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

Yong Chen (), Yinhui Cai (), Longcheng Liu (), Guangting Chen (), Randy Goebel (), Guohui Lin (), Bing Su () and An Zhang ()
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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

Journal of Combinatorial Optimization, 2022, vol. 43, issue 3, No 4, 588 pages

Abstract: 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.

Keywords: Path cover; Flow-shop scheduling; Conflict graph; b-matching; Approximation algorithm (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10878-021-00793-3

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