Results 1  10
of
34
Symmetry and approximability of submodular maximization problems
"... A number of recent results on optimization problems involving submodular functions have made use of the ”multilinear relaxation” of the problem [3], [8], [24], [14], [13]. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of ”symmetry ..."
Abstract

Cited by 47 (3 self)
 Add to MetaCart
A number of recent results on optimization problems involving submodular functions have made use of the ”multilinear relaxation” of the problem [3], [8], [24], [14], [13]. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of ”symmetry gap”. Our main result is that for any fixed instance that exhibits a certain ”symmetry gap ” in its multilinear relaxation, there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries. This unifies several known hardness results for submodular maximization, e.g. the optimality of (1 − 1/e)approximation for monotone submodular maximization under a cardinality constraint [20], [7], and the impossibility of ( 1 +ɛ)approximation for uncon2 strained (nonmonotone) submodular maximization [8]. It follows from our result that ( 1 + ɛ)approximation is also impossible for 2 nonmonotone submodular maximization subject to a (nontrivial) matroid constraint. On the algorithmic side, we present a 0.309approximation for this problem, improving the previously known factor of 1 − o(1) [14]. 4 As another application, we consider the problem of maximizing a nonmonotone submodular function over the bases of a matroid. A ( 1 − o(1))approximation has been developed for this problem, 6 assuming that the matroid contains two disjoint bases [14]. We show that the best approximation one can achieve is indeed related to packings of bases in the matroid. Specifically, for any k ≥ 2, there is a class of matroids of fractional base packing number k k−1 ν = , such that any algorithm achieving a better than (1 − 1)approximation for this class would require exponentially many
The Exponential Mechanism for Social Welfare: Private, Truthful, and Nearly Optimal
, 2012
"... In this paper, we show that for any mechanism design problem, the exponential mechanism can be implemented as a truthful mechanism while still preserving differential privacy, if the objective is to maximize social welfare. Our instantiation of the exponential mechanism can be interpreted as a gener ..."
Abstract

Cited by 19 (2 self)
 Add to MetaCart
In this paper, we show that for any mechanism design problem, the exponential mechanism can be implemented as a truthful mechanism while still preserving differential privacy, if the objective is to maximize social welfare. Our instantiation of the exponential mechanism can be interpreted as a generalization of the VCG mechanism in the sense that the VCG mechanism is the extreme case when the privacy parameter goes to infinity. To our knowledge, this is the first general tool for designing mechanisms that are both truthful and differentially private.
Limitations of randomized mechanisms for combinatorial auctions
 In Proceedings of the 52nd IEEE Symposium on Foundations of Computer Science (FOCS
, 2011
"... Abstract — The design of computationally efficient and incentive compatible mechanisms that solve or approximate fundamental resource allocation problems is the main goal of algorithmic mechanism design. A central example in both theory and practice is welfaremaximization in combinatorial auctions. ..."
Abstract

Cited by 18 (4 self)
 Add to MetaCart
Abstract — The design of computationally efficient and incentive compatible mechanisms that solve or approximate fundamental resource allocation problems is the main goal of algorithmic mechanism design. A central example in both theory and practice is welfaremaximization in combinatorial auctions. Recently, a randomized mechanism has been discovered for combinatorial auctions that is truthful in expectation and guarantees a (1 − 1/e)approximation to the optimal social welfare when players have coverage valuations [11]. This approximation ratio is the best possible even for nontruthful algorithms, assuming P ̸ = NP [16]. Given the recent sequence of negative results for combinatorial auctions under more restrictive notions of incentive compatibility [7], [2], [9], this development raises a natural question: Are truthfulinexpectation mechanisms compatible with polynomialtime approximation in a way that deterministic or universally truthful
The Computational Complexity of Truthfulness in Combinatorial Auctions
 In Proceedings of the ACM Conference on Electronic Commerce (EC
"... ar ..."
Representation, approximation and learning of submodular functions using lowrank decision trees
 In Proceedings of the Conference on Learning Theory (COLT
, 2013
"... We study the complexity of approximate representation and learning of submodular functions over the uniform distribution on the Boolean hypercube {0, 1}n. Our main result is the following structural theorem: any submodular function is close in `2 to a realvalued decision tree (DT) of depth O(1/2) ..."
Abstract

Cited by 13 (8 self)
 Add to MetaCart
(Show Context)
We study the complexity of approximate representation and learning of submodular functions over the uniform distribution on the Boolean hypercube {0, 1}n. Our main result is the following structural theorem: any submodular function is close in `2 to a realvalued decision tree (DT) of depth O(1/2). This immediately implies that any submodular function is close to a function of at most 2O(1/ 2) variables and has a spectral `1 norm of 2O(1/ 2). It also implies the closest previous result that states that submodular functions can be approximated by polynomials of degree O(1/2) (Cheraghchi et al., 2012). Our result is proved by constructing an approximation of a submodular function by a DT of rank 4/2 and a proof that any rankr DT can be approximated by a DT of depth 52 (r + log(1/)). We show that these structural results can be exploited to give an attributeefficient PAC learning algorithm for submodular functions running in time Õ(n2) · 2O(1/4). The best previous algorithm for the problem requires nO(1/ 2) time and examples (Cheraghchi et al., 2012) but works also in the agnostic setting. In addition, we give improved learning algorithms for a number of related settings. We also prove that our PAC and agnostic learning algorithms are essentially optimal via two lower bounds: (1) an informationtheoretic lower bound of 2Ω(1/ 2/3) on the complexity of learning monotone submodular functions in any reasonable model (including learning with value queries); (2) computational lower bound of nΩ(1/ 2/3) based on a reduction to learning of sparse parities with noise, widelybelieved to be intractable. These are the first lower bounds for learning of submodular functions over the uniform distribution.
A Truthful Randomized Mechanism for Combinatorial Public Projects via Convex Optimization
, 2011
"... In Combinatorial Public Projects, there is a set of projects that may be undertaken, and a set of selfinterested players with a stake in the set of projects chosen. A public planner must choose a subset of these projects, subject to a resource constraint, with the goal of maximizing social welfare. ..."
Abstract

Cited by 11 (6 self)
 Add to MetaCart
In Combinatorial Public Projects, there is a set of projects that may be undertaken, and a set of selfinterested players with a stake in the set of projects chosen. A public planner must choose a subset of these projects, subject to a resource constraint, with the goal of maximizing social welfare. Combinatorial Public Projects has emerged as one of the paradigmatic problems in Algorithmic Mechanism Design, a field concerned with solving fundamental resource allocation problems in the presence of both selfish behavior and the computational constraint of polynomialtime. We design a polynomialtime, truthfulinexpectation,(1−1/e)approximation mechanism for welfare maximization in a fundamental variant of combinatorial public projects. Our results apply to combinatorial public projects when players have valuations that are matroid rank sums (MRS), which encompass most concrete examples of submodular functions studied in this context, including coverage functions, matroid weightedrank functions, and convex combinations thereof. Our approximation factor is the best possible, assuming P ̸ = NP. Ours is the first mechanism that achieves a constant factor approximation for a natural NPhard variant of combinatorial public projects.
Combinatorial Auctions with Restricted Complements
"... a thorn in the side of algorithmic mechanism designers. On the one hand, complements are common in the standard motivating applications for combinatorial auctions, like spectrum license auctions. On the other, welfare maximization in the presence of complements is notoriously difficult, and this int ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
(Show Context)
a thorn in the side of algorithmic mechanism designers. On the one hand, complements are common in the standard motivating applications for combinatorial auctions, like spectrum license auctions. On the other, welfare maximization in the presence of complements is notoriously difficult, and this intractability has stymiedtheoretical progress in the area. For example, there are no known positive results for combinatorial auctionsinwhichbiddervaluationsaremultiparameterandnoncomplementfree,otherthantherelatively weak results known for general valuations. To make inroads on the problem of combinatorial auction design in the presence of complements, we propose a model for valuations with complements that is parameterized by the “size ” of the complements. The model permits a succinct representation, a variety of computationally efficient queries, and nontrivial welfaremaximizationalgorithmsandmechanisms.Specifically,ahypergraphr valuation v foragoodsetM is represented by a hypergraph H = (M,E), where every (hyper)edge e ∈ E contains at most r vertices and has a nonnegative weight we. Each good j ∈ M also has a nonnegative weight wj. The value v(S) for a subset S ⊆ M of goods is defined as the sum of the weights of the goods and edges entirely contained in S. We design the following polynomialtime approximation algorithms and truthful mechanismsfor welfare maximization with bidders with hypergraph valuations.
On the Limits of BlackBox Reductions in Mechanism Design
"... We consider the problem of converting an arbitrary approximation algorithm for a singleparameter optimization problem into a computationally efficient truthful mechanism. We ask for reductions that are blackbox, meaning that they require only oracle access to the given algorithm and in particular ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
(Show Context)
We consider the problem of converting an arbitrary approximation algorithm for a singleparameter optimization problem into a computationally efficient truthful mechanism. We ask for reductions that are blackbox, meaning that they require only oracle access to the given algorithm and in particular do not require explicit knowledge of the problem constraints. Such a reduction is known to be possible, for example, for the social welfare objective when the goal is to achieve Bayesian truthfulness and preserve social welfare in expectation. We show that a blackbox reduction for the social welfare objective is not possible if the resulting mechanism is required to be truthful in expectation and to preserve the worstcase approximation ratio of the algorithm to within a subpolynomial factor. Further, we prove that for other objectives such as makespan, no blackbox reduction is possible even if we only require Bayesian truthfulness and an averagecase performance guarantee.
Testing coverage functions
, 2012
"... A coverage function f over a ground set [m] is associated with a universe U of weighted elements and m sets A1,..., Am ⊆ U, and for any T ⊆ [m], f(T) is defined as the total weight of the elements in the union ∪j∈T Aj. Coverage functions are an important special case of submodular functions, and ar ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
A coverage function f over a ground set [m] is associated with a universe U of weighted elements and m sets A1,..., Am ⊆ U, and for any T ⊆ [m], f(T) is defined as the total weight of the elements in the union ∪j∈T Aj. Coverage functions are an important special case of submodular functions, and arise in many applications, for instance as a class of utility functions of agents in combinatorial auctions. Set functions such as coverage functions often lack succinct representations, and in algorithmic applications, an access to a value oracle is assumed. In this paper, we ask whether one can test if a given oracle is that of a coverage function or not. We demonstrate an algorithm which makes O(mU) queries to an oracle of a coverage function and completely reconstructs it. This gives a polytime tester for succinct coverage functions for which U  is polynomially bounded in m. In contrast, we demonstrate a set function which is “far ” from coverage, but requires 2 ˜ Θ(m) queries to distinguish it from the class of coverage functions.
Mechanisms for Risk Averse Agents, Without Loss
, 2012
"... Auctions in which agents ’ payoffs are random variables have received increased attention in recent years. In particular, recent work in algorithmic mechanism design has produced mechanisms employing internal randomization, partly in response to limitations on deterministic mechanisms imposed by co ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
Auctions in which agents ’ payoffs are random variables have received increased attention in recent years. In particular, recent work in algorithmic mechanism design has produced mechanisms employing internal randomization, partly in response to limitations on deterministic mechanisms imposed by computational complexity. For many of these mechanisms, which are often referred to as truthfulinexpectation, incentive compatibility is contingent on the assumption that agents are riskneutral. These mechanisms have been criticized on the grounds that this assumption is too strong, because “real ” agents are typically risk averse, and moreover their precise attitude towards risk is typically unknown apriori. In response, researchers in algorithmic mechanism design have sought the design of universallytruthful mechanisms — mechanisms for which incentivecompatibility makes no assumptions regarding agents ’ attitudes towards risk. Starting with the observation that universal truthfulness is strictly stronger than incentive compatibility in the presence of risk aversion, we show that any truthfulinexpectation mechanism can be generically transformed into a mechanism that is incentive compatible even when agents are risk averse, without modifying the mechanism’s allocation rule. The transformed mechanism does not require reporting of agents ’ risk profiles. Equivalently, our result can be stated as follows: Every (randomized) allocation rule that is implementable in dominant strategies when players are risk neutral is also implementable when players are endowed with an arbitrary and unknown concave utility function for money. Our result has two main implications: (1) A mechanism designer concerned with an objective which depends only on the allocation rule of the mechanism can feel free to design a truthfulinexpectation mechanism, knowing that the riskneutrality assumption can be removed by a generic blackbox transformation. (2) Studying universallytruthful mechanisms under the pretense of robustness to risk aversion is no longer justified. 1