Maximization of k-Submodular Function with a Matroid Constraint

Sun, Yan; Liu, Yuezhu; Li, Min

doi:10.1007/978-3-031-20350-3_1

Cited by 11 publications

(3 citation statements)

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“…For monotone k-submodular maximization under a total size constraint (i.e., at most a given number of items can be selected), Ohsaka and Yoshida [9] proposed a 1 2 -approximation algorithm, and for the problem under individual size constraints (i.e., in each dimension at most a given number of items can be selected), they gave a 1 3 -approximation algorithm. Under a matroid constraint, Sakaue [13] shown that fully greedy algorithm is 1 2 -approximation for the monotone case, and Sun et al [15] gave a 1 3 -approximation algorithm for the non-monotone case.…”

Section: Related Workmentioning

confidence: 99%

“…The problem of maximizing a k-submodular function is known to be NP-hard, since it is a generalization of the NP-hard submodular maximization problem. Extensive research has been dedicated to developing efficient algorithms with desirable approximation ratios for the problem under different constraints, for example, cardinality constraints [2,9,18], matroid constraints [13,15], knapsack constraints [17,12], and the unconstrained setting [19,6,14].…”

Section: Introductionmentioning

confidence: 99%

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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: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Improved Analysis of Greedy Algorithm on k-Submodular Knapsack

Tang,

Wang

2023

Frontiers in Artificial Intelligence and Applications

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“…Regarding monotone k-SMM under matroid constraints, Sakaue [20] obtained an approximation result of 1/2 through the greedy algorithm. And in the non-monotone case, Sun et al [22] proposed an algorithm that ensures a 1/3-approximation. Niu et al [18] obtained approximation ratios of (1/2 − ϵ) and (1/3 − ϵ) using a threshold-decreasing algorithm for monotone and non-monotone k-SMM, respectively.…”

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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