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122
Combination Can Be Hard: Approximability of the Unique Coverage Problem
 In Proceedings of the 17th Annual ACMSIAM Symposium on Discrete Algorithms
, 2006
"... Abstract We prove semilogarithmic inapproximability for a maximization problem called unique coverage:given a collection of sets, find a subcollection that maximizes the number of elements covered exactly once. Specifically, assuming that NP 6 ` BPTIME(2n " ) for an arbitrary "> ..."
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Cited by 77 (2 self)
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Abstract We prove semilogarithmic inapproximability for a maximization problem called unique coverage:given a collection of sets, find a subcollection that maximizes the number of elements covered exactly once. Specifically, assuming that NP 6 ` BPTIME(2n &quot; ) for an arbitrary &quot;> 0, we prove O(1 / logoe n) inapproximability for some constant oe = oe(&quot;). We also prove O(1 / log1/3 &quot; n) inapproximability, forany &quot;> 0, assuming that refuting random instances of 3SAT is hard on average; and prove O(1 / log n)inapproximability under a plausible hypothesis concerning the hardness of another problem, balanced bipartite independent set. We establish an \Omega (1 / log n)approximation algorithm, even for a moregeneral (budgeted) setting, and obtain an \Omega (1 / log B)approximation algorithm when every set hasat most B elements. We also show that our inapproximability results extend to envyfree pricing, animportant problem in computational economics. We describe how the (budgeted) unique coverage problem, motivated by realworld applications, has close connections to other theoretical problemsincluding max cut, maximum coverage, and radio broadcasting. 1 Introduction In this paper we consider the approximability of the following natural maximization analog of set cover: Unique Coverage Problem. Given a universe U = {e1,..., en} of elements, and given a collection S = {S1,..., Sm} of subsets of U. Find a subcollection S0 ` S to maximize the number of elements that are uniquely covered, i.e., appear in exactly one set of S 0.
Approximation algorithms and online mechanisms for item pricing
, 2007
"... We present approximation and online algorithms for problems of pricing a collection of items for sale so as to maximize the seller’s revenue in an unlimited supply setting. Our first result is an O(k)approximation algorithm for pricing items to singleminded bidders who each want at most k items. ..."
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Cited by 76 (9 self)
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We present approximation and online algorithms for problems of pricing a collection of items for sale so as to maximize the seller’s revenue in an unlimited supply setting. Our first result is an O(k)approximation algorithm for pricing items to singleminded bidders who each want at most k items. This improves over work of Briest and Krysta (2006) who achieve an O(k2) bound. For the case k = 2, where we obtain a 4approximation, this can be viewed as the following graph vertex pricing problem: given a (multi) graph G with valuations wi j on the edges, find prices pi ≥ 0 for the vertices to maximize {(i, j):wi j≥pi+p j} (pi + p j). We also improve the approximation of Guruswami et al. (2005) for the “highway problem” in which all desired subsets are intervals on a line, from O(logm+ logn) to O(logn), where m is the number of bidders and n is the number of items. Our approximation algorithms can
Multiparameter mechanism design and sequential posted pricing
 CoRR
"... We study the classic mathematical economics problem of Bayesian optimal mechanism design where a principal aims to optimize expected revenue when allocating resources to selfinterested agents with preferences drawn from a known distribution. In single parameter settings (i.e., where each agent’s pr ..."
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Cited by 65 (6 self)
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We study the classic mathematical economics problem of Bayesian optimal mechanism design where a principal aims to optimize expected revenue when allocating resources to selfinterested agents with preferences drawn from a known distribution. In single parameter settings (i.e., where each agent’s preference is given by a single private value for being served and zero for not being served) this problem is solved [20]. Unfortunately, these single parameter optimal mechanisms are impractical and rarely employed [1], and furthermore the underlying economic theory fails to generalize to the important, relevant, and unsolved multidimensional setting (i.e., where each agent’s preference is given by multiple values for each of the multiple services available) [25]. In contrast to the theory of optimal mechanisms we develop a theory of sequential posted price mechanisms, where agents in sequence are offered takeitorleaveit prices. We prove that these
Algorithmic pricing via virtual valuations
 In Proc. of 8th EC
, 2007
"... Algorithmic pricing is the computational problem that sellers (e.g., in supermarkets) face when trying to set prices for their items to maximize their profit in the presence of a known demand. Guruswami et al. [9] propose this problem and give logarithmic approximations (in the number of consumers) ..."
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Cited by 56 (5 self)
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Algorithmic pricing is the computational problem that sellers (e.g., in supermarkets) face when trying to set prices for their items to maximize their profit in the presence of a known demand. Guruswami et al. [9] propose this problem and give logarithmic approximations (in the number of consumers) for the unitdemand and singleparameter cases where there is a specific set of consumers and their valuations for bundles are known precisely. Subsequently several versions of the problem have been shown to have polylogarithmic inapproximability. This problem has direct ties to the important open question of better understanding the Bayesian optimal mechanism in multiparameter agent settings; however, for this purpose approximation factors logarithmic in the number of agents are inadequate. It is therefore of vital interest to consider special cases where constant approximations are possible. We consider the unitdemand variant of this pricing problem. Here a consumer has a valuation for each different item and their value for a set of items is simply the maximum value they have for any item in the set. Instead of considering a set of consumers with precisely known preferences, like the prior algorithmic pricing literature, we assume that the preferences of the consumers are drawn from a distribution. This is the standard assumption in economics; furthermore, the setting of a specific set of customers with specific preferences, which is employed in all of the prior work in algorithmic pricing, is a special case of this general Bayesian pricing problem, where there is a discrete Bayesian distribution for preferences specified by picking one consumer uniformly from the given set of consumers. Notice that the distribution over the valuations for the individual items that this generates is obviously correlated. Our work complements these existing works by considering the case where the consumer’s valuations for the different items are independent random variables. Our main
Mechanism Design via Machine Learning
 IN PROC. OF THE 46TH IEEE SYMP. ON FOUNDATIONS OF COMPUTER SCIENCE
, 2005
"... We use techniques from samplecomplexity in machine learning to reduce problems of incentivecompatible mechanism design to standard algorithmic questions, for a broad class of revenuemaximizing pricing problems. Our reductions imply that for these problems, given an optimal (or #approximation) al ..."
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Cited by 45 (9 self)
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We use techniques from samplecomplexity in machine learning to reduce problems of incentivecompatible mechanism design to standard algorithmic questions, for a broad class of revenuemaximizing pricing problems. Our reductions imply that for these problems, given an optimal (or #approximation) algorithm for the standard algorithmic problem, we can convert it into a (1 + #)approximation (or #(1 + #)approximation) for the incentivecompatible mechanism design problem, so long as the number of bidders is sufficiently large as a function of an appropriate measure of complexity of the comparison class of solutions. We apply these results to the problem of auctioning a digital good, to the attribute auction problem which includes a wide variety of discriminatory pricing problems, and to the problem of itempricing in unlimitedsupply combinatorial auctions. From a machine learning perspective, these settings present several challenges: in particular, the loss function is discontinuous and asymmetric, and the range of bidders' valuations may be large.
Item Pricing for Revenue Maximization
"... We consider the problem of pricing n items to maximize revenue when faced with a series of unknown buyers with complex preferences, and show that a simple pricing scheme achieves surprisingly strong guarantees. We show that in the unlimited supply setting, a random single price achieves expected rev ..."
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Cited by 41 (4 self)
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We consider the problem of pricing n items to maximize revenue when faced with a series of unknown buyers with complex preferences, and show that a simple pricing scheme achieves surprisingly strong guarantees. We show that in the unlimited supply setting, a random single price achieves expected revenue within a logarithmic factor of the total social welfare for customers with general valuation functions, which may not even necessarily be monotone. This generalizes work of Guruswami et. al [18], who show a logarithmic factor for only the special cases of singleminded and unitdemand customers. In the limited supply setting, we show that for subadditive valuations, a random single price achieves revenue within a factor of 2 O( √ log n log log n) of the total social welfare, i.e., the optimal revenue the seller could hope to extract even if the seller could price each bundle differently for every buyer. This is the best approximation known for any item pricing scheme for subadditive (or even submodular) valuations, even using multiple prices. We complement this result with a lower bound showing a sequence of subadditive (in fact, XOS) buyers for which any single price has approximation ratio 2 Ω(log1/4 n), thus showing that single price schemes cannot achieve a polylogarithmic ratio. This lower bound demonstrates a clear distinction between revenue maximization and social welfare maximization in this setting, for which [12, 10] show that a fixed price achieves a logarithmic approximation in the case of XOS [12], and more generally subadditive [10], customers.
Approximating revenuemaximizing combinatorial auctions
 In AAAI
, 2005
"... Designing revenuemaximizing combinatorial auctions (CAs) is a recognized open problem in mechanism design. It is unsolved even for two bidders and two items for sale. Rather than attempting to characterize the optimal auction, we focus on designing approximations (suboptimal auction mechanisms whic ..."
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Cited by 37 (3 self)
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Designing revenuemaximizing combinatorial auctions (CAs) is a recognized open problem in mechanism design. It is unsolved even for two bidders and two items for sale. Rather than attempting to characterize the optimal auction, we focus on designing approximations (suboptimal auction mechanisms which yield high revenue). Our approximations belong to the family of virtual valuations combinatorial auctions (VVCA). VVCA is a VickreyClarkeGroves (VCG) mechanism run on virtual valuations that are linear transformations of the bidders ’ real valuations. We pursue two approaches to constructing approximately optimal CAs. The first is to construct a VVCA with worstcase and averagecase performance guarantees. We give a logarithmic approximation auction for basic important special cases of the problem: 1) limited supply of items on sale with additive valuations and 2) unlimited supply. The second approach is to search the parameter space of VVCAs in order to obtain highrevenue mechanisms for the general problem. We introduce a series of increasingly sophisticated algorithms that use economic insights to guide the search and thus reduce the computational complexity. Our experiments demonstrate that in many cases these algorithms perform almost as well as the optimal VVCA, yield a substantial increase in revenue over the VCG mechanism and drastically outperform the straightforward algorithms in runtime. 1
Uniform budgets and the envyfree pricing problem
 In Proceedings of the 35th International Colloquium on Automata, Languages and Programming
, 2008
"... We consider the unitdemand minbuying pricing problem, in which we want to compute revenue maximizing prices for a set of products P assuming that each consumer from a set of consumer samples C will purchase her cheapest affordable product once prices are fixed. We focus on the special uniformbudg ..."
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Cited by 31 (5 self)
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We consider the unitdemand minbuying pricing problem, in which we want to compute revenue maximizing prices for a set of products P assuming that each consumer from a set of consumer samples C will purchase her cheapest affordable product once prices are fixed. We focus on the special uniformbudget case, in which every consumer has only a single nonzero budget for some set of products. This constitutes a special case also of the unitdemand envyfree pricing problem. We show that, assuming specific hardness of the balanced bipartite independent set problem in constant degree graphs or hardness of refuting random 3CNF formulas, the unitdemand minbuying pricing problem with uniform budgets cannot be approximated in polynomial time within O(log ε C) for some ε> 0. This is the first result giving evidence that unitdemand envyfree pricing, as well, might be hard to approximate essentially better than within the known logarithmic ratio. We then introduce a slightly more general problem definition in which consumers are given as an explicit probability distribution and show that in this case the envyfree pricing problem can be shown to be inapproximable within O(P  ε) assuming NP � T δ>0 BPTIME(2O(nδ)). Finally, we briefly argue that all the results apply to the important setting of pricing with singleminded consumers as well. 1
NearOptimal Pricing in NearLinear Time
 In 9th Workshop on Algorithms and Data Structures
, 2005
"... Abstract. We present efficient approximation algorithms for a number of problems that call for computing the prices that maximize the revenue of the seller on a set of items. Algorithms for such problems enable the design of auctions and related pricing mechanisms [3]. In light of the fact that the ..."
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Cited by 31 (4 self)
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Abstract. We present efficient approximation algorithms for a number of problems that call for computing the prices that maximize the revenue of the seller on a set of items. Algorithms for such problems enable the design of auctions and related pricing mechanisms [3]. In light of the fact that the problems we address are APXhard in general [5], we design nearlinear and nearcubic time approximation schemes under the assumption that the number of distinct items for sale is constant. 1
Multiunit auctions with unknown supply
 ACM Conference on Electronic Commerce
, 2006
"... We study multiunit auctions for perishable goods, in a setting where the supply arrives online. This is motivated by its application in advertisement auctions on the internet. We give a 1competitive algorithm for computing the op4 timal single price auction assuming that all the agents report the ..."
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Cited by 25 (3 self)
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We study multiunit auctions for perishable goods, in a setting where the supply arrives online. This is motivated by its application in advertisement auctions on the internet. We give a 1competitive algorithm for computing the op4 timal single price auction assuming that all the agents report their bids truthfully. We use that algorithm to develop a truthful auction with a constant competitive ratio compared to the optimum offline singleprice auction. 1.