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


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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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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Correspondence to Guohui Lin or Bing Su.

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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).

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  • Path cover
  • Flow-shop scheduling
  • Conflict graph
  • b-matching
  • Approximation algorithm