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117
Maximizing a Submodular Set Function subject to a Matroid Constraint (Extended Abstract)
 PROC. OF 12 TH IPCO
, 2007
"... Let f: 2 N → R + be a nondecreasing submodular set function, and let (N, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2approximation [9] for this problem. It is also known, via a reduction from the maxkcover problem, that there is no (1 ..."
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Cited by 114 (12 self)
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Let f: 2 N → R + be a nondecreasing submodular set function, and let (N, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2approximation [9] for this problem. It is also known, via a reduction from the maxkcover problem, that there is no (1 − 1/e + ɛ)approximation for any constant ɛ> 0, unless P = NP [6]. In this paper, we improve the 1/2approximation to a (1−1/e)approximation, when f is a sum of weighted rank functions of matroids. This class of functions captures a number of interesting problems including set coverage type problems. Our main tools are the pipage rounding technique of Ageev and Sviridenko [1] and a probabilistic lemma on monotone submodular functions that might be of independent interest. We show that the generalized assignment problem (GAP) is a special case of our problem; although the reduction requires N  to be exponential in the original problem size, we are able to interpret the recent (1 − 1/e)approximation for GAP by Fleischer et al. [10] in our framework. This enables us to obtain a (1 − 1/e)approximation for variants of GAP with more complex constraints.
A class of submodular functions for document summarization
 IN THE 49TH ANNUAL MEETING OF THE ASSOCIATION FOR COMPUTATIONAL LINGUISTICS: HUMAN LANGUAGE TECHNOLOGIES (ACLHLT
, 2011
"... We design a class of submodular functions meant for document summarization tasks. These functions each combine two terms, one which encourages the summary to be representative of the corpus, and the other which positively rewards diversity. Critically, our functions are monotone nondecreasing and su ..."
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Cited by 89 (21 self)
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We design a class of submodular functions meant for document summarization tasks. These functions each combine two terms, one which encourages the summary to be representative of the corpus, and the other which positively rewards diversity. Critically, our functions are monotone nondecreasing and submodular, which means that an efficient scalable greedy optimization scheme has a constant factor guarantee of optimality. When evaluated on DUC 20042007 corpora, we obtain better than existing stateofart results in both generic and queryfocused document summarization. Lastly, we show that several wellestablished methods for document summarization correspond, in fact, to submodular function optimization, adding further evidence that submodular functions are a natural fit for document summarization.
Approximating minimum cost connectivity problems
 58 in Approximation algorithms and Metaheuristics, Editor
, 2007
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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.
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
Algorithms for dynamic geometric problems over data streams
 In STOC ’04: Proceedings of the thirtysixth annual ACM symposium on Theory of computing
, 2004
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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 nonnegative, 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.
Robust submodular observation selection
, 2008
"... In many applications, one has to actively select among a set of expensive observations before making an informed decision. For example, in environmental monitoring, we want to select locations to measure in order to most effectively predict spatial phenomena. Often, we want to select observations wh ..."
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Cited by 44 (4 self)
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In many applications, one has to actively select among a set of expensive observations before making an informed decision. For example, in environmental monitoring, we want to select locations to measure in order to most effectively predict spatial phenomena. Often, we want to select observations which are robust against a number of possible objective functions. Examples include minimizing the maximum posterior variance in Gaussian Process regression, robust experimental design, and sensor placement for outbreak detection. In this paper, we present the Submodular Saturation algorithm, a simple and efficient algorithm with strong theoretical approximation guarantees for cases where the possible objective functions exhibit submodularity, an intuitive diminishing returns property. Moreover, we prove that better approximation algorithms do not exist unless NPcomplete problems admit efficient algorithms. We show how our algorithm can be extended to handle complex cost functions (incorporating nonunit observation cost or communication and path costs). We also show how the algorithm can be used to nearoptimally trade off expectedcase (e.g., the Mean Square Prediction Error in Gaussian Process regression) and worstcase (e.g., maximum predictive variance) performance. We show that many important machine learning problems fit our robust submodular observation selection formalism, and provide extensive empirical evaluation on several realworld problems. For Gaussian Process regression, our algorithm compares favorably with stateoftheart heuristics described in the geostatistics literature, while being simpler, faster and providing theoretical guarantees. For robust experimental design, our algorithm performs favorably compared to SDPbased algorithms.
Covering problems with hard capacities
 IN PROC OF. FOCS’02
, 2002
"... We consider the classical vertex cover and set cover problems with the addition of hard capacity constraints. This means that a set (vertex) can only cover a limited number of its elements (adjacent edges) and the number of available copies of each set (vertex) is bounded. This is a natural generali ..."
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Cited by 41 (1 self)
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We consider the classical vertex cover and set cover problems with the addition of hard capacity constraints. This means that a set (vertex) can only cover a limited number of its elements (adjacent edges) and the number of available copies of each set (vertex) is bounded. This is a natural generalization of the classical problems that also captures resource limitations in practical scenarios. We obtain the following results. For the unweighted vertex cover problem with hard capacities we give aapproximation algorithm which is based on randomized rounding with alterations. We prove that the weighted version is at least as hard as the set cover problem. This is an interesting separation between the approximability of weighted and unweighted versions of a “natural ” graph problem. A logarithmic approximation factor for both the set cover and the weighted vertex cover problem with hard capacities follows from the work of Wolsey [23] on submodular set cover. We provide in this paper a simple and intuitive proof for this bound.