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Submodular function maximization via the multilinear relaxation and contention resolution schemes
 IN ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2011
"... We consider the problem of maximizing a nonnegative submodular set function f: 2 N → R+ over a ground set N subject to a variety of packing type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that all ..."
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Cited by 38 (2 self)
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We consider the problem of maximizing a nonnegative submodular set function f: 2 N → R+ over a ground set N subject to a variety of packing type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that allows us to derive a number of new results, in particular when f may be a nonmonotone function. Our algorithms are based on (approximately) solving the multilinear extension F of f [5] over a polytope P that represents the constraints, and then effectively rounding the fractional solution. Although this approach has been used quite successfully in some settings [6, 22, 24, 13, 3], it has been limited in some important ways. We overcome these limitations as follows. First, we give constant factor approximation algorithms to maximize
Submodular Secretary Problem and Extensions
"... Online auction is the essence of many modern markets, particularly networked markets, in which information about goods, agents, and outcomes is revealed over a period of time, and the agents must make irrevocable decisions without knowing future information. Optimal stopping theory, especially the c ..."
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Cited by 24 (1 self)
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Online auction is the essence of many modern markets, particularly networked markets, in which information about goods, agents, and outcomes is revealed over a period of time, and the agents must make irrevocable decisions without knowing future information. Optimal stopping theory, especially the classic secretary problem, is a powerful tool for analyzing such online scenarios which generally require optimizing an objective function over the input. The secretary problem and its generalization the multiplechoice secretary problem were under a thorough study in the literature. In this paper, we consider a very general setting of the latter problem called the submodular secretary problem, in which the goal is to select k secretaries so as to maximize the expectation of a (not necessarily monotone) submodular function which defines efficiency of the selected secretarial group based on their overlapping skills. We present the first constantcompetitive algorithm for this case. In a more general setting in which selected secretaries should form an independent (feasible) set in each of l given matroids as well, we obtain an O(l log² r)competitive algorithm generalizing several previous results, where r is the maximum rank of the matroids. Another generalization is to consider l knapsack constraints (i.e., a knapsack constraint assigns a nonnegative cost to each secretary, and requires that the total cost of all the secretaries employed be no more than a budget value) instead of the matroid constraints, for which we present an O(l)competitive algorithm. In a sharp contrast, we show for a more general setting of subadditive secretary problem, there is no õ ( √ n)competitive algorithm and thus submodular functions are the most general functions to consider for constantcompetitiveness in our setting. We complement this result by giving a matching O ( √ n)competitive algorithm for the subadditive case. At the end, we consider some special cases of our general setting as well.
Fast greedy algorithms in mapreduce and streaming
 In SPAA
, 2013
"... Greedy algorithms are practitioners ’ best friends—they are intuitive, simple to implement, and often lead to very good solutions. However, implementing greedy algorithms in a distributed setting is challenging since the greedy choice is inherently sequential, and it is not clear how to take advant ..."
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Cited by 22 (1 self)
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Greedy algorithms are practitioners ’ best friends—they are intuitive, simple to implement, and often lead to very good solutions. However, implementing greedy algorithms in a distributed setting is challenging since the greedy choice is inherently sequential, and it is not clear how to take advantage of the extra processing power. Our main result is a powerful sampling technique that aids in parallelization of sequential algorithms. We then show how to use this primitive to adapt a broad class of greedy algorithms to the MapReduce paradigm; this class includes maximum cover and submodular maximization subject to psystem constraints. Our method yields efficient algorithms that run in a logarithmic number of rounds, while obtaining solutions that are arbitrarily close to those produced by the standard sequential greedy algorithm. We begin with algorithms for modular maximization subject to a matroid constraint, and then extend this approach to obtain approximation algorithms for submodular maximization subject to knapsack or psystem constraints. Finally, we empirically validate our algorithms, and show that they achieve the same quality of the solution as standard greedy algorithms but run in a substantially fewer number of rounds. Categories and Subject Descriptors
Submodular Function Maximization
, 2012
"... Submodularity is a property of set functions with deep theoretical consequences and far–reaching applications. At first glance it appears very similar to concavity, in other ways it resembles convexity. It appears in a wide variety of applications: in Computer Science it has recently been identifie ..."
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Cited by 19 (5 self)
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Submodularity is a property of set functions with deep theoretical consequences and far–reaching applications. At first glance it appears very similar to concavity, in other ways it resembles convexity. It appears in a wide variety of applications: in Computer Science it has recently been identified and utilized in domains such as viral marketing (Kempe et al., 2003), information gathering (Krause and Guestrin, 2007), image segmentation (Boykov and
NearOptimal MAP Inference for Determinantal Point Processes
"... Determinantal point processes (DPPs) have recently been proposed as computationally efficient probabilistic models of diverse sets for a variety of applications, including document summarization, image search, and pose estimation. Many DPP inference operations, including normalization and sampling, ..."
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Cited by 14 (3 self)
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Determinantal point processes (DPPs) have recently been proposed as computationally efficient probabilistic models of diverse sets for a variety of applications, including document summarization, image search, and pose estimation. Many DPP inference operations, including normalization and sampling, are tractable; however, finding the most likely configuration (MAP), which is often required in practice for decoding, is NPhard, so we must resort to approximate inference. This optimization problem, which also arises in experimental design and sensor placement, involves finding the largest principal minor of a positive semidefinite matrix. Because the objective is logsubmodular, greedy algorithms have been used in the past with some empirical success; however, these methods only give approximation guarantees in the special case of monotone objectives, which correspond to a restricted class of DPPs. In this paper we propose a new algorithm for approximating the MAP problem based on continuous techniques for submodular function maximization. Our method involves a novel continuous relaxation of the logprobability function, which, in contrast to the multilinear extension used for general submodular functions, can be evaluated and differentiated exactly and efficiently. We obtain a practical algorithm with a 1/4approximation guarantee for a more general class of nonmonotone DPPs; our algorithm also extends to MAP inference under complex polytope constraints, making it possible to combine DPPs with Markov random fields, weighted matchings, and other models. We demonstrate that our approach outperforms standard and recent methods on both synthetic and realworld data. 1
Nonmonotone submodular maximization via a structural continuous greedy algorithm (Extended Abstract)
 IN ICALP
, 2011
"... Consider a suboptimal solution S for a maximization problem. Suppose S’s value is small compared to an optimal solution OP T to the problem, yet S is structurally similar to OP T. A natural question in this setting is whether there is a way of improving S based solely on this information. In this p ..."
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Cited by 14 (3 self)
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Consider a suboptimal solution S for a maximization problem. Suppose S’s value is small compared to an optimal solution OP T to the problem, yet S is structurally similar to OP T. A natural question in this setting is whether there is a way of improving S based solely on this information. In this paper we introduce the Structural Continuous Greedy Algorithm, answering this question affirmatively in the setting of the Nonmonotone Submodular Maximization Problem. We improve on the best approximation factor known for this problem. In the Nonmonotone Submodular Maximization Problem we are given a nonnegative submodular function f, and the objective is to find a subset maximizing f. Our method yields an 0.42approximation for this problem, improving on the current best approximation factor of 0.41 given by Gharan and Vondrák [5]. On the other hand, Feige et al. [4] showed a lower bound of 0.5 for this problem.
Improved Competitive Ratios for Submodular Secretary Problems (Extended Abstract)
, 2011
"... The Classical Secretary Problem was introduced during the 60’s of the 20 th century, nobody is sure exactly when. Since its introduction, many variants of the problem have been proposed and researched. In the classical secretary problem, and many of its variant, the input (which is a set of secret ..."
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Cited by 10 (2 self)
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The Classical Secretary Problem was introduced during the 60’s of the 20 th century, nobody is sure exactly when. Since its introduction, many variants of the problem have been proposed and researched. In the classical secretary problem, and many of its variant, the input (which is a set of secretaries, or elements) arrives in a random order. In this paper we apply to the secretary problem a simple observation which states that the random order of the input can be generated by independently choosing a random continuous arrival time for each secretary. Surprisingly, this simple observation enables us to improve the competitive ratio of several known and studied variants of the secretary problem. In addition, in some cases the proofs we provide assuming random arrival times are shorter and simpler in comparison to existing proofs. In this work we consider three variants of the secretary problem, all of which have the same objective of maximizing the value of the chosen set of secretaries given a monotone submodular function f. In the first variant we are allowed to hire a set of secretaries only if it is an independent set of a given partition matroid. The second variant allows us to choose any set of up to k secretaries. In the last and third variant, we can hire any set of secretaries satisfying a given knapsack constraint.
Advances on Matroid Secretary Problems: Free Order Model and Laminar Case
, 2012
"... The most wellknown conjecture in the context of matroid secretary problems claims the existence of a constantfactor approximation applicable to any matroid. Whereas this conjecture remains open, modified forms of it were shown to be true, when assuming that the assignment of weights to the secreta ..."
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Cited by 7 (1 self)
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The most wellknown conjecture in the context of matroid secretary problems claims the existence of a constantfactor approximation applicable to any matroid. Whereas this conjecture remains open, modified forms of it were shown to be true, when assuming that the assignment of weights to the secretaries is not adversarial but uniformly random [19, 17]. However, so far, there was no variant of the matroid secretary problem with adversarial weight assignment for which a constantfactor approximation was found. We address this point by presenting a 9approximation for the free order model, a model suggested shortly after the introduction of the matroid secretary problem, and for which no constantfactor approximation was known so far. The free order model is a relaxed version of the original matroid secretary problem, with the only difference that one can choose the order in which secretaries are interviewed. Furthermore, we consider the classical matroid secretary problem for the special case of laminar matroids. Only recently, a constantfactor approximation has been found for this case, using a clever but rather involved method and analysis [12] that leads to a 16000/3approximation. This is arguably the most involved special case of the matroid secretary problem for which a constantfactor approximation is known. We present a considerably simpler and stronger 3 √ 3e ≈ 14.12approximation, based on reducing the problem to a matroid secretary problem on a partition matroid.
Pricing Tasks in Online Labor Markets
 Proceedings of HCOMP11: The 3rd Workshop on Human Computation
, 2011
"... In this paper we present a mechanism for determining nearoptimal prices for tasks in online labor markets, often used for crowdsourcing. In particular, the mechanisms are designed to handle the intricacies of markets like Mechanical Turk where workers arrive online and requesters have budget constra ..."
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Cited by 7 (0 self)
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In this paper we present a mechanism for determining nearoptimal prices for tasks in online labor markets, often used for crowdsourcing. In particular, the mechanisms are designed to handle the intricacies of markets like Mechanical Turk where workers arrive online and requesters have budget constraints. The mechanism is incentive compatible, budget feasible, and has competitive ratio performance and also performs well in practice. To demonstrate the mechanism’s practical effectiveness we conducted experiments on the Mechanical Turk platform.
Streaming Submodular Maximization: Massive Data Summarization on the Fly
, 2014
"... How can one summarize a massive data set “on the fly”, i.e., without even having seen it in its entirety? In this paper, we address the problem of extracting representative elements from a large stream of data. I.e., we would like to select a subset of say k data points from the stream that are most ..."
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Cited by 7 (3 self)
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How can one summarize a massive data set “on the fly”, i.e., without even having seen it in its entirety? In this paper, we address the problem of extracting representative elements from a large stream of data. I.e., we would like to select a subset of say k data points from the stream that are most representative according to some objective function. Many natural notions of “representativeness ” satisfy submodularity, an intuitive notion of diminishing returns. Thus, such problems can be reduced to maximizing a submodular set function subject to a cardinality constraint. Classical approaches to submodular maximization require full access to the data set. We develop the first efficient streaming algorithm with constant factor 1/2 − ε approximation guarantee to the optimum solution, requiring only a single pass through the data, and memory independent of data size. In our experiments, we extensively evaluate the effectiveness of our approach on several applications, including training largescale kernel methods and exemplarbased clustering, on millions of data points. We observe that our streaming method, while achieving practically the same utility value, runs about 100 times faster than previous work.