Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines - Furmańczyk, H.; Kubale, M. (Bulletin of the Polish Academy of Sciences. Technical sciences, 2017)
In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with speeds s ≥ s ≥ s ≥ s , respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incompatibility graph of degree Δ, where two incompatible jobs cannot be processed on the same machine. We show that the general problem is NP-hard even if s = s = s . If, however, Δ ≤ 4 and s ≥ 12s , s = s = s , then the problem can be solved to optimality in time O(n ). The same algorithm returns a solution of value at most 2 times optimal provided that s ≥ 2s . Finally, we study the case s ≥ s ≥ s = s and give a 32/15-approximation algorithm running also in O(n ) time.