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17
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 85 (13 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
Submodular Maximization Over Multiple Matroids via Generalized Exchange Properties
, 2009
"... Submodularfunction maximization is a central problem in combinatorial optimization, generalizing many important NPhard problems including Max Cut in digraphs, graphs and hypergraphs, certain constraint satisfaction problems, maximumentropy sampling, and maximum facilitylocation problems. Our mai ..."
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Cited by 29 (7 self)
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Submodularfunction maximization is a central problem in combinatorial optimization, generalizing many important NPhard problems including Max Cut in digraphs, graphs and hypergraphs, certain constraint satisfaction problems, maximumentropy sampling, and maximum facilitylocation problems. Our main result is that for any k ≥ 2 and any ε> 0, there is a natural localsearch algorithm which has approximation guarantee of 1/(k + ε) for the problem of maximizing a monotone submodular function subject to k matroid constraints. This improves a 1/(k + 1)approximation of Nemhauser, Wolsey and Fisher, obtained more than 30 years ago. Also, our analysis can be applied to the problem of maximizing a linear objective function and even a general nonmonotone submodular function subject to k matroid constraints. We show that in these cases the approximation guarantees of our algorithms are 1/(k − 1 + ε) and 1/(k + 1 + 1/k + ε), respectively.
Capacitated facility location: separation algorithms and computational experience
 Mathematical Programming
, 1998
"... We consider the polyhedral approach to solving the capacitated facility location problem. The valid inequalities considered are the knapsack, flow cover, effective capacity, single depot, and combinatorial inequalities. The flow cover, effective capacity, and single depot inequalities form subfamili ..."
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Cited by 20 (2 self)
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We consider the polyhedral approach to solving the capacitated facility location problem. The valid inequalities considered are the knapsack, flow cover, effective capacity, single depot, and combinatorial inequalities. The flow cover, effective capacity, and single depot inequalities form subfamilies of the general family of submodular inequalities. The separation problem based on the family of submodular inequalities is NPhard in general. For the wellknown subclass of flow cover inequalities, however, we show that if the client set is fixed, and if all capacities are equal, then the separation problem can be solved in polynomial time. For the flow cover inequalities based on an arbitrary client set, and for the effective capacity and single depot inequalities we develop separation heuristics. An important part of all these heuristic is based on constructive proofs that two specific conditions are necessary for the effective capacity inequalities to be facet defining. The proofs show precisely how structures that violate the two conditions can be modified to produce stronger inequalities. The family of combinatorial inequalities was originally developed for the uncapacitated facility location problem, but is also valid for the capacitated problem. No computational experience using the combinatorial inequalities has been reported so far. Here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities. We report on computational results from solving 60 small and medium size problems.
An 0.828 Approximation algorithm for the uncapacitated facility location problem, Discrete Applied Mathematics 93(23
, 1999
"... The uncapacitated facility location problem in the following formulation is considered: max Z(S) = S⊆I max i∈S bij − ∑ ci, j∈J where I and J are finite sets, and bij, ci ≥ 0 are rational numbers. Let Z ∗ denote the optimal value of the problem and ZR = ∑ j∈J mini∈I bij − ∑ i∈I ci. Cornuejols, Fisher ..."
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Cited by 17 (1 self)
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The uncapacitated facility location problem in the following formulation is considered: max Z(S) = S⊆I max i∈S bij − ∑ ci, j∈J where I and J are finite sets, and bij, ci ≥ 0 are rational numbers. Let Z ∗ denote the optimal value of the problem and ZR = ∑ j∈J mini∈I bij − ∑ i∈I ci. Cornuejols, FisherandNemhauser(1977) provethat fortheproblemwiththeadditional cardinality constraint S  ≤ K, a simple greedy algorithm finds a feasible solution S such that (Z(S)−ZR)/(Z ∗−ZR) ≥ 1−e−1 ≈ 0.632. Wesuggestapolynomialtimeapproximation algorithmfortheunconstrainedversionoftheproblem,basedontheideaofrandomized rounding due to Goemans and Williamson (1994). It is proved that the algorithm delivers a solution S such that (Z(S)−ZR)/(Z ∗ −ZR) ≥ 2 ( √ 2−1) ≈ 0.828. We also show that there exists ε> 0 such that it is NPhard to find an approximate solution S with (Z(S)−ZR)/(Z ∗ −ZR) ≥ 1−ε.
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 17 (2 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
On the TwoLevel Uncapacitated Facility Location Problem
 INFORMS J. COMPUT
, 1996
"... We study the twolevel uncapacitated facility location (TUFL) problem. Given two types of facilities, which we call yfacilities and zfacilities, the problem is to decide which facilities of both types to open, and to which pair of y and zfacilities each client should be assigned, in order to sat ..."
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Cited by 16 (3 self)
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We study the twolevel uncapacitated facility location (TUFL) problem. Given two types of facilities, which we call yfacilities and zfacilities, the problem is to decide which facilities of both types to open, and to which pair of y and zfacilities each client should be assigned, in order to satisfy the demand at maximum profit. We first present two multicommodity flow formulations of TUFL and investigate the relationship between these formulations and similar formulations of the onelevel uncapacitated facility location (UFL) problem. In particular, we show that all nontrivial facets for UFL define facets for the twolevel problem, and derive conditions when facets of TUFL are also facets for UFL. For both formulations of TUFL, we introduce new families of facets and valid inequalities and discuss the associated separation problems. We also characterize the extreme points of the LPrelaxation of the first formulation. While the LPrelaxation of a multicommodity formulation provi...
The Optimal Diversity Management Problem
 Operations Research
, 2004
"... In some industries, a certain part can be needed in a very large number of different configurations. This is the case, e.g., for the electrical wirings in european car factories. Fortunately, a given configuration can be replaced by a more complete, therefore also more expensive, one. The diversity ..."
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Cited by 13 (0 self)
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In some industries, a certain part can be needed in a very large number of different configurations. This is the case, e.g., for the electrical wirings in european car factories. Fortunately, a given configuration can be replaced by a more complete, therefore also more expensive, one. The diversity management problem consists in choosing an optimal set of some given number $k$ of configurations that will be produced, any non produced configuration being replaced by the cheapest produced one compatible with it. We model the problem as an integer linear program close to the one commonly used for the $k$median problem. Our aim is to solve those problems to optimality. The large scale instances we are interested in lead to difficult LP relaxations, which seem to be intractable by the best direct methods currently available. Most of this paper deals with the use of Lagrangean Relaxation to reduce the size of the problem in order to be able subsequently to solve it to optimality via classical integer optimization.
Valid inequalities and facets of the capacitated plant location problem
 Mathematical Programming
, 1989
"... Recently, several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods, and polyhedral characterizations, of integer programming problems. In this paper, we investigate the polyhedral structure of the capacitate ..."
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Cited by 9 (1 self)
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Recently, several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods, and polyhedral characterizations, of integer programming problems. In this paper, we investigate the polyhedral structure of the capacitated plant location problem. Our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure. The first part of the paper introduces a family of facets for a version of the capacitated plant location problem with constant capacity K for all plants. These facet inequalities depend on K and thus differ fundamentally from the valid inequalities for the uncapacitated version of the problem. We also introduce a second formulation for a model with indivisible customer demand and show that it is equivalent to a vertex packing problem on a derived graph. We identify facets and valid inequalities for this version of the problem by applying known results for the vertex packing polytope.
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 9 (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.
Graphbased Submodular Selection for Extractive Summarization
"... Abstract—We propose a novel approach for unsupervised extractive summarization. Our approach builds a semantic graph for the document to be summarized. Summary extraction is then formulated as optimizing submodular functions defined on the semantic graph. The optimization is theoretically guaranteed ..."
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Cited by 7 (4 self)
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Abstract—We propose a novel approach for unsupervised extractive summarization. Our approach builds a semantic graph for the document to be summarized. Summary extraction is then formulated as optimizing submodular functions defined on the semantic graph. The optimization is theoretically guaranteed to be nearoptimal under the framework of submodularity. Extensive experiments on the ICSI meeting summarization task on both human transcripts and automatic speech recognition (ASR) outputs show that the graphbased submodular selection approach consistently outperforms the maximum marginal relevance (MMR) approach, a conceptbased approach using integer linear programming (ILP), and a recursive graphbased ranking algorithm using Google’s PageRank. I.