  Modern platforms are using accelerators in conjunction with standard
processing units in order to reduce the running time of specific operations,
such as matrix operations, and improve their performance. Scheduling on such
hybrid platforms is a challenging problem since the algorithms used for the
case of homogeneous resources do not adapt well. In this paper we consider the
problem of scheduling a set of tasks subject to precedence constraints on
hybrid platforms, composed of two types of processing units. We propose a
$(3+2\sqrt{2})$-approximation algorithm and a conditional lower bound of 3 on
the approximation ratio. These results improve upon the 6-approximation
algorithm proposed by Kedad-Sidhoum et al. as well as the lower bound of 2 due
to Svensson for identical machines. Our algorithm is inspired by the former one
and distinguishes the allocation and the scheduling phases. However, we propose
a different allocation procedure which, although is less efficient for the
allocation sub-problem, leads to an improved approximation ratio for the whole
scheduling problem. This approximation ratio actually decreases when the number
of processing units of each type is close and matches the conditional lower
bound when they are equal.
