Local ratio method on partial set multi-cover

Ran, Yingli; Shi, Yishuo; Zhang, Zhao

doi:10.1007/s10878-016-0066-0

Cited by 22 publications

(3 citation statements)

References 24 publications

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“…This is called a partial set multi-cover problem and is less studied than the previous one. Ran et al address the difficulty of the problem [14], while recent works [15], [16] provide heuristics with bounded error that still require solving an integer linear problem at each step, which is computationally impossible in our context.…”

Section: B Set Covering Algorithmsmentioning

confidence: 99%

On Cost-Effective, Reliable Coverage for LoS Communications in Urban Areas

Gemmi

Cigno

Maccari

2022

IEEE Trans. Netw. Serv. Manage.

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The use of ultra high frequencies in 5G and future networks to improve transmission speeds and capacity requires that users' equipment remain in Line of Sight with the access antennas most of the service time. This requirement implies a change in perspective to plan the coverage: Antennas cannot be placed on roofs or remote antenna sites, and a robust coverage is based on multi-antenna visibility from any point. This paper tackles the problem of public street coverage in urban areas with a data-driven methodology. Starting from 3D digital maps, we formalize the problem of antenna placement as a set coverage problem and leverage powerful heuristics to implement a general algorithm that allows the exploration of different policies, returning the detailed coverage, the antenna placement, and the cost of the coverage. Results on 15 areas in 3 Italian cities show the properties of different policies and confirm for the first time on large scale real data the feasibility of Line of Sight communications with a sustainable number of antennas per km 2 .

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Section: B Set Covering Algorithmsmentioning

confidence: 99%

On Cost-Effective, Reliable Coverage for LoS Communications in Urban Areas

Gemmi

Cigno

Maccari

2022

IEEE Trans. Netw. Serv. Manage.

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“…However, their ratio is meaningful only when the covering percentage q is very close to 1. In paper [24], the authors presented a simple greedy algorithm achieving performance ratio ∆. They also presented a local ratio algorithm, which reveals a what they called "shock wave" phenomenon: their performance ratio is f for both PSC and SMC , however, when q is smaller than 1 by a very small constant, the ratio jumps abruptly to O(n).…”

Section: Related Workmentioning

confidence: 99%

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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“…This ratio is meaningful only when the covering percentage q is very close to 1. In [20], Ran et al presented a simple greedy algorithm achieving performance ratio ∆. Recall that in terms of ∆, greedy algorithm for Set Cover achieves performance ratio ln ∆.…”

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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Local ratio method on partial set multi-cover

Cited by 22 publications

References 24 publications

On Cost-Effective, Reliable Coverage for LoS Communications in Urban Areas

On Cost-Effective, Reliable Coverage for LoS Communications in Urban Areas

Approximation algorithm for the partial set multi-cover problem

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

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