An Approximation Algorithm for the Two-Node-Connected Star Problem with Steiner Nodes

Q2 Mathematics
Graciela Ferreira, Franco Robledo, Pablo Romero
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引用次数: 0

Abstract

The goal in topological network design is to build a minimum-cost topology meeting specific real-life constraints. There is a cost-robustness trade-off under single and multiple failures.

Previous works in the field suggest that a backbone composed by a two-node-connected toplogy provides savings with respect to elementary cycles. Consequently, we introduce the Two-Node Connected Star Problem with Steiner Nodes (2NCSP-SN). The goal is to design a minimum-cost topology, where the backbone is two-node connected, the access network is connected in a star topology or by direct links to the backbone, and optional nodes (called Steiner nodes) could be included in the solution. The 2NCSP-SN belongs to the class of NP-Hard problems. This promotes the development of heuristics and approximation algorithms.

An approximation algorithm of factor 4α for the 2NCSP-SN is introduced, being α1/2 the cost-ratio between backbone and access links. This is a generalization of the well-known factor 2 for the design of minimum-cost two-connected spanning networks (if we fix α=1/2). Finally, an exact Integer Linear Programming (ILP) formulation is proposed in order to highlight the effectiveness of the approximation algorithm. The results confirm a small gap between the globally optimum solution and the topology offered by our approximation algorithm when the ratio α is close to 1/2.

具有Steiner节点的两节点连通星型问题的近似算法
拓扑网络设计的目标是构建满足特定现实约束的最小成本拓扑。在单个和多个故障下存在成本-鲁棒性权衡。该领域以前的工作表明,由两个节点连接的拓扑组成的主干提供了相对于基本循环的节省。因此,我们引入了具有斯坦纳节点的两节点连通星型问题(2NCSP-SN)。目标是设计一个最小成本的拓扑,其中主干网是双节点连接,接入网以星形拓扑连接或通过直接链接连接到主干网,并且可选节点(称为Steiner节点)可以包含在解决方案中。2NCSP-SN属于NP-Hard问题。这促进了启发式和近似算法的发展。介绍了2NCSP-SN的因子4α近似算法,即α≥1/2主干网与接入链路之间的成本比。这是众所周知的最小成本双连接跨越网络设计因子2的推广(如果我们固定α=1/2)。最后,为了突出逼近算法的有效性,提出了一个精确整数线性规划(ILP)公式。结果表明,当比值α接近1/2时,全局最优解与近似算法给出的拓扑之间存在很小的差距。
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来源期刊
Electronic Notes in Discrete Mathematics
Electronic Notes in Discrete Mathematics Mathematics-Discrete Mathematics and Combinatorics
CiteScore
1.30
自引率
0.00%
发文量
0
期刊介绍: Electronic Notes in Discrete Mathematics is a venue for the rapid electronic publication of the proceedings of conferences, of lecture notes, monographs and other similar material for which quick publication is appropriate. Organizers of conferences whose proceedings appear in Electronic Notes in Discrete Mathematics, and authors of other material appearing as a volume in the series are allowed to make hard copies of the relevant volume for limited distribution. For example, conference proceedings may be distributed to participants at the meeting, and lecture notes can be distributed to those taking a course based on the material in the volume.
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