Abstract
Given a graph \(G = (V, E)\), we seek for a collection of vertex disjoint paths each of order at most 3 that together cover all the vertices of V. The problem is called 3-path partition, and it has close relationships to the well-known path cover problem and the set cover problem. The general k-path partition problem for a constant \(k \ge 3\) is NP-hard, and it admits a trivial k-approximation. When \(k = 3\), the previous best approximation ratio is 1.5 due to Monnot and Toulouse (Oper Res Lett 35:677–684, 2007), based on two maximum matchings. In this paper we first show how to compute in polynomial time a 3-path partition with the least 1-paths, and then apply a greedy approach to merge three 2-paths into two 3-paths whenever possible. Through an amortized analysis, we prove that the proposed algorithm is a 13 / 9-approximation. We also show that the performance ratio 13 / 9 is tight for our algorithm.
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Notes
An alternating path starts with a singleton and its edges are alternately outside and inside \(E^*\).
Such a path is called an augmenting path, that is, an alternating path that ends on an endpoint of a 2-path in \(\mathcal{Q}\).
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Acknowledgements
The authors would like to thank the anonymous reviewers for their many suggestions and comments that help improve the paper presentation. YC and AZ were supported by the NSFC Grants 11771114 and 11571252; YC was also supported by the China Scholarship Council Grant 201508330054. RG, GL and YX were supported by the NSERC Canada.
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Chen, Y., Goebel, R., Lin, G. et al. An improved approximation algorithm for the minimum 3-path partition problem. J Comb Optim 38, 150–164 (2019). https://doi.org/10.1007/s10878-018-00372-z
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DOI: https://doi.org/10.1007/s10878-018-00372-z