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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.14278 |
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| _version_ | 1866909228130631680 |
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| author | Wan, Zongqi Zhang, Jialin Sun, Xiaoming Zhang, Zhijie |
| author_facet | Wan, Zongqi Zhang, Jialin Sun, Xiaoming Zhang, Zhijie |
| contents | Symmetric submodular maximization is an important class of combinatorial optimization problems, including MAX-CUT on graphs and hyper-graphs. The state-of-the-art algorithm for the problem over general constraints has an approximation ratio of $0.432$. The algorithm applies the canonical continuous greedy technique that involves a sampling process. It, therefore, suffers from high query complexity and is inherently randomized. In this paper, we present several efficient deterministic algorithms for maximizing a symmetric submodular function under various constraints. Specifically, for the cardinality constraint, we design a deterministic algorithm that attains a $0.432$ ratio and uses $O(kn)$ queries. Previously, the best deterministic algorithm attains a $0.385-ε$ ratio and uses $O\left(kn (\frac{10}{9ε})^{\frac{20}{9ε}-1}\right)$ queries. For the matroid constraint, we design a deterministic algorithm that attains a $1/3-ε$ ratio and uses $O(kn\log ε^{-1})$ queries. Previously, the best deterministic algorithm can also attain $1/3-ε$ ratio but it uses much larger $O(ε^{-1}n^4)$ queries. For the packing constraints with a large width, we design a deterministic algorithm that attains a $0.432-ε$ ratio and uses $O(n^2)$ queries. To the best of our knowledge, there is no deterministic algorithm for the constraint previously. The last algorithm can be adapted to attain a $0.432$ ratio for single knapsack constraint using $O(n^4)$ queries. Previously, the best deterministic algorithm attains a $0.316-ε$ ratio and uses $\widetilde{O}(n^3)$ queries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14278 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Efficient Deterministic Algorithms for Maximizing Symmetric Submodular Functions Wan, Zongqi Zhang, Jialin Sun, Xiaoming Zhang, Zhijie Data Structures and Algorithms Symmetric submodular maximization is an important class of combinatorial optimization problems, including MAX-CUT on graphs and hyper-graphs. The state-of-the-art algorithm for the problem over general constraints has an approximation ratio of $0.432$. The algorithm applies the canonical continuous greedy technique that involves a sampling process. It, therefore, suffers from high query complexity and is inherently randomized. In this paper, we present several efficient deterministic algorithms for maximizing a symmetric submodular function under various constraints. Specifically, for the cardinality constraint, we design a deterministic algorithm that attains a $0.432$ ratio and uses $O(kn)$ queries. Previously, the best deterministic algorithm attains a $0.385-ε$ ratio and uses $O\left(kn (\frac{10}{9ε})^{\frac{20}{9ε}-1}\right)$ queries. For the matroid constraint, we design a deterministic algorithm that attains a $1/3-ε$ ratio and uses $O(kn\log ε^{-1})$ queries. Previously, the best deterministic algorithm can also attain $1/3-ε$ ratio but it uses much larger $O(ε^{-1}n^4)$ queries. For the packing constraints with a large width, we design a deterministic algorithm that attains a $0.432-ε$ ratio and uses $O(n^2)$ queries. To the best of our knowledge, there is no deterministic algorithm for the constraint previously. The last algorithm can be adapted to attain a $0.432$ ratio for single knapsack constraint using $O(n^4)$ queries. Previously, the best deterministic algorithm attains a $0.316-ε$ ratio and uses $\widetilde{O}(n^3)$ queries. |
| title | Efficient Deterministic Algorithms for Maximizing Symmetric Submodular Functions |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2406.14278 |