Abstract
Scheduling problem with agreement graph on two identical machines is addressed. The problem consists in scheduling a set of non-preemptive jobs on two identical machines in a minimum time. Agreement constraints are imposed over the jobs. They express that only specific jobs can be scheduled concurrently on different machines. These constraints are represented by an agreement graph. This problem can be seen as a problem of partitioning a vertex-weighted graph into cliques. The problem is NP-hard for an arbitrary agreement graph, but for some particular graphs the problem is still open. For this reason, we study the complexity of the problem for some specific graphs. In particular, we prove the NP-hardness of the problem if the agreement graph is a tree. We also propose polynomial time algorithms to solve the problem for the cases of caterpillars and cycles.
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