SIAM Journal on Discrete Mathematics
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
On Local Search for Weighted K-Set Packing
Mathematics of Operations Research
A combinatorial algorithm minimizing submodular functions in strongly polynomial time
Journal of Combinatorial Theory Series B
Greedy local improvement and weighted set packing approximation
Journal of Algorithms
A combinatorial strongly polynomial algorithm for minimizing submodular functions
Journal of the ACM (JACM)
A d/2 approximation for maximum weight independent set in d-claw free graphs
Nordic Journal of Computing
On the complexity of approximating k-set packing
Computational Complexity
Evolutionary algorithms and matroid optimization problems
Proceedings of the 9th annual conference on Genetic and evolutionary computation
Maximizing Non-Monotone Submodular Functions
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Optimal approximation for the submodular welfare problem in the value oracle model
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
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
Submodular Maximization over Multiple Matroids via Generalized Exchange Properties
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
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
Submodular maximization by simulated annealing
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Maximizing a Monotone Submodular Function Subject to a Matroid Constraint
SIAM Journal on Computing
Improved approximations for k-exchange systems
ESA'11 Proceedings of the 19th European conference on Algorithms
Concentration inequalities for nonlinear matroid intersection
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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Submodular function maximization is a central problem in combinatorial optimization, generalizing many important NP-hard problems including max cut in digraphs, graphs, and hypergraphs; certain constraint satisfaction problems; maximum entropy sampling; and maximum facility location problems. Our main result is that for any k ≥ 2 and any ε 0, there is a natural local search algorithm that has approximation guarantee of 1/(k + ε) for the problem of maximizing a monotone submodular function subject to k matroid constraints. This improves upon the 1/(k + 1)-approximation of Fisher, Nemhauser, and Wolsey obtained in 1978 [Fisher, M., G. Nemhauser, L. Wolsey. 1978. An analysis of approximations for maximizing submodular set functions---II. Math. Programming Stud.8 73--87]. 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-1) + ε), respectively. Our analyses are based on two new exchange properties for matroids. One is a generalization of the classical Rota exchange property for matroid bases, and another is an exchange property for two matroids based on the structure of matroid intersection.