Scheduling with Contact Restrictions – A Problem Arising in Pandemics
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We study a scheduling problem arising in pandemic times where jobs should keep sufficient distance during transit times to machines. To model this, each job has a transit time before and after its processing time, i.e., three parameters. We seek conflict-free schedules in which the transit times of no two jobs on machines in close proximity intersect. More formally, for the problem "Scheduling with Contact Restrictions" (SCR), we are given a set of jobs, a set of machines, and a conflict graph on the machines. The goal is to find a conflict-free schedule of minimum makespan. We show that, unless P=NP, the problem does not allow for a constant factor approximation even for identical jobs and every choice of fixed positive parameters, including the unit case. However, given an oracle for a maximum collection of disjoint independent sets , we provide a 2-approximation algorithm for identical jobs. Moreover, we present optimal and near optimal schedules for unit jobs on several graph classes. For bipartite graphs, we give a 4/3-approximation and an asymptotic 7/6-approximation. Furthermore, we solve the problem on complete graphs, complete bipartite graphs, and traceable bipartite graphs in polynomial time. For trees, we compute schedules with highest load density yielding near optimal solutions.
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