Maximizing a Submodular Function with Bounded Curvature under an Unknown Knapsack Constraint
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909362231967744 |
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| author | Klimm, Max Knaack, Martin |
| author_facet | Klimm, Max Knaack, Martin |
| contents | This paper studies the problem of maximizing a monotone submodular function under an unknown knapsack constraint. A solution to this problem is a policy that decides which item to pack next based on the past packing history. The robustness factor of a policy is the worst case ratio of the solution obtained by following the policy and an optimal solution that knows the knapsack capacity. We develop a policy with a robustness factor that is decreasing in the curvature $c$ of the submodular function. For the extreme cases $c=0$ corresponding to an additive objective function, it matches a previously known and best possible robustness factor of $1/2$. For the other extreme case of $c=1$ it yields a robustness factor of $\approx 0.35$ improving over the best previously known robustness factor of $\approx 0.06$.
The analysis of our policy relies on a greedy algorithm that is a slight modification of Wolsey's greedy algorithm for the submodular knapsack problem with a known knapsack constraint. We obtain tight approximation guarantees for both of these algorithms in the setting of a submodular objective function with curvature $c$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_09668 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Maximizing a Submodular Function with Bounded Curvature under an Unknown Knapsack Constraint Klimm, Max Knaack, Martin Data Structures and Algorithms Discrete Mathematics This paper studies the problem of maximizing a monotone submodular function under an unknown knapsack constraint. A solution to this problem is a policy that decides which item to pack next based on the past packing history. The robustness factor of a policy is the worst case ratio of the solution obtained by following the policy and an optimal solution that knows the knapsack capacity. We develop a policy with a robustness factor that is decreasing in the curvature $c$ of the submodular function. For the extreme cases $c=0$ corresponding to an additive objective function, it matches a previously known and best possible robustness factor of $1/2$. For the other extreme case of $c=1$ it yields a robustness factor of $\approx 0.35$ improving over the best previously known robustness factor of $\approx 0.06$. The analysis of our policy relies on a greedy algorithm that is a slight modification of Wolsey's greedy algorithm for the submodular knapsack problem with a known knapsack constraint. We obtain tight approximation guarantees for both of these algorithms in the setting of a submodular objective function with curvature $c$. |
| title | Maximizing a Submodular Function with Bounded Curvature under an Unknown Knapsack Constraint |
| topic | Data Structures and Algorithms Discrete Mathematics |
| url | https://arxiv.org/abs/2209.09668 |