Non-monotone k-Submodular Function Maximization with Individual Size Constraints

Hao, Xiao‐Jiang; Liu, Qian; Zhou, Yang; Li, Min

doi:10.1007/978-3-031-26303-3_24

Cited by 4 publications

(4 citation statements)

References 13 publications

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“…The following useful lemma says that the optimal value f (T ) is no more than twice the value of any partial greedy solution S t , plus the total marginal gain of other item-dimension pairs in the optimal solution. This conclusion is firstly noticed by Ward and Živnỳ (implicitly in Theorem 5.1 [19]) and formalized by Xiao et al [22]. For completeness, we write down the proof in our notations.…”

Section: Algorithm 1 Greedysupporting

confidence: 63%

“…One decade ago, k-submodular functions were introduced by Huber and Kolmogorov [5] to express submodularity on choosing k disjoint sets of elements instead of a single set. Subsequently, ksubmodular functions have become a popular research direction [3,4,8,14,21], especially the problem of maximizing k-submodular functions.…”

Section: Related Workmentioning

confidence: 99%

“…Pham et al [12] proposed streaming algorithms with approximation ratios 1 4 − and 1 5 − for the monotone and non-monotone cases, respectively, which requires O( n log n) queries of the k-submodular function. Other works related to kSKM include [11,16,22,23,24]. While all the above mentioned are combinatorial algorithms, the best known result for kSKM comes from extension-based methods.…”

Section: Related Workmentioning

confidence: 99%

See 2 more Smart Citations

Improved Analysis of Greedy Algorithm on k-Submodular Knapsack

Tang,

Wang

2023

Frontiers in Artificial Intelligence and Applications

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A k-submodular function is a generalization of submodular functions that takes k disjoint subsets as input and outputs a real value. It captures many problems in combinatorial optimization and machine leaning such as influence maximization, sensor placement, feature selection, etc. In this paper, we consider the monotone k-submodular maximization problem under a knapsack constraint, and explore the performance guarantee of a greedy-based algorithm: enumerating all size-2 solutions and extending every singleton solution greedily; the best outcome is returned. We provide a novel analysis framework and prove that this algorithm achieves an approximation ratio of at least 0.328. This is the best-known result of combinatorial algorithms on k-submodular knapsack maximization. In addition, within the framework, we can further improve the approximation ratio to a value approaching 1/3 with any desirable accuracy, by enumerating sufficiently large base solutions. The results can even be extended to non-monotone k-submodular functions.

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Section: Algorithm 1 Greedysupporting

confidence: 63%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Improved Analysis of Greedy Algorithm on k-Submodular Knapsack

Tang,

Wang

2023

Frontiers in Artificial Intelligence and Applications

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“…For non-monotone k-SMM, Nguyen and Thai [16] obtained a 1/3-approximation ratio under total size constraints and Shi et al [21] achieved a (1 − 1 e − ϵ)-approximation ratio under individual size constraints. Under the individual size constraints, Xiao et al [25] obtained 1/(B m + 4) and 1/4 approximation guarantees through two algorithms, respectively, where B m = max{B 1 , . .…”

mentioning

confidence: 99%

Maximizing a $ k $-submodular function with $ p $-system constraints

Ye,

Li,

Zhou

et al. 2024

MFC

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Maximizing a k-submodular function has received widespread attention in recent years. For k-submodular maximization (k-SMM) with psystem constraints, we propose two algorithms. We first propose a greedy algorithm. Through this algorithm, we can obtain an approximation result of 1/(1 + p) if the function is monotone, and a 1/(2 + p)-approximation ratio in the non-monotone case with O(n 2 (IO + kEO)) time complexity. The second algorithm is a threshold descent algorithm. We obtain a 1 (1+ϵ)(1+p)+ϵ 2approximation ratio in the monotone case and a 1 (1+ϵ)(2+p)+ϵ 2 -approximation ratio in the non-monotone case under O( n(IO+kEO) ϵ log n ϵ ) time complexity. Although the approximation ratio is slightly reduced, the time complexity is also greatly reduced. 2020 Mathematics Subject Classification. 90C27.

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