Scheduling of unit-length jobs with cubic incompatibility graphs on three uniform machines

In the paper we consider the problem of scheduling n identical jobs on 3 uniform machines with speeds s(1),s(2), and s(3) to minimize the schedule.length. We assume that jobs are subjected to some kind of mutual exclusion constraints, modeled by a cubic incompatibility graph. We show that if the graph is 2-chromatic then. the problem can be solved in 0(n(2)) time. If the graph is 3-chromatic, the problem becomes NP-hard even if s1 > s(2) = s(3). However, in this case there exists a 10/7-approximation algorithm running in 0(n(3)) time. Moreover, this algorithm solves the problem almost surely to optimality if 3(s1)/4 <= s(2) = s(3). (C) 2016 Elsevier B.V. All rights reserved.