Scheduling with Machine Conflicts

We study the scheduling problem of makespan minimization with machine conflicts that arise in various settings, e.g., shared resources for pre- and post-processing of tasks or spatial restrictions. In this context, each job has a blocking time before and after its processing time, i.e., three parameters. Given a set of jobs, a set of machines, and a graph representing machine conflicts, the problem SchedulingWithMachineConflicts (smc), asks for a conflict-free schedule of minimum makespan in which the blocking times of no two jobs intersect on conflicting machines. We show that, unless P = NP , smc on m machines does not allow for a O ( m 1 - ε ) -approximation algorithm for any ε > 0 , even in the case of identical jobs and every choice of fixed positive parameters, including the unit case. Complementary, we provide approximation algorithms when a suitable collection of independent sets is given. Finally, we present polynomial time algorithms to solve the problem for the case of unit jobs smc-Unit on special graph classes. As our main result, we solve smc-Unit for bipartite graphs by using structural insights for conflict graphs of star forests. As the set of active machines at each point in time induces a bipartite graph, the insights yield a local optimality criterion.