Multiple knapsack-constrained monotone DR-submodular maximization on distributive lattice
We consider a problem of maximizing a monotone DR-submodular function under multiple order-consistent knapsack constraints on a distributive lattice. Because a distributive lattice is used to represent a dependency constraint, the problem can represent a dependency constrained version of a submodula...
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| Veröffentlicht in: | Mathematical programming Jg. 194; H. 1-2; S. 85 - 119 |
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| Sprache: | Englisch |
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Springer
01.07.2022
Springer Nature B.V |
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| ISSN: | 0025-5610, 1436-4646 |
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| Abstract | We consider a problem of maximizing a monotone DR-submodular function under multiple order-consistent knapsack constraints on a distributive lattice. Because a distributive lattice is used to represent a dependency constraint, the problem can represent a dependency constrained version of a submodular maximization problem on a set. We propose a ( [Formula omitted])-approximation algorithm for this problem. To achieve this result, we generalize the continuous greedy algorithm to distributive lattices: We choose a median complex as a continuous relaxation of the distributive lattice and define the multilinear extension on it. We show that the median complex admits special curves, named uniform linear motions. The multilinear extension of a DR-submodular function is concave along a positive uniform linear motion, which is a key property used in the continuous greedy algorithm. |
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| AbstractList | We consider a problem of maximizing a monotone DR-submodular function under multiple order-consistent knapsack constraints on a distributive lattice. Because a distributive lattice is used to represent a dependency constraint, the problem can represent a dependency constrained version of a submodular maximization problem on a set. We propose a (1-1/e)-approximation algorithm for this problem. To achieve this result, we generalize the continuous greedy algorithm to distributive lattices: We choose a median complex as a continuous relaxation of the distributive lattice and define the multilinear extension on it. We show that the median complex admits special curves, named uniform linear motions. The multilinear extension of a DR-submodular function is concave along a positive uniform linear motion, which is a key property used in the continuous greedy algorithm. We consider a problem of maximizing a monotone DR-submodular function under multiple order-consistent knapsack constraints on a distributive lattice. Because a distributive lattice is used to represent a dependency constraint, the problem can represent a dependency constrained version of a submodular maximization problem on a set. We propose a ( [Formula omitted])-approximation algorithm for this problem. To achieve this result, we generalize the continuous greedy algorithm to distributive lattices: We choose a median complex as a continuous relaxation of the distributive lattice and define the multilinear extension on it. We show that the median complex admits special curves, named uniform linear motions. The multilinear extension of a DR-submodular function is concave along a positive uniform linear motion, which is a key property used in the continuous greedy algorithm. |
| Audience | Academic |
| Author | Maehara, Takanori Nakashima, So Yamaguchi, Yutaro |
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| Snippet | We consider a problem of maximizing a monotone DR-submodular function under multiple order-consistent knapsack constraints on a distributive lattice. Because a... |
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| SubjectTerms | Algorithms Constraints Greedy algorithms Lattices Maximization Optimization |
| Title | Multiple knapsack-constrained monotone DR-submodular maximization on distributive lattice |
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