Prize-Collecting Survivable Network Design in Node-Weighted Graphs

Chekuri, Chandra; Ene, Alina; Vakilian, Ali

doi:10.1007/978-3-642-32512-0_9

Cited by 16 publications

(20 citation statements)

References 17 publications

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“…The first constrained says that if y e = 1 then at least r e sets containing e must be selected and thus e is fully covered. Relaxing (5) and by a scaling, we have the following linear program:…”

Section: S: E∈smentioning

confidence: 99%

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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: S: E∈smentioning

confidence: 99%

“…It was shown by Chekuri et.al in paper [5] that the integrality gap for linear program (12) is O(r max log n). Combining this with Claim 2 and Claim 3, the output S ′ of Algorithm 2 has cost c(S ′ ) = 2 i 0 +1 O(r max log n)opt M DSC .…”

Section: Theoretical Analysismentioning

confidence: 99%

Approximation algorithm for the partial set multi-cover problem

et al. 2019

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“…Our algorithm repeats growing several dual variables simultaneously in an LP relaxation. This approach has been used in the augmentation problem with node weights [5,12], but its approximation factor depends on the number of nodes. This is because the approximation factor is decided by the number of dual variables that are grown simultaneously in a single constraint.…”

Section: Algorithm For the Augmentation Problemmentioning

confidence: 99%

“…Hence, when T = k −1 i=0 S i , for each demand cut X, Γ(X) \ T includes at least k − k + 1 nodes in S * . This implies that x * /(k − k + 1) is a feasible solution to (5) when the connectivity requirement is k , and thus w(S k ) = O(k 2 ) · w(S * )/(k − k + 1). The weight of the (k, m)-CDS output by our algorithm is at most k k =0 w(S k ) = O(k 2 ) · w(S * ) k k =0 1 k−k +1 = O(k 2 log k) · w(S * ).…”

Section: Combined Decomposition and Covering Algorithmmentioning

confidence: 99%

Approximation Algorithms for Highly Connected Multi-dominating Sets in Unit Disk Graphs

Fukunaga

2017

Algorithmica

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Given an undirected graph on a node set V and positive integers k and m, a k-connected m-dominating set ((k, m)-CDS) is defined as a subset S of V such that each node in V \ S has at least m neighbors in S, and a k-connected subgraph is induced by S. The weighted (k, m)-CDS problem is to find a minimum weight (k, m)-CDS in a given node-weighted graph. The problem is called the unweighted (k, m)-CDS problem if the objective is to minimize the cardinality of a (k, m)-CDS. These problems have been actively studied for unit disk graphs, motivated by the application of constructing a virtual backbone in a wireless ad hoc network. In this paper, we consider the case in which k ≤ m, and we present a simple O(k5 k )-approximation algorithm for the unweighted (k, m)-CDS problem, and a primal-dual O(k 2 log k)-approximation algorithm for the weighted (k, m)-CDS problem.A CDS does not give a fault-tolerant virtual backbone network. This is because a CDS is only required to be connected, and each node outside a CDS is required to have only one neighbor in the CDS. Hence, if a backbone node fails, the virtual backbone network may be disconnected, or a nonbackbone node may lose access to the virtual backbone network. To overcome this disadvantage, Dai and Wu [10] proposed replacing a CDS by a k-connected k-dominating set, and they addressed the problem of finding a minimum k-connected k-dominating set in a unit disk graph. For a graph with the node set V , a subset S of V is called k-connected if the subgraph induced by S is kconnected (i.e., it is connected even if any k − 1 nodes are removed), and is called k-dominating if each node v ∈ V \ S has k neighbors in S. Triggered by their study, much attention has been paid to this problem, extending the notion of a k-connected k-dominating set to a more-general k-connected m-dominating set ((k, m)-CDS).The problem of finding a minimum cardinality (k, m)-CDS in a unit disk graph is called the unweighted (k, m)-CDS problem. If each node is given a nonnegative weight, and the objective is to minimize the weight of a (k, m)-CDS, then this is called the weighted (k, m)-CDS problem. As for the unweighted (k, m)-CDS problem, several constant-approximation algorithms were given for k ≤ 3 [21,22,26,27,30]. As for the weighted (k, m)-CDS problem, there are several constantapproximation algorithms for k = m = 1 [2, 31], but no approximation algorithm was known for the case of (k, m) = (1, 1) before our study (see Section 2 for more literature reviews).After these previous studies, a natural question arises as to whether there is a constantapproximation algorithm for the unweighted (k, m)-CDS problem with k ≥ 4, and for the weighted problem with (k, m) = (1, 1). For the unweighted problem, this question has been already addressed in both [26] and [27].

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“…The ST problem is a special case of SF where all demands share an endpoint. Very recently, Chekuri et al [2] give an algorithm with an approximation ratio of O(log n) w.r.t. to the fractional solution for SF and higher connectivity problems.…”

Section: Contributions and Techniquesmentioning

confidence: 99%

Improved Approximation Algorithms for (Budgeted) Node-Weighted Steiner Problems

Bateni

Hajiaghayi

Liaghat

2013

Automata, Languages, and Programming

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Abstract. Moss and Rabani [12] study constrained node-weighted Steiner tree problems with two independent weight values associated with each node, namely, cost and prize (or penalty). They give an O(log n)-approximation algorithm for the prize-collecting node-weighted Steiner tree problem (PCST)-where the goal is to minimize the cost of a tree plus the penalty of vertices not covered by the tree. They use the algorithm for PCST to obtain a bicriteria (2, O(log n))-approximation algorithm for the Budgeted node-weighted Steiner tree problemwhere the goal is to maximize the prize of a tree with a given budget for its cost. Their solution may cost up to twice the budget, but collects a factor Ω( 1 log n ) of the optimal prize. We improve these results from at least two aspects. Our first main result is a primal-dual O(log h)-approximation algorithm for a more general problem, prize-collecting node-weighted Steiner forest (PCSF), where we have h demands each requesting the connectivity of a pair of vertices. Our approximation guarantee is tight within constant factors. Our algorithm as well as its analysis is much simpler than that of Moss and Rabani [12] for PCST. Incidentally, Könemann et al. [11] have shown very recently that the prize-collecting algorithm in [12] is flawed. Our algorithm can be seen as a greedy algorithm which reduces the number of demands by choosing a structure with minimum cost-to-reduction ratio. We believe this natural style of argument (also used in [10,8]) may lead to progress on other node-weighted settings. Our second main contribution is for the Budgeted node-weighted Steiner tree problem, which is also an improvement to Moss and Rabani [12] and Guha et al. [8]. In the unrooted case, we improve upon an O(log 2 n)-approximation of [8], and present an O(log n)-approximation algorithm without any budget violation. For the rooted case, where a specified vertex has to appear in the solution tree, we improve the bicriteria result of [12] to a bicriteria approximation ratio of (1 + , O(log n)/ 2 ) for any positive (possibly subconstant) . That is, for any permissible budget violation 1 + , we present an algorithm achieving a tradeoff in the guarantee for prize. Indeed, we show that this is almost tight for the natural linear-programming relaxation used by us as well as in [12].

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Prize-Collecting Survivable Network Design in Node-Weighted Graphs

Cited by 16 publications

References 17 publications

Approximation algorithm for the partial set multi-cover problem

Approximation algorithm for the partial set multi-cover problem

Approximation Algorithms for Highly Connected Multi-dominating Sets in Unit Disk Graphs

Improved Approximation Algorithms for (Budgeted) Node-Weighted Steiner Problems

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