Randomized rounding: a technique for provably good algorithms and algorithmic proofs
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Probabilistic construction of deterministic algorithms: approximating packing integer programs
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A threshold of ln n for approximating set cover
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Randomized rounding without solving the linear program
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The budgeted maximum coverage problem
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Improved Approximation Guarantees for Packing and Covering Integer Programs
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On Multidimensional Packing Problems
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Approximation techniques for utilitarian mechanism design
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Faster and Simpler Algorithms for Multicommodity Flow and Other Fractional Packing Problems
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Maximizing Non-Monotone Submodular Functions
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Optimal approximation for the submodular welfare problem in the value oracle model
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Maximizing a Submodular Set Function Subject to a Matroid Constraint (Extended Abstract)
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
Maximizing submodular set functions subject to multiple linear constraints
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Symmetry and Approximability of Submodular Maximization Problems
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Maximizing Nonmonotone Submodular Functions under Matroid or Knapsack Constraints
SIAM Journal on Discrete Mathematics
Dependent Randomized Rounding via Exchange Properties of Combinatorial Structures
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Submodular function maximization via the multilinear relaxation and contention resolution schemes
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Nonmonotone submodular maximization via a structural continuous greedy algorithm
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
A Unified Continuous Greedy Algorithm for Submodular Maximization
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Submodular maximization by simulated annealing
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On k-column sparse packing programs
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From query complexity to computational complexity
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
A note on maximizing a submodular set function subject to a knapsack constraint
Operations Research Letters
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We study the problem of maximizing a monotone submodular set function subject to linear packing constraints. An instance of this problem consists of a matrix A∈[0,1]m ×n, a vector b∈[1,∞)m, and a monotone submodular set function f: 2[n]→ℝ+. The objective is to find a set S that maximizes f(S) subject to AxS≤b, where xS stands for the characteristic vector of the set S. A well-studied special case of this problem is when f is linear. This special linear case captures the class of packing integer programs. Our main contribution is an efficient combinatorial algorithm that achieves an approximation ratio of Ω(1 / m1/W), where W= min {bi / Aij : Aij0} is the width of the packing constraints. This result matches the best known performance guarantee for the linear case. One immediate corollary of this result is that the algorithm under consideration achieves constant factor approximation when the number of constraints is constant or when the width of the constraints is sufficiently large. This motivates us to study the large width setting, trying to determine its exact approximability. We develop an algorithm that has an approximation ratio of (1−ε)(1−1/e) when W=Ω(ln m / ε2). This result essentially matches the theoretical lower bound of 1−1/e. We also study the special setting in which the matrix A is binary and k-column sparse. A k-column sparse matrix has at most k non-zero entries in each of its column. We design a fast combinatorial algorithm that achieves an approximation ratio of Ω(1 / (Wk1/W)), that is, its performance guarantee only depends on the sparsity and width parameters.