms are simple and their approximation ratios improve over those previously known, in some cases by orders of magnitude. We develop a general analysis framework to bound the approximation ratios of our algorithms. This framework is based on a novel connection between random sampling and game-theoretic cost sharing.

In the multicommodity rent-or-buy (MROB) network design problems, we are given a network together with a set of k terminal pairs ( s 1 , t 1 ), …, ( s k , t k . The goal is to provision the network so that a given amount of flow can be shipped between s i and t i for all 1 ≤ i ≤ k simultaneously. In order to provision the network, one can either rent capacity on edges at some cost per unit of flow, or buy them at some larger fixed cost. Bought edges have no incremental, flow-dependent cost. The overall objective is to minimize the total provisioning cost. Recently, Gupta et al. [2003a] presented a 12-approximation for the MROB problem. Their algroithm chooses a subset of the terminal pairs in the graph at random and then buys the edges of an approximate Steiner forest for these pairs. This technique had previously been introduced [Gupta et al. 2003b] for the single-sink rent-or-buy network design problem. In this article we give a 6.828-approximation for the MROB problem by refining the algorithm of Gupta et al. and simplifying their analysis. The improvement in our article is based on a more careful adaptation and simplified analysis of the primal-dual algorithm for the Steiner forest problem due to Agrawal et al. [1995]. Our result significantly reduces the gap between the single-sink and multisink case.

Despite improvements in wireless access technologies such as 3G or 802.11x, ubiquitous data access has remained a challenge, mainly due to the lack of inexpensive, pervasive backhaul connections from access points to the Internet. With the recent WiMAX standard for high-speed, non-line-of-sight fixed wireless links, multihop wireless backhauls might now overcome this bottleneck. However an important remaining challenge is to provide rate and delay guarantees for customer connections similar to wired backhauls. We provide several schemes for performing admission control for connections with QoS requirements over a multihop wireless backhaul. This is the first work to address both rate and delay requirements for connections. Our admission control algorithms first construct appropriate tree-based topologies connecting wireless backhaul nodes to a wired gateway and then admit the best subset of connections while respecting their rate and delay requirements. Alternately, we admit all the connections with appropriate degradation of their QoS requirements

Given an arc-weighted directed graph G = (V, A, /spl lscr/) and a pair of nodes s, t, we seek to find an s-t walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasi-polynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [Bansal, N et al. (2004)] is O(log/sup 2/ OPT). The second application is an O(log/sup 2/ k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first non-trivial approximation algorithm for this problem. The third application is an O(log/sup 2/ k) approximation (in quasi-poly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios (Garg, N et al.) by a logarithmic factor and almost matches the inapproximability threshold on trees (Halperin and Krauthgamer, 2003). This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than /spl Omega/(log/sup 1-/spl epsi// OPT), even in undirected graphs. Even though our algorithm runs in quasi-poly time, we believe that the implications for the approximability of several basic optimization problems are interesting.

Several combinatorial optimization problems choose elements to minimize the total cost of constructing a feasible solution that satisfies requirements of clients. In the Steiner Tree problem, for example, edges must be chosen to connect terminals (clients); in Vertex Cover, vertices must be chosen to cover edges (clients); in Facility Location, facilities must be chosen and demand vertices (clients) connected to these chosen facilities. We consider a stochastic version of such a problem where the solution is constructed in two stages: Before the actual requirements materialize, we can choose elements in a first stage. The actual requirements are then revealed, drawn from a pre-specified probability distribution π thereupon, some more elements may be chosen to obtain a feasible solution for the actual requirements. However, in this second (recourse) stage, choosing an element is costlier by a factor of σ> 1. The goal is to minimize the first stage cost plus the expected second stage cost.We give a general yet simple technique to adapt approximation algorithms for several deterministic problems to their stochastic versions via the following method. First stage: Draw σ independent sets of clients from the distribution π and apply the approximation algorithm to construct a feasible solution for the union of these sets. Second stage: Since the actual requirements have now been revealed, augment the first-stage solution to be feasible for these requirements. We use this framework to derive constant factor approximations for stochastic versions of Vertex Cover, Steiner Tree and Uncapacitated Facility Location for arbitrary distributions π in one fell swoop. For special (product) distributions, we obtain additional and improved results. Our techniques adapt and use the notion of strict cost-shares introduced in [5].

We study the multicommodity rent-or-buy problem, a type of network design problem with economies of scale. In this problem, capacity on an edge can be rented, with cost incurred on a per-unit of capacity basis, or bought, which allows unlimited use after payment of a large fixed cost. Given a graph and a set of source-sink pairs, we seek a minimum-cost way of installing sufficient capacity on edges so that a prescribed amount of flow can be sent simultaneously from each source to the corresponding sink. The first constant-factor approximation algorithm for this problem was recently given by Kumar et al.; however, this algorithm and its analysis are both quite complicated, and its performance guarantee is extremely large. In this paper, we give a conceptually simple 12-approximation algorithm for this problem. Our analysis of this algorithm makes crucial use of cost sharing, the task of allocating the cost of an object to many users of the object in a "fair" manner. While techniques from approximation algorithms have recently yielded new progress on cost sharing problems, our work is the first to show the converse - those ideas from cost sharing can be fruitfully applied in the design and analysis of approximation algorithms.

We develop a general method for turning a primal-dual algorithm into a group strategy proof cost-sharing mechanism. We use our method to design approximately budget balanced cost sharing mechanisms for two NP-complete problems: metric facility location, and single source rent-or-buy network design. Both mechanisms are competitive, group strategyproof and recover a constant fraction of the cost. For the facility location game our cost-sharing method recovers a 1/3rd of the total cost, while in the network design game the cost shares pay for a 1/15 fraction of the cost of the solution.

The authors give the first constant factor approximation algorithm for the facility location problem with nonuniform, hard capacities. Facility location problems have received a great deal of attention in recent years. Approximation algorithms have been developed for many variants. Most of these algorithms are based on linear programming, but the LP techniques developed thus far have been unsuccessful in dealing with hard capacities. A local-search based approximation algorithm (M. Korupolu et al., 1998; F.A. Chudak and D.P. Williamson, 1999) is known for the special case of hard but uniform capacities. We present a local-search heuristic that yields an approximation guarantee of 9 + /spl epsi/ for the case of nonuniform hard capacities. To obtain this result, we introduce new operations that are natural in this context. Our proof is based on network flow techniques.