Abstract
Given a graph \(G = (V, E)\), the 3-path partition problem is to find a minimum collection of vertex-disjoint paths each of order at most 3 to cover all the vertices of V. It is different from but closely related to the well-known 3-set cover problem. The best known approximation algorithm for the 3-path partition problem was proposed recently and has a ratio 13/9. Here we present a local search algorithm and show, by an amortized analysis, that it is a 4/3-approximation. This ratio matches up to the best approximation ratio for the 3-set cover problem.
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References
Chen, Y., et al.: A local search 4/3-approximation algorithm for the minimum \(3\)-path partition problem. arXiv:1812.09353 (2018)
Chen, Y., Goebel, R., Lin, G., Su, B., Xu, Y., Zhang, A.: An improved approximation algorithm for the minimum \(3\)-path partition problem. J. Comb. Optim. (2018). https://doi.org/10.1007/s10878-018-00372-z
Duh, R., Fürer, M.: Approximation of \(k\)-set cover by semi-local optimization. In: Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing (STOC 1997), pp. 256–264 (1997)
Feige, U.: A threshold of for approximating set cover. J. ACM 45, 634–652 (1998)
Franzblau, D.S., Raychaudhuri, A.: Optimal hamiltonian completions and path covers for trees, and a reduction to maximum flow. ANZIAM J. 44, 193–204 (2002)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman and Company, San Francisco (1979)
Goldberg, A.V., Karzanov, A.V.: Maximum skew-symmetric flows and matchings. Math. Program. 100, 537–568 (2004)
Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations, pp. 85–103. Springer, Boston (1972). https://doi.org/10.1007/978-1-4684-2001-2_9
Levin, A.: Approximating the unweighted k-set cover problem: greedy meets local search. In: Erlebach, T., Kaklamanis, C. (eds.) WAOA 2006. LNCS, vol. 4368, pp. 290–301. Springer, Heidelberg (2007). https://doi.org/10.1007/11970125_23
Monnot, J., Toulouse, S.: The path partition problem and related problems in bipartite graphs. Oper. Res. Lett. 35, 677–684 (2007)
Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and sub-constant error-probability PCP characterization of NP. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC 1997), pp. 475–484 (1997)
Vazirani, V.: Approximation Algorithms. Springer, Heidelberg (2001). https://doi.org/10.1007/978-3-662-04565-7
Yan, J.-H., Chang, G.J., Hedetniemi, S.M., Hedetniemi, S.T.: \(k\)-path partitions in trees. Discrete Appl. Math. 78, 227–233 (1997)
Acknowledgement
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. LL was supported by the China Scholarship Council Grant No. 201706315073, and the Fundamental Research Funds for the Central Universities Grant No. 20720160035. WT was supported in part by funds from the College of Engineering and Computing at the Georgia Southern University.
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Chen, Y. et al. (2019). A Local Search 4/3-approximation Algorithm for the Minimum 3-path Partition Problem. In: Chen, Y., Deng, X., Lu, M. (eds) Frontiers in Algorithmics. FAW 2019. Lecture Notes in Computer Science(), vol 11458. Springer, Cham. https://doi.org/10.1007/978-3-030-18126-0_2
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