Covering Problems in Edge- and Node-Weighted Graphs

Fukunaga, Takuro

doi:10.1007/978-3-319-08404-6_19

Cited by 4 publications

(4 citation statements)

References 34 publications

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“…Let us mention that the idea on formulating our LP relaxation is potentially useful for other covering problems. The author pointed out in his recent work [6] that a natural LP relaxation has a large integrality gap for many covering problems in node-weighted graphs. He also presented several tight approximation algorithms using the LP relaxations designed based on the idea we propose in the present paper.…”

Section: Lp Relaxationmentioning

confidence: 99%

“…Note: In [6], the author applied a similar idea of lifting LP relaxations for solving several covering problems in edge-and node-weighted graphs. He defined a new LP relaxation by replacing edge variables by variables corresponding to pairs of edges and constraints, and showed that the new LP relaxation has better integrality gap than the original one.…”

Section: Lp Relaxation For Prize-collecting Augmentation Problemmentioning

confidence: 99%

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Spider covers for prize-collecting network activation problem

Fukunaga

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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In the network activation problem, each edge in a graph is associated with an activation function that decides whether the edge is activated from weights assigned to its end nodes. The feasible solutions of the problem are node weights such that the activated edges form graphs of required connectivity, and the objective is to find a feasible solution minimizing its total weight. In this paper, we consider a prize-collecting version of the network activation problem and present the first nontrivial approximation algorithms. Our algorithms are based on a new linear programming relaxation of the problem. They round optimal solutions for the relaxation by repeatedly computing node weights activating subgraphs, called spiders. For the problem with element-and nodeconnectivity requirements, we also present a new potential function on uncrossable biset families and use it to analyze our algorithms. v∈V w(v), denoted by w(V ). We assume throughout the paper that G is undirected even though the problem

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Section: Lp Relaxationmentioning

confidence: 99%

Section: Lp Relaxation For Prize-collecting Augmentation Problemmentioning

confidence: 99%

Spider covers for prize-collecting network activation problem

Fukunaga

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In 2008, PAREKH [11] presented an approximate algorithm to solve the prize-collecting edge dominating set problem (PCEDS) problem, algorithm approximate ratio is 8 / 3. In 2010, KAMIYAMA [12] proposed polynomial time algorithm for solving PCEDS problem on edge-weighted tree graphs, and time complexity is 2 ( ) O V ; in 2016, FUKUNAGA [13] promoted the conclusion of KAMIYAMA, in the problem of PCEDS, considering the simultaneous weighting of vertexes and edges on the tree graph, a polynomial time algorithm with time complexity (log ) O V was proposed.…”

Section: Introductionmentioning

confidence: 99%

Algorithm research for prize-collecting dominating set problem on special graph

Gao

Chen²

2022

6th International Workshop on Advanced Algorithms and Control Engineering (IWAACE 2022)

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The prize-collecting dominating set (PCDS) problem is the generalization of the minimum dominating set (MDS) problem. The MDS problem requires a dominating set in a given graph, namely a subset of vertexes. Any vertex in the graph belongs to the closed neighborhood of the subset of vertexes, where the subset of vertexes with the smallest cardinality is the MDS. In the PCDS problem: given an undirected graph, its vertex set V has nonnegative weighting function  and nonnegative penalty function  . Finding a vertex subset D V  , for vertexes do not belong to the closed neighborhood of D , we need to pay penalties for them. The objective of this issue is to minimize the sum of weights and penalties. As we all know that the MDS problem is NP-hard in general graph, obviously, PCDS problem is NP-hard, too., In certain cases, the PCDS problem can achieve better results than the MDS problem, which has a certain value in practical applications such as facility location and logistics management. Whether the problem is limited to a special graph will get useful structural characterization and corresponding algorithm results is considered, designed polynomial time exact algorithm on the star and according to the r -LMP approximation algorithm, designed the 2-LMP approximation algorithm on the path and cycle.

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“…Most previous investigations of the interaction topology have focused on edge-weighted graphs, and node weights have been largely neglected. In [31], an algorithmic technique was proposed to deal with the graph covering problem in nodeweighted graphs.…”

Section: Introductionmentioning

confidence: 99%

Weighted Consensus Problem for Multiagent Systems with Edge‐ and Node‐Weighted Directed Graphs

et al. 2019

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In this paper, the concept of consensus is generalized to weighted consensus, by which the conventional consensus, the bipartite consensus, and the cluster consensus problems can be unified in the proposed weighted consensus frame. The dynamics of agents are modeled by the general linear time-invariant systems. The interaction topology is modeled by edge- and node-weighted directed graphs. Under both state feedback and output feedback control strategies, the weighted consensus problems are transformed into the equivalent conventional consensus problems. Then, some distributed state feedback and output feedback protocols are proposed to solve the weighted consensus problems. For output feedback case, a unified frame to construct the state-observer-based weighted consensus protocols is proposed, and different design approaches are discussed. As special cases, some related results for bipartite consensus and cluster consensus can be obtained directly. Finally, a simple example is given to illustrate the effectiveness of our proposed approaches.

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Covering Problems in Edge- and Node-Weighted Graphs

Cited by 4 publications

References 34 publications

Spider covers for prize-collecting network activation problem

Spider covers for prize-collecting network activation problem

Algorithm research for prize-collecting dominating set problem on special graph

Weighted Consensus Problem for Multiagent Systems with Edge‐ and Node‐Weighted Directed Graphs

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