The 2-valued case of makespan minimization with assignment constraints

We consider the following special case of minimizing makespan. A set of jobs J and a set of machines M are given. Each job $j\in J$ can be scheduled on a machine from a subset $M_j$ of M. The processing time of j is the same on all machines in $M_j$ . The jobs are of two sizes, namely b (big) and s (small). We present a polynomial‐time algorithm that approximates the value of the optimal makespan within a factor of 1.883 and some further improvements when every job can be scheduled on at most two machines.