Two-machine routing open shop on a tree: instance reduction and efficiently solvable subclass
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The routing open shop problem is a natural combination of the metric traveling salesman problem and the classical open shop scheduling problem. Both counterparts generally are NP-hard, being however polynomially solvable in special cases considered: TSP is trivial on a tree, and open shop is solvable in linear time in case of two machines by a well-known Gonzalez-Sahni algorithm (while being NP-hard for three and more machines). Surprisingly, the combination of those problem becomes NP-hard even in a simplest case of a two-machine routing open shop on a link. We describe an instance reduction procedure for the two-machine problem on arbitrary tree. The procedure preserves the standard lower bound on the makespan and allows to describe wide polynomially solvable subclasses of the problem. Conditions, describing these classes, allow building an optimal schedule in linear time. For any instance from those classes optimal makespan coincides with the standard lower bound.
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