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Path cover with minimum nontrivial paths and its application in two-machine flow-shop scheduling with a conflict graph

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Abstract

Path cover is a well-known intractable problem that finds a minimum number of vertex disjoint paths in a given graph to cover all the vertices. We show that a variant, in which the objective is to minimize the number of length-0 paths, is polynomial-time solvable. We further show that another variant, to minimize the total number of length-0 and length-1 paths, is also polynomial-time solvable. Both variants find applications in approximating the two-machine flow-shop scheduling problem in which job processing has constraints that are formulated as a conflict graph. For the unit jobs, we present a 4/3-approximation for the scheduling problem with an arbitrary conflict graph, based on the exact algorithm for the above second variant of the path cover problem. For arbitrary jobs where the conflict graph is the union of two disjoint cliques, we present a simple 3/2-approximation algorithm.

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Acknowledgements

Y. Chen, G. Chen, and A. Zhang are supported by the NSFC Grants 11771114, 11971139 and the Zhejiang Provincial NSFC Grant LY21A010014; Y. Chen and A. Zhang are also supported by the China Scholarship Council Grants 201508330054 and 201908330090, respectively. R. Goebel, G. Lin, and L. Liu are supported by NSERC Canada; L. Liu is also supported by the Fundamental Research Funds for the Central Universities (Grant No. 20720160035) and the China Scholarship Council Grant No. 201706315073.

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Author notes

  1. Yong Chen, Yinhui Cai, Longcheng Liu are co-first authors.

Authors and Affiliations

  1. Department of Mathematics, Hangzhou Dianzi University, Hangzhou, China

    Yong Chen, Yinhui Cai & An Zhang

  2. School of Mathematical Sciences, Xiamen University, Xiamen, China

    Longcheng Liu

  3. Taizhou University, Taizhou, China

    Guangting Chen

  4. Department of Computing Science, University of Alberta, Edmonton, AB, T6G 2E8, Canada

    Randy Goebel & Guohui Lin

  5. School of Economics and Management, Xi’an Technological University, Xi’an, China

    Bing Su

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  1. Yong Chen
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  2. Yinhui Cai
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  4. Guangting Chen
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Correspondence to Guohui Lin or Bing Su.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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An extended abstract appears in Proceedings of COCOON 2018 Cai et al. (2018)

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Chen, Y., Cai, Y., Liu, L. et al. Path cover with minimum nontrivial paths and its application in two-machine flow-shop scheduling with a conflict graph. J Comb Optim 43, 571–588 (2022). https://doi.org/10.1007/s10878-021-00793-3

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