A Grid-Based Approximation Algorithm for the Minimum Weight Triangulation Problem

AbstractConsider a vertex-weighted graph G with a source s and a target t. Tracking Paths requires finding a minimum weight set of vertices (trackers) such that the sequence of trackers in each path from s to t is unique. In this work, we derive a factor 66-approximation algorithm for Tracking Paths in weighted graphs and a factor 4-approximation algorithm if the input is unweighted. This is the first constant factor approximation for this problem. While doing so, we also study approximation of the closely related r-Fault Tolerant Feedback Vertex Set problem. There, for a fixed integer r and a given vertex-weighted graph G, the task is to find a minimum weight set of vertices intersecting every cycle of G in at least $$r+1$$ r + 1 vertices. We give a factor $$\mathcal {O}(r^2)$$ O ( r 2 ) approximation algorithm for r-Fault Tolerant Feedback Vertex Set if r is a constant.

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Minimum Weight Triangulation

Encyclopedia of Algorithms ◽

10.1007/978-0-387-30162-4_241 ◽

2008 ◽

pp. 546-548

Author(s):

Christos Levcopoulos

Keyword(s):

Minimum Weight ◽

Minimum Weight Triangulation

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Minimum weight triangulation is NP-hard

Proceedings of the twenty-second annual symposium on Computational geometry - SCG '06 ◽

10.1145/1137856.1137859 ◽

2006 ◽

Cited By ~ 2

Author(s):

Unknown

Keyword(s):

Minimum Weight ◽

Np Hard ◽

Minimum Weight Triangulation

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A BETTER APPROXIMATION FOR MINIMUM AVERAGE ROUTING PATH CLUSTERING PROBLEM IN 2-D UNDERWATER SENSOR NETWORKS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830909000142 ◽

2009 ◽

Vol 01 (02) ◽

pp. 175-191 ◽

Cited By ~ 6

Author(s):

WEI WANG ◽

DONGHYUN KIM ◽

JAMES WILLSON ◽

BHAVANI THURAISINGHAM ◽

WEILI WU

Keyword(s):

Sensor Networks ◽

Approximation Algorithm ◽

Dominating Set ◽

Minimum Weight ◽

Performance Ratio ◽

Underwater Sensor Networks ◽

Clustering Problem ◽

Np Completeness ◽

Dominating Set Problem ◽

Special Case

Previously, we proposed Minimum Average Routing Path Clustering Problem (MARPCP) in multi-hop USNs. The goal of this problem is to find a clustering of a USN so that the average clustering-based routing path from a node to it nearest underwater sink is minimized. We relaxed MARPCP to a special case of Minimum Weight Dominating Set Problem (MWDSP), namely MWDSP-R. In addition, we showed the Performance Ratio (PR) of α-approximation algorithm for MWDSP-R is 3α for MARPCP. Based on this result, we showed the existence of a (15 + ∊)-approximation algorithm for MARPCP. In this paper, we first establish the NP-completeness of both MARPCP and MWDSP-R. Then, we propose a PTAS for MWDSP-R. By combining this result with our previous one, we have a (3 + ∊)-approximation algorithm for MARPCP.

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