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 $O(n^2)$ time. If the graph is 3-chromatic, the problem becomes NP-hard even if $s_1>s_2=s_3$. However, in this case there exists a $4\mathrm{/}3$-approximation algorithm running in $O(n^3)$ time. Moreover, this algorithm solves the problem almost surely to optimality if $3s_1\mathrm{/}4\le s_2=s_3$.