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131
Truthful and Near-Optimal Mechanism Design via Linear Programming
"... We give a general technique to obtain approximation mechanisms that are truthful in expectation.We show that for packing domains, any ff-approximation algorithm that also bounds the integrality gapof the LP relaxation of the problem by ff can be used to construct an ff-approximation mechanismthat is ..."
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Cited by 141 (12 self)
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We give a general technique to obtain approximation mechanisms that are truthful in expectation.We show that for packing domains, any ff-approximation algorithm that also bounds the integrality gapof the LP relaxation of the problem by ff can be used to construct an ff-approximation mechanismthat is truthful in expectation. This immediately yields a variety of new and significantly improved results for various problem domains and furthermore, yields truthful (in expectation) mechanisms withguarantees that match the best known approximation guarantees when truthfulness is not required. In particular, we obtain the first truthful mechanisms with approximation guarantees for a variety of multi-parameter domains. We obtain truthful (in expectation) mechanisms achieving approximation guarantees of O( p m) for combinatorial auctions (CAs), (1 + ffl) for multi-unit CAs with B = \Omega (log m) copies ofeach item, and 2 for multi-parameter knapsack problems (multi-unit auctions). Our construction is based on considering an LP relaxation of the problem and using the classicVCG [25, 9, 12] mechanism to obtain a truthful mechanism in this fractional domain. We argue that the (fractional) optimal solution scaled down by ff, where ff is the integrality gap of the problem, canbe represented as a convex combination of integer solutions, and by viewing this convex combination as specifying a probability distribution over integer solutions, we get a randomized, truthful in expectationmechanism. Our construction can be seen as a way of exploiting VCG in a computational tractable way even when the underlying social-welfare maximization problem is NP-hard.
Optimal Approximation for the Submodular Welfare Problem in the value oracle model
- STOC'08
, 2008
"... In the Submodular Welfare Problem, m items are to be distributed among n players with utility functions wi: 2 [m] → R+. The utility functions are assumed to be monotone and submodular. Assuming that player i receives a set of items Si, we wish to maximize the total utility Pn i=1 wi(Si). In this pap ..."
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Cited by 122 (11 self)
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In the Submodular Welfare Problem, m items are to be distributed among n players with utility functions wi: 2 [m] → R+. The utility functions are assumed to be monotone and submodular. Assuming that player i receives a set of items Si, we wish to maximize the total utility Pn i=1 wi(Si). In this paper, we work in the value oracle model where the only access to the utility functions is through a black box returning wi(S) for a given set S. Submodular Welfare is in fact a special case of the more general problem of submodular maximization subject to a matroid constraint: max{f(S) : S ∈ I}, where f is monotone submodular and I is the collection of independent sets in some matroid. For both problems, a greedy algorithm is known to yield a 1/2-approximation [21, 16]. In special cases where the matroid is uniform (I = {S: |S | ≤ k}) [20] or the submodular function is of a special type [4, 2], a (1 − 1/e)approximation has been achieved and this is optimal for these problems in the value oracle model [22, 6, 15]. A (1 − 1/e)-approximation for the general Submodular Welfare Problem has been known only in a stronger demand oracle model [4], where in fact 1 − 1/e can be improved [9]. In this paper, we develop a randomized continuous greedy algorithm which achieves a (1 − 1/e)-approximation for the Submodular Welfare Problem in the value oracle model. We also show that the special case of n equal players is approximation resistant, in the sense that the optimal (1 − 1/e)approximation is achieved by a uniformly random solution. Using the pipage rounding technique [1, 2], we obtain a (1 − 1/e)-approximation for submodular maximization subject to any matroid constraint. The continuous greedy algorithm has a potential of wider applicability, which we demonstrate on the examples of the Generalized Assignment Problem and the AdWords Assignment Problem.
Truthful randomized mechanisms for combinatorial auctions
- IN STOC
, 2006
"... We design two computationally-efficient incentive-compatible mechanisms for combinatorial auctions with general bidder preferences. Both mechanisms are randomized, and are incentive-compatible in the universal sense. This is in contrast to recent previous work that only addresses the weaker notion o ..."
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Cited by 108 (19 self)
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We design two computationally-efficient incentive-compatible mechanisms for combinatorial auctions with general bidder preferences. Both mechanisms are randomized, and are incentive-compatible in the universal sense. This is in contrast to recent previous work that only addresses the weaker notion of incentive compatibility in expectation. The first mechanism obtains an O(pm)-approximation of the optimal social welfare for arbitrary bidder valuations -- this is the best approximation possible in polynomial time. The second one obtains an O(log2 m)- approximation for a subclass of bidder valuations that includes all submodular bidders. This improves over the best previously obtained incentive-compatible mechanism for this class which only provides an O(pm)-approximation.
On the Hardness of Being Truthful
- In 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS
, 2008
"... The central problem in computational mechanism design is the tension between incentive compatibility and computational ef ciency. We establish the rst significant approximability gap between algorithms that are both truthful and computationally-ef cient, and algorithms that only achieve one of these ..."
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Cited by 64 (8 self)
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The central problem in computational mechanism design is the tension between incentive compatibility and computational ef ciency. We establish the rst significant approximability gap between algorithms that are both truthful and computationally-ef cient, and algorithms that only achieve one of these two desiderata. This is shown in the context of a novel mechanism design problem which we call the COMBINATORIAL PUB-LIC PROJECT PROBLEM (CPPP). CPPP is an abstraction of many common mechanism design situations, ranging from elections of kibbutz committees to network design. Our result is actually made up of two complementary results – one in the communication-complexity model and one in the computational-complexity model. Both these hardness results heavily rely on a combinatorial characterization of truthful algorithms for our problem. Our computational-complexity result is one of the rst impossibility results connecting mechanism design to complexity theory; its novel proof technique involves an application of the Sauer-Shelah Lemma and may be of wider applicability, both within and without mechanism design. 1
Maximizing a Monotone Submodular Function subject to a Matroid Constraint
, 2008
"... Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max |S|≤k f(S), the greedy algorithm yields a (1 − 1/e)-app ..."
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Cited by 63 (0 self)
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Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max |S|≤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) [28], and also for explicitly posed instances assuming P � = NP [10]. 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 [1], and a continuous greedy process that might be of independent interest. As a special case, our algorithm implies an optimal approximation for the Submodular Welfare Problem in the value oracle model [32]. 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.
Approximation algorithms for allocation problems: Improving the Factor
- of 1 − 1/e. Proc. of IEEE FOCS
, 2006
"... Combinatorial allocation problems require allocating items to players in a way that maximizes the total utility. Two such problems received attention recently, and were addressed using the same linear programming (LP) relaxation. In the Maximum Submodular Welfare (SMW) problem, utility functions of ..."
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Cited by 61 (9 self)
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Combinatorial allocation problems require allocating items to players in a way that maximizes the total utility. Two such problems received attention recently, and were addressed using the same linear programming (LP) relaxation. In the Maximum Submodular Welfare (SMW) problem, utility functions of players are submodular, and for this case Dobzinski and Schapira [SODA 2006] showed an approximation ratio of 1 − 1/e. In the Generalized Assignment Problem (GAP) utility functions are linear but players also have capacity constraints. GAP admits a (1 − 1/e)approximation as well, as shown by Fleischer, Goemans, Mirrokni and Sviridenko [SODA 2006]. In both cases, the approximation ratio was in fact shown for a more general version of the problem, for which improving 1 − 1/e is NPhard. In this paper, we show how to improve the 1 − 1/e approximation ratio, both for SMW and for GAP. A common theme in both improvements is the use of a new and optimal Fair Contention Resolution technique. However, each of the improvements involves a different rounding procedure for the above mentioned LP. In addition, we prove APX-hardness results for SMW (such results were known for GAP). An important feature of our hardness results is that they apply even in very restricted settings, e.g. when every player has nonzero utility only for a constant number of items. 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 51 (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 multi-prover proof system for MAX-3-COLORING. 1
Bayesian combinatorial auctions
- Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP), I
, 2008
"... Abstract. We study the following Bayesian setting: m items are sold to n sel¯sh bidders in m independent second-price auctions. Each bidder has a private valuation function that expresses complex preferences over all subsets of items. Bidders only have beliefs about the valuation func-tions of the o ..."
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Cited by 48 (1 self)
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Abstract. We study the following Bayesian setting: m items are sold to n sel¯sh bidders in m independent second-price auctions. Each bidder has a private valuation function that expresses complex preferences over all subsets of items. Bidders only have beliefs about the valuation func-tions of the other bidders, in the form of probability distributions. The objective is to allocate the items to the bidders in a way that provides a good approximation to the optimal social welfare value. We show that if bidders have submodular valuation functions, then every Bayesian Nash equilibrium of the resulting game provides a 2-approximation to the op-timal social welfare. Moreover, we show that in the full-information game a pure Nash always exists and can be found in time that is polynomial in both m and n. 1
Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions
- Proc. of ACM EC
, 2007
"... We provide tight information-theoretic lower bounds for the welfare maximization problem in combinatorial auctions. In this problem the goal is to partition m items between k bidders in a way that maximizes the sum of bidders ’ values for their allocated items. Bidders have complex preferences over ..."
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Cited by 45 (7 self)
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We provide tight information-theoretic lower bounds for the welfare maximization problem in combinatorial auctions. In this problem the goal is to partition m items between k bidders in a way that maximizes the sum of bidders ’ values for their allocated items. Bidders have complex preferences over items expressed by valuation functions that assign values to all subsets of items. We study the “black box ” setting in which the auctioneer has oracle access to the valuation functions of the bidders. In particular, we explore the well-known value queries model in which the permitted query to a valuation function is in the form of a subset of items, and the reply is the value assigned to that subset of items by the valuation function. We consider different classes of valuation functions: Submodular, subadditive, and superadditive. For these classes it has been shown that one can achieve approximation ratios of 1 − 1 e, √1, and m √ log m m, respectively, via a polynomial (in n and m) number of value queries. We prove that these approximation factors are essentially the best possible: For any fixed ɛ> 0, a (1 − 1/e + ɛ)-approximation for submodular valuations or an 1 m 1/2−ɛ-approximation for subad-ditive valuations would require exponentially many value queries, and a log1+ɛ m m-approximation for superadditive valuations would require a superpolynomial number of value queries.
Approximating Submodular Functions Everywhere
"... Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e.g., ..."
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Cited by 45 (4 self)
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Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e.g., exact minimization or approximate maximization. In this paper, we consider the problem of approximating a non-negative, monotone, submodular function f on a ground set of size n everywhere, after only poly(n) oracle queries. Our main result is a deterministic algorithm that makes poly(n) oracle queries and derives a function ˆ f such that, for every set S, ˆ f(S) approximates f(S) within a factor α(n), where α(n) = √ n + 1 for rank functions of matroids and α(n) = O ( √ n log n) for general monotone submodular functions. Our result is based on approximately finding a maximum volume inscribed ellipsoid in a symmetrized polymatroid, and the analysis involves various properties of submodular functions and polymatroids. Our algorithm is tight up to logarithmic factors. Indeed, we show that no algorithm can achieve a factor better than Ω ( √ n / log n), even for rank functions of a matroid.
