Abstract
Given a digraph \(G = (V, E)\), the k-path partition problem is to find a minimum collection of vertex-disjoint directed paths each of order at most k to cover all the vertices of V. The problem has various applications in facility location, network monitoring, transportation and others. Its special case on undirected graphs has received much attention recently, but the general version is seemingly untouched in the literature. We present the first k/2-approximation algorithm, for any \(k \ge 3\), based on a novel concept of augmenting path to minimize the number of singletons in the partition. When \(k \ge 7\), we present an improved \((k+2)/3\)-approximation algorithm based on the maximum path-cycle cover followed by a careful 2-cycle elimination process. When \(k = 3\), we define the second novel kind of augmenting paths to reduce the number of 2-paths and propose an improved 13/9-approximation algorithm.
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References
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Acknowledgments
This research is supported by the NSFC Grants 11771114 and 11971139 (YC and AZ), the Zhejiang Provincial NSFC Grant LY21A010014 (YC and AZ), the CSC Grants 201508330054 (YC) and 201908330090 (AZ), the Grant-in-Aid for Scientific Research of the Ministry of Education, Science, Sports and Culture of Japan Grant No. 18K11183 (ZZC), and the NSERC Canada (GL).
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Chen, Y., Chen, ZZ., Kennedy, C., Lin, G., Xu, Y., Zhang, A. (2022). Approximation Algorithms for the Directed Path Partition Problems. In: Chen, J., Li, M., Zhang, G. (eds) Frontiers of Algorithmics. IJTCS-FAW 2021. Lecture Notes in Computer Science(), vol 12874. Springer, Cham. https://doi.org/10.1007/978-3-030-97099-4_2
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DOI: https://doi.org/10.1007/978-3-030-97099-4_2
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