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38
Maximizing nonmonotone submodular functions
 In Proceedings of 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS
, 2007
"... Submodular maximization generalizes many important problems including Max Cut in directed/undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular fu ..."
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Cited by 145 (17 self)
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Submodular maximization generalizes many important problems including Max Cut in directed/undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is NPhard. In this paper, we design the first constantfactor approximation algorithms for maximizing nonnegative submodular functions. In particular, we give a deterministic local search 1 2approximation and a randomizedapproximation algo
Limiting the Spread of Misinformation in Social Networks
"... In this work, we study the notion of competing campaigns in a social network. By modeling the spread of influence in the presence of competing campaigns, we provide necessary tools for applications such as emergency response where the goal is to limit the spread of misinformation. We study the probl ..."
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Cited by 53 (2 self)
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In this work, we study the notion of competing campaigns in a social network. By modeling the spread of influence in the presence of competing campaigns, we provide necessary tools for applications such as emergency response where the goal is to limit the spread of misinformation. We study the problem of influence limitation where a “bad ” campaign starts propagating from a certain node in the network and use the notion of limiting campaigns to counteract the effect of misinformation. The problem can be summarized as identifying a subset of individuals that need to be convinced to adopt the competing (or “good”) campaign so as to minimize the number of people that adopt the “bad ” campaign at the end of both propagation processes. We show that this optimization problem is NPhard and provide approximation guarantees for a greedy solution for various definitions of this problem by proving that they are submodular. Although the greedy algorithm is a polynomial time algorithm, for today’s large scale social networks even this solution is computationally very expensive. Therefore, we study the performance of the degree centrality heuristic as well as other heuristics that have implications on our specific problem. The experiments on a number of closeknit regional networks obtained from the Facebook social network show that in most cases inexpensive heuristics do in fact compare well with the greedy approach.
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.
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.
Nonmonotone submodular maximization under matroid and knapsack constraints
 In Proc. 41th ACM Symp. on Theory of Computing
, 2009
"... Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Un ..."
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Cited by 37 (1 self)
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Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NPhard. In this paper, we give the first constantfactor approximation algorithm for maximizing any nonnegative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for nonmonotone submodular functions. In particular, for any constant k, we present a 1 k+2+ 1 k +ǫapproximation for the submodular maximization problem under k matroid constraints, 1 k+ǫ and a ( 1 5 − ǫ)approximation algorithm for this problem subject to k knapsack constraints (ǫ> 0 is 1 any constant). We improve the approximation guarantee of our algorithm to k+1+ 1 for k ≥ 2 k−1 +ǫ partition matroid constraints. This idea also gives aapproximation for maximizing a monotone submodular function subject to k ≥ 2 partition matroids, which improves over the previously best known guarantee of
A tight linear time (1/2)approximation for unconstrained submodular maximization
 in: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, IEEE
"... Abstract—We consider the Unconstrained Submodular Maximization problem in which we are given a nonnegative submodular function f: 2 N → R +, and the objective is to find a subset S ⊆ N maximizing f(S). This is one of the most basic submodular optimization problems, having a wide range of applicatio ..."
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Cited by 36 (2 self)
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Abstract—We consider the Unconstrained Submodular Maximization problem in which we are given a nonnegative submodular function f: 2 N → R +, and the objective is to find a subset S ⊆ N maximizing f(S). This is one of the most basic submodular optimization problems, having a wide range of applications. Some well known problems captured by Unconstrained Submodular Maximization include MaxCut, MaxDiCut, and variants of MaxSAT and maximum facility location. We present a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige et al. [11]. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem. Our method might seem counterintuitive, since it is known that the greedy algorithm fails to achieve any bounded approximation factor for the problem. KeywordsSubmodular Functions, Approximation Algorithms I.
Approximability of Combinatorial Problems with Multiagent Submodular Cost Functions
"... Abstract — Applications in complex systems such as the Internet have spawned recent interest in studying situations involving multiple agents with their individual cost or utility functions. In this paper, we introduce an algorithmic framework for studying combinatorial problems in the presence of m ..."
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Cited by 33 (6 self)
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Abstract — Applications in complex systems such as the Internet have spawned recent interest in studying situations involving multiple agents with their individual cost or utility functions. In this paper, we introduce an algorithmic framework for studying combinatorial problems in the presence of multiple agents with submodular cost functions. We study several fundamental covering problems (Vertex Cover, Shortest Path, Perfect Matching, and Spanning Tree) in this setting and establish tight upper and lower bounds for the approximability of these problems. 1.
Learning submodular functions
 In Proceedings of the 43rd annual ACM symposium on Theory of computing
, 2011
"... Abstract. Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications that have been used in many areas, including combinatorial optimization, machine learning, and economics. In this work we use a learning theoretic angle for study ..."
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Cited by 28 (3 self)
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Abstract. Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications that have been used in many areas, including combinatorial optimization, machine learning, and economics. In this work we use a learning theoretic angle for studying submodular functions. We provide algorithms for learning submodular functions, as well as lower bounds on their learnability. In doing so, we uncover several novel structural results revealing both extremal properties as well as regularities of submodular functions, of interest to many areas. Submodular functions are a discrete analog of convex functions that enjoy numerous applications and have structural properties that can be exploited algorithmically. They arise naturally in the study of graphs, matroids, covering problems, facility location problems, etc., and they have been extensively studied in operations research and combinatorial optimization for many years [8]. More recently submodular functions have become key concepts both in the machine
Submodular Maximization by Simulated Annealing
"... We consider the problem of maximizing a nonnegative (possibly nonmonotone) submodular set function with or without constraints. Feige et al. [9] showed a 2/5approximation for the unconstrained problem and also proved that no approximation better than 1/2 is possible in the value oracle model. Cons ..."
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Cited by 19 (2 self)
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We consider the problem of maximizing a nonnegative (possibly nonmonotone) submodular set function with or without constraints. Feige et al. [9] showed a 2/5approximation for the unconstrained problem and also proved that no approximation better than 1/2 is possible in the value oracle model. Constantfactor approximation has been also known for submodular maximization subject to a matroid independence constraint (a factor of 0.309 [33]) and for submodular maximization subject to a matroid base constraint, provided that the fractional base packing number ν is bounded away from 1 (a 1/4approximation assuming that ν ≥ 2 [33]). In this paper, we propose a new algorithm for submodular maximization which is based on the idea of simulated annealing. We prove that this algorithm achieves improved approximation for two problems: a 0.41approximation for unconstrained submodular maximization, and a 0.325approximation for submodular maximization subject to a matroid independence constraint. On the hardness side, we show that in the value oracle model it is impossible to achieve a 0.478approximation for submodular maximization subject to a matroid independence constraint, or a 0.394approximation subject to a matroid base constraint in matroids with two disjoint bases. Even for the special case of cardinality constraint, we prove it is impossible to achieve a 0.491approximation. (Previously it was conceivable that a 1/2approximation exists for these problems.) It is still an open question whether a 1/2approximation is possible for unconstrained submodular maximization. 1