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144
Algorithms in Discrete Convex Analysis
 Math. Programming
, 2000
"... this paper is to describe the f#eA damental results on M and Lconvex f#24L2A+ with special emphasis on algorithmic aspects. ..."
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Cited by 161 (34 self)
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this paper is to describe the f#eA damental results on M and Lconvex f#24L2A+ with special emphasis on algorithmic aspects.
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 1/2approximation [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.
Multidocument summarization via budgeted maximization of submodular functions
 In Proceedings of Human Language Technologies: The Conference of the North American Chapter of the Association for Computational Linguistics (NAACLHLT
, 2010
"... We treat the text summarization problem as maximizing a submodular function under a budget constraint. We show, both theoretically and empirically, a modified greedy algorithm can efficiently solve the budgeted submodular maximization problem nearoptimally, and we derive new approximation bounds in ..."
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Cited by 68 (13 self)
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We treat the text summarization problem as maximizing a submodular function under a budget constraint. We show, both theoretically and empirically, a modified greedy algorithm can efficiently solve the budgeted submodular maximization problem nearoptimally, and we derive new approximation bounds in doing so. Experiments on DUC’04 task show that our approach is superior to the bestperforming method from the DUC’04 evaluation on ROUGE1 scores. 1
Crowdsourcing to smartphones: incentive mechanism design for mobile phone sensing
 in Proceedings of the 18th annual international conference on Mobile computing and networking (Mobicom 2012
, 2012
"... Mobile phone sensing is a new paradigm which takes advantage of the pervasive smartphones to collect and analyze data beyond the scale of what was previously possible. In a mobile phone sensing system, the platform recruits smartphone users to provide sensing service. Existing mobile phone sensing a ..."
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Cited by 64 (0 self)
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Mobile phone sensing is a new paradigm which takes advantage of the pervasive smartphones to collect and analyze data beyond the scale of what was previously possible. In a mobile phone sensing system, the platform recruits smartphone users to provide sensing service. Existing mobile phone sensing applications and systems lack good incentive mechanisms that can attract more user participation. To address this issue, we design incentive mechanisms for mobile phone sensing. We consider two system models: the platformcentric model where the platform provides a reward shared by participating users, and the usercentric model where users have more control over the payment they will receive. For the platformcentric model, we design an incentive mechanism using a Stackelberg game, where the platform is the leader while the users are the followers. We show how to compute the unique Stackelberg Equilibrium, at which the utility of the platform is maximized, and none of the users can improve its utility by unilaterally deviating from its current strategy. For the usercentric model, we design an auctionbased incentive mechanism, which is computationally efficient, individually rational, profitable, and truthful. Through extensive simulations, we evaluate the performance and validate the theoretical properties of our incentive mechanisms.
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 computationallyef 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 computationallyef 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 PUBLIC 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 communicationcomplexity model and one in the computationalcomplexity model. Both these hardness results heavily rely on a combinatorial characterization of truthful algorithms for our problem. Our computationalcomplexity result is one of the rst impossibility results connecting mechanism design to complexity theory; its novel proof technique involves an application of the SauerShelah 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.
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 51 (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.
Submodular Function Minimization under Covering Constraints
, 2009
"... This paper addresses the problems of minimizing nonnegative submodular functions under covering constraints, which generalize the vertex cover, edge cover, and set cover problems. We give approximation algorithms for these problems exploiting the discrete convexity of submodular functions. We first ..."
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Cited by 48 (1 self)
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This paper addresses the problems of minimizing nonnegative submodular functions under covering constraints, which generalize the vertex cover, edge cover, and set cover problems. We give approximation algorithms for these problems exploiting the discrete convexity of submodular functions. We first present a rounding 2approximation algorithm for the submodular vertex cover problem based on the halfintegrality of the continuous relaxation problem, and show that the rounding algorithm can be performed by one application of submodular function minimization on a ring family. We also show that a rounding algorithm and a primaldual algorithm for the submodular cost set cover problem are both constant factor approximation algorithms if the maximum frequency is fixed. In addition, we give an essentially tight lower bound on the approximability of the submodular edge cover problem.
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 ..."
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Cited by 47 (3 self)
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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