A Unified Approach to Approximating Partial Covering Problems

Könemann, Jochen; Parekh, Ojas; Segev, Danny

doi:10.1007/s00453-009-9317-0

Cited by 66 publications

(44 citation statements)

References 42 publications

(43 reference statements)

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“…Extending a folklore reduction from set cover type problems to node-weighted Steiner tree problems, we argue that our algorithm may be interpreted as a nontrivial generalization of the above-outlined algorithm by Könemann et al [13]. First of all, the following reduction shows that the partial covering problem can be encoded as the quota node-weighted Steiner tree problem.…”

Section: Generalizing the Algorithm Of Könemann Et Almentioning

confidence: 76%

“…Let α = α 2 and β = c(T1) OP T . With this notation we obtain in a similar way as Könemann et al [13] c…”

Section: A Proof Of Lemmamentioning

confidence: 91%

“…In the prize-collecting version of the problem every element has a penalty and the objective is to minimize the sum of costs of the chosen sets and the penalties of the elements that are not covered. Könemann et al [13] describe a unified framework for partial cover. They show how to obtain an approximation algorithm for a class I of partial cover instances if there is an r-approximate LMP algorithm for the corresponding prize-collecting version.…”

Section: Related Workmentioning

confidence: 99%

“…Here we draw connections to the recent work on the partial cover problems. Könemann et al [13] showed how to obtain a ( 4 /3 + ε)r-approximation algorithm for the partial cover problems using an r-approximate LMP algorithm for the corresponding prize-collecting version as a black-box. Their approach is roughly as follows.…”

Section: Trying To Beat the Factor Of 4: Relation To The Partial Covermentioning

confidence: 99%

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Approximating Node-Weighted k-MST on Planar Graphs

2020

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We study the problem of finding a minimum weight connected subgraph spanning at least k vertices on planar, node-weighted graphs. We give a (4 + ε)-approximation algorithm for this problem. We achieve this by utilizing the recent LMP primal-dual 3-approximation for the node-weighted prize-collecting Steiner tree problem by Byrka et al (SWAT'16) and adopting an approach by Chudak et al. (Math. Prog. '04) regarding Lagrangian relaxation for the edge-weighted variant. In particular, we improve the procedure of picking additional vertices (tree merging procedure) given by Sadeghian (2013) by taking a constant number of recursive steps and utilizing the limited guessing procedure of Arora and Karakostas (Math. Prog. '06). More generally, our approach readily gives a ( 4 /3 · r + ε)-approximation on any graph class where the algorithm of Byrka et al. for the prizecollecting version gives an r-approximation. We argue that this can be interpreted as a generalization of an analogous result by Könemann et al. (Algorithmica '11) for partial cover problems. Together with a lower bound construction by Mestre (STACS'08) for partial cover this implies that our bound is essentially best possible among algorithms that utilize an LMP algorithm for the Lagrangian relaxation as a black box. In addition to that, we argue by a more involved lower bound construction that even using the LMP algorithm by Byrka et al. in a non-black-box fashion could not beat the factor 4 /3 · r when the tree merging step relies only on the solutions output by the LMP algorithm.

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Section: Generalizing the Algorithm Of Könemann Et Almentioning

confidence: 76%

“…Let α = α 2 and β = c(T1) OP T . With this notation we obtain in a similar way as Könemann et al [13] c…”

Section: A Proof Of Lemmamentioning

confidence: 91%

Section: Related Workmentioning

confidence: 99%

Section: Trying To Beat the Factor Of 4: Relation To The Partial Covermentioning

confidence: 99%

See 2 more Smart Citations

Approximating Node-Weighted k-MST on Planar Graphs

2020

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“…Using local ratio method, he also obtained performance ratio f . Konemann et al [19] presented a Lagrangian relaxation framework and obtained performance ratio ( 4 3 +ε)H(∆) for the generalized partial set cover problem.…”

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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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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A Unified Approach to Approximating Partial Covering Problems

Cited by 66 publications

References 42 publications

Approximating Node-Weighted k-MST on Planar Graphs

Approximating Node-Weighted k-MST on Planar Graphs

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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