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

Author

Listed:
  • Yong Chen

    (Hangzhou Dianzi University)

  • Yinhui Cai

    (Hangzhou Dianzi University)

  • Longcheng Liu

    (Xiamen University)

  • Guangting Chen

    (Taizhou University)

  • Randy Goebel

    (University of Alberta)

  • Guohui Lin

    (University of Alberta)

  • Bing Su

    (Xi’an Technological University)

  • An Zhang

    (Hangzhou Dianzi University)

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.

Suggested Citation

  • Yong Chen & Yinhui Cai & Longcheng Liu & Guangting Chen & Randy Goebel & Guohui Lin & Bing Su & An Zhang, 2022. "Path cover with minimum nontrivial paths and its application in two-machine flow-shop scheduling with a conflict graph," Journal of Combinatorial Optimization, Springer, vol. 43(3), pages 571-588, April.
    • Handle: RePEc:spr:jcomop:v:43:y:2022:i:3:d:10.1007_s10878-021-00793-3
    DOI: 10.1007/s10878-021-00793-3
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    References listed on IDEAS

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