Approximation algorithm for partial positive influence problem in social network

Ran, Yingli; Zhang, Zhao; Du, Hongwei; Zhu, Yuqing

doi:10.1007/s10878-016-0005-0

Cited by 25 publications

(6 citation statements)

References 18 publications

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“…However, there is a problem of which sets should be chosen in each iteration. As indicated by previous studies in paper [23], a natural generalization of the classic greedy algorithm cannot yield good results. One reason might be that the number of elements fully covered by a sub-collection of sets is not submodular.…”

Section: Our Contribution and Techniquesmentioning

confidence: 89%

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Approximation algorithm for the partial set multi-cover problem

et al. 2019

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Partial set cover problem and set multi-cover problem are two generalizations of set cover problem. In this paper, we consider the partial set multi-cover problem which is a combination of them: given an element set E, a collection of sets S ⊆ 2 E , a total covering ratio q which is a constant between 0 and 1, each set S ∈ S is associated with a cost c S , each element e ∈ E is associated with a covering requirement r e , the goal is to find a minimum cost sub-collection S ′ ⊆ S to fully cover at least q|E| elements, where element e is fully covered if it belongs to at least r e sets of S ′ . Denote by r max = max{r e : e ∈ E} the maximum covering requirement. We present an (O( rmax log 2 n ε ), 1 − ε)-bicriteria approximation algorithm, that is, the output of our algorithm has cost at most O( rmax log 2 n ε ) times of the optimal value while the number of fully covered elements is at least (1 − ε)q|E|.

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Section: Our Contribution and Techniquesmentioning

confidence: 89%

“…However, combining partial set cover with set multi-cover has enormously increased the difficulty of studies. Ran et al [23] were the first to study approximation algorithms for PSMC, using greedy strategy and dual-fitting analysis. However, their ratio is meaningful only when the covering percentage q is very close to 1.…”

Section: Related Workmentioning

confidence: 99%

Approximation algorithm for the partial set multi-cover problem

et al. 2019

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“…Given a hypergraph G = (V, E) and an integer 1 ≤ k ≤ n where n is the number of vertices, each vertex v ∈ V has a covering requirement r v , the goal of MinPSMC is to select the minimum number of hyperedges to fully cover at least k vertices, where a vertex v is fully covered if it belongs to at least r v selected hyperedges. Ran et al were the first to study the MinPSMC problem [17]. It was shown that MinpU is a special case of the MinPSMC problem, and the MinPSMC problem is at least as hard as the DkS problem [16].…”

Section: Related Workmentioning

confidence: 99%

“…For a point p covered by O * , among the squares in O * containing p, denote by s t the square with the largest (x(s) − x(g s ))-value, (17) where g s is the grid point contained in s. We say that vertex u covers point p if p is contained in some square of S u . Note that when going along Q, every time we meet a…”

Section: The Dynamic Programmingmentioning

confidence: 99%

Approximation algorithm for minimum partial multi-cover under a geometric setting

et al. 2021

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In a minimum p union problem (MinpU), given a hypergraph G = (V, E) and an integer p, the goal is to find a set of p hyperedges E ′ ⊆ E such that the number of vertices covered by E ′ (that is | e∈E ′ e|) is minimized. It was known that MinpU is at least as hard as the densest k-subgraph problem. A question is: how about the problem in some geometric settings? In this paper, we consider the unit square MinpU problem (MinpU-US) in which V is a set of points on the plane, and each hyperedge of E consists of a set of points in a unit square. A ( 11+ε , 4)-bicriteria approximation algorithm is presented, that is, the algorithm finds at least p 1+ε unit squares covering at most 4opt points, where opt is the optimal value for the MinpU-US instance (the minimum number of points that can be covered by p unit squares).

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“…However, combining partial cover with multi-cover seems to enormously increase the difficulty of studies. Ran et al [19] were the first to study approximation algorithm for the minimum partial multi-cover problem (PMC). Using greedy strategy and a delicate dual fitting analysis, they gave a γH ∆ -approximation algorithm, where γ = 1/(1 − (1 − q)η), η = ∆ cmax c min rmax r min , and c max , c min are the maximum and the minimum cost of set, r max , r min are the maximum and the minimum covering requirement of element, respectively.…”

Section: Related Workmentioning

confidence: 99%

A Bicriteria Approximation Algorithm for Minimum Submodular Cost Partial Multi-Cover Problem

Shi

Zhang

2018

Algorithmic Aspects in Information and Management

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This paper studies randomized approximation algorithm for a variant of the set cover problem called minimum submodular cost partial multi-cover (SCPMC).In a partial set cover problem, the goal is to find a minimum cost sub-collection of sets covering at least a required fraction of elements. In a multi-cover problem, each element e has a covering requirement r e , and the goal is to find a minimum cost sub-collection of sets S ′ which fully covers all elements, where an element e is fully covered by S ′ if e belongs to at least r e sets of S ′ . In a minimum submodular cost set cover problem (SCSC), the cost function on sub-collection of sets is submodular and the goal is to find a set cover with the minimum cost.The SCPMC problem studied in this paper is a combination of the above three problems, in which the cost function on sub-collection of sets is submodular and the goal is to find a minimum cost sub-collection of sets which fully covers at least q-percentage of all elements. Previous work shows that such a combination enormously increases the difficulty of studies, even when the cost function is linear.In this paper, assuming that the maximum covering requirement r max = max e r e is a constant and the cost function is nonnegative, monotone nondecreasing, and submodular, we give the first randomized bicriteria algorithm for SCPMC the output of which fully covers at least (q−ε)-percentage of all elements and the performance ratio is O(b/ε) with a high probability, where b = max e f re and f is the maximum number of sets containing a common element. The algorithm is based on a novel non-linear program. Furthermore, in the case when the * Corresponding Author: Zhao Zhang, hxhzz@sina.com. covering requirement r ≡ 1, a bicriteria O(f /ε)-approximation can be achieved even when monotonicity requirement is dropped off from the cost function.

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Approximation algorithm for partial positive influence problem in social network

Cited by 25 publications

References 18 publications

Approximation algorithm for the partial set multi-cover problem

Approximation algorithm for the partial set multi-cover problem

Approximation algorithm for minimum partial multi-cover under a geometric setting

A Bicriteria Approximation Algorithm for Minimum Submodular Cost Partial Multi-Cover Problem

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