Results 1  10
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27
Adwords and generalized online matching
 In FOCS ’05: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
, 2005
"... How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competit ..."
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Cited by 98 (5 self)
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How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competitive ratios of 1 − 1/e for this problem. 1
Approximation algorithms for combinatorial auctions with complementfree bidders
 In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC
, 2005
"... We exhibit three approximation algorithms for the allocation problem in combinatorial auctions with complement free bidders. The running time of these algorithms is polynomial in the number of items m and in the number of bidders n, even though the “input size ” is exponential in m. The first algori ..."
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Cited by 93 (22 self)
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We exhibit three approximation algorithms for the allocation problem in combinatorial auctions with complement free bidders. The running time of these algorithms is polynomial in the number of items m and in the number of bidders n, even though the “input size ” is exponential in m. The first algorithm provides an O(log m) approximation. The second algorithm provides an O ( √ m) approximation in the weaker model of value oracles. This algorithm is also incentive compatible. The third algorithm provides an improved 2approximation for the more restricted case of “XOS bidders”, a class which strictly contains submodular bidders. We also prove lower bounds on the possible approximations achievable for these classes of bidders. These bounds are not tight and we leave the gaps as open problems. 1
Allocating online advertisement space with unreliable estimates
 In Proceedings of the 8th ACM Conference on Electronic Commerce (EC
, 2007
"... We study the problem of optimally allocating online advertisement space to budgetconstrained advertisers. This problem was defined and studied from the perspective of worstcase online competitive analysis by Mehta et al. Our objective is to find an algorithm that takes advantage of the given estim ..."
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Cited by 44 (6 self)
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We study the problem of optimally allocating online advertisement space to budgetconstrained advertisers. This problem was defined and studied from the perspective of worstcase online competitive analysis by Mehta et al. Our objective is to find an algorithm that takes advantage of the given estimates of the frequencies of keywords to compute a near optimal solution when the estimates are accurate, while at the same time maintaining a good worstcase competitive ratio in case the estimates are totally incorrect. This is motivated by realworld situations where search engines have stochastic information that provide reasonably accurate estimates of the frequency of search queries except in certain highly unpredictable yet economically valuable spikes in the search pattern. Our approach is a blackbox approach: we assume we have access to an oracle that uses the given estimates to recommend an advertiser every time a query arrives. We use this oracle to design an algorithm that provides two performance guarantees: the performance guarantee in the case that the oracle gives an accurate estimate, and its worstcase performance guarantee. Our algorithm can be fine tuned by adjusting a parameter α, giving a tradeoff curve between the two performance measures with the best competitive ratio for the worstcase scenario at one end of the curve and the optimal solution for the scenario where estimates are accurate at the other end. Finally, we demonstrate the applicability of our framework by applying it to two classical online problems, namely the lost cow and the ski rental problems.
Tight approximation algorithms for maximum general assignment problems
 Proc. of ACMSIAM SODA
, 2006
"... A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin – i.e. for bin i, a family Ii of subsets of items that fit in bin i. The goal is to pack items into bin ..."
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Cited by 43 (7 self)
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A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin – i.e. for bin i, a family Ii of subsets of items that fit in bin i. The goal is to pack items into bins to maximize the aggregate value. This class of problems includes the maximum generalized assignment problem (GAP) 1) and a distributed caching problem (DCP) described in this paper. Given a βapproximation algorithm for finding the highest value packing of a single bin, we give 1. A polynomialtime LProunding based ((1 − 1 e)β)approximation algorithm. 2. A simple polynomialtime local search ( β approximation algorithm, for any ɛ> 0. β+1 − ɛ)Therefore, for all examples of SAP that admit an approximation scheme for the singlebin problem, we obtain an LPbased algorithm with (1 − 1 e − ɛ)approximation and a local search algorithm with ( 1 2 −ɛ)approximation guarantee. Furthermore, for cases in which the subproblem admits a fully polynomial approximation scheme (such as for GAP), the LPbased algorithm analysis can be strengthened to give a guarantee of 1 − 1 e. The best previously known approximation algorithm for GAP is a 1 2approximation by Shmoys and Tardos; and Chekuri and Khanna. Our LP algorithm is based on rounding a new linear programming relaxation, with a provably better integrality gap. To complement these results, we show that SAP and DCP cannot be approximated within a factor better than 1 − 1 e unless NP ⊆ DTIME(n O(log log n)), even if there exists a polynomialtime exact algorithm for the singlebin problem.
Online budgeted matching in random input models with applications to adwords
 In SODA 2008
"... We study an online assignment problem, motivated by Adwords Allocation, in which queries are to be assigned to bidders with budget constraints. We analyze the performance of the Greedy algorithm (which assigns each query to the highest bidder) in a randomized input model with queries arriving in a r ..."
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Cited by 43 (8 self)
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We study an online assignment problem, motivated by Adwords Allocation, in which queries are to be assigned to bidders with budget constraints. We analyze the performance of the Greedy algorithm (which assigns each query to the highest bidder) in a randomized input model with queries arriving in a random permutation. Our main result is a tight analysis of Greedy in this model showing that it has a competitive ratio of 1 − 1/e for maximizing the value of the assignment. We also consider the more standard i.i.d. model of input, and show that our analysis holds there as well. This is to be contrasted with the worst case analysis of [MSVV05] which shows that Greedy has a ratio of 1/2, and that the optimal algorithm presented there has a ratio of 1 − 1/e. The analysis of Greedy is important in the Adwords setting because it is the natural allocation algorithm for an auctionstyle process. From a theoretical perspective, our result simplifies and generalizes the classic algorithm of Karp, Vazirani and Vazirani for online bipartite matching. Our results include a new proof to show that the Ranking algorithm of [KVV90] has a ratio of 1 − 1/e in the worst case. It has been recently discovered [KV07] (independent of our results) that one of the crucial lemmas in [KVV90], related to a certain reduction, is incorrect. Our proof is direct, in that it does not go via such a reduction, which also enables us to generalize the analysis to our online assignment problem. 1
Inapproximability results for combinatorial auctions with submodular utility functions
 in Proceedings of WINE 2005
, 2005
"... We consider the following allocation problem arising in the setting of combinatorial auctions: a set of goods is to be allocated to a set of players so as to maximize the sum of the utilities of the players (i.e., the social welfare). In the case when the utility of each player is a monotone submodu ..."
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Cited by 38 (0 self)
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We consider the following allocation problem arising in the setting of combinatorial auctions: a set of goods is to be allocated to a set of players so as to maximize the sum of the utilities of the players (i.e., the social welfare). In the case when the utility of each player is a monotone submodular function, we prove that there is no polynomial time approximation algorithm which approximates the maximum social welfare by a factor better than 1 − 1/e � 0.632, unless P = NP. Our result is based on a reduction from a multiprover proof system for MAX3COLORING. 1
Maximizing Submodular Set Functions Subject to Multiple Linear Constraints
, 2009
"... The concept of submodularity plays a vital role in combinatorial optimization. In particular, many important optimization problems can be cast as submodular maximization problems, including maximum coverage, maximum facility location and max cut in directed/undirected graphs. In this paper we presen ..."
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Cited by 29 (1 self)
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The concept of submodularity plays a vital role in combinatorial optimization. In particular, many important optimization problems can be cast as submodular maximization problems, including maximum coverage, maximum facility location and max cut in directed/undirected graphs. In this paper we present the first known approximation algorithms for the problem of maximizing a nondecreasing submodular set function subject to multiple linear constraints. Given a ddimensional budget vector ¯ L, for some d ≥ 1, and an oracle for a nondecreasing submodular set function f over a universe U, where each element e ∈ U is associated with a ddimensional cost vector, we seek a subset of elements S ⊆ U whose total cost is at most ¯ L, such that f(S) is maximized. We develop a framework for maximizing submodular functions subject to d linear constraints that yields a (1 − ε)(1 − e−1)approximation to the optimum for any ε> 0, where d> 1 is some constant. Our study is motivated by a variant of the classical maximum coverage problem that we call maximum coverage with multiple packing constraints. We use our framework to obtain the same approximation ratio for this problem. To the best of our knowledge, this is the first time the theoretical bound of 1 − e−1 is (almost) matched for both of these problems.
On the approximability of budgeted allocations and improved lower bounds for submodular welfare maximization and GAP
 In Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer ScienceVolume 00
, 2008
"... In this paper we consider the following maximum budgeted allocation(MBA) problem: Given a set of m indivisible items and n agents; each agent i willing to pay bij on item j and with a maximum budget of Bi, the goal is to allocate items to agents to maximize revenue. The problem naturally arises as a ..."
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Cited by 19 (1 self)
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In this paper we consider the following maximum budgeted allocation(MBA) problem: Given a set of m indivisible items and n agents; each agent i willing to pay bij on item j and with a maximum budget of Bi, the goal is to allocate items to agents to maximize revenue. The problem naturally arises as auctioneer revenue maximization in budgetconstrained auctions and as winner determination problem in combinatorial auctions when utilities of agents are budgetedadditive. Our main results are: • We give a 3/4approximation algorithm for MBA improving upon the previous best of ≃ 0.632[AM04, FV06]. Our techniques are based on a natural LP relaxation of MBA and our factor is optimal in the sense that it matches the integrality gap of the LP. • We prove it is NPhard to approximate MBA to any factor better than 15/16, previously only NPhardness was known [SS06, LLN01]. Our result also implies NPhardness of approximating maximum submodular welfare with demand oracle to a factor better than 15/16, improving upon the best known hardness of 275/276[FV06]. • Our hardness techniques can be modified to prove that it is NPhard to approximate the Generalized Assignment Problem (GAP) to any factor better than 10/11. This improves upon the 422/423 hardness of [CK00, CC02]. We use iterative rounding on a natural LP relaxation of MBA to obtain the 3/4approximation. We also give a (3/4 − ɛ)factor algorithm based on the primaldual schema which runs in Õ(nm) time, for any constant ɛ> 0. 1
Inapproximability for vcgbased combinatorial auctions
"... The existence of incentivecompatible, computationallyefficient mechanisms for combinatorial auctions with good approximation ratios is the paradigmatic problem in algorithmic mechanism design. It is believed that, in many cases, good approximations for combinatorial auctions may be unattainable due ..."
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Cited by 14 (6 self)
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The existence of incentivecompatible, computationallyefficient mechanisms for combinatorial auctions with good approximation ratios is the paradigmatic problem in algorithmic mechanism design. It is believed that, in many cases, good approximations for combinatorial auctions may be unattainable due to an inherent clash between truthfulness and computational efficiency. In this paper, we prove the first computationalcomplexity inapproximability results for incentivecompatible mechanisms for combinatorial auctions. Our results are tight, hold for the important class of VCGbased mechanisms, and are based on the complexity assumption that NP has no polynomialsize circuits. We show two different techniques to obtain such lower bounds: one for deter
Algorithmic game theory
, 2007
"... I dedicate this thesis to my parents, Anjani and Sitanshu Mehta. iii ACKNOWLEDGEMENTS I would like to thank Dick Lipton and Vijay Vazirani, for their continuous support and guidance throughout these five years. Thanks also to the rest of the theory and ACO group at Georgia Tech, faculty and students ..."
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Cited by 8 (0 self)
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I dedicate this thesis to my parents, Anjani and Sitanshu Mehta. iii ACKNOWLEDGEMENTS I would like to thank Dick Lipton and Vijay Vazirani, for their continuous support and guidance throughout these five years. Thanks also to the rest of the theory and ACO group at Georgia Tech, faculty and students, who have created such a motivating and enriching research environment. Thanks, of course, to my collaborators and coauthors the results presented here are products of our joint work. Finally, thanks to Swati, the fixed point of the map of my life, without whose support and patience I would never have finished this thesis.