  The problem of scheduling jobs on parallel machines (identical, uniform, or
unrelated), under incompatibility relation modeled as a block graph, under the
makespan optimality criterion, is considered in this paper. No two jobs that
are in the relation (equivalently in the same block) may be scheduled on the
same machine in this model.
  The presented model stems from a well-established line of research combining
scheduling theory with methods relevant to graph coloring. Recently, cluster
graphs and their extensions like block graphs were given additional attention.
We complement hardness results provided by other researchers for block graphs
by providing approximation algorithms. In particular, we provide a
$2$-approximation algorithm for $P|G = block\ graph|C_{max}$ and a PTAS for the
case when the jobs are unit time in addition. In the case of uniform machines,
we analyze two cases. The first one is when the number of blocks is bounded,
i.e. $Q|G = k-block\ graph|C_{max}$. For this case, we provide a PTAS,
improving upon results presented by D. Page and R. Solis-Oba. The improvement
is two-fold: we allow richer graph structure, and we allow the number of
machine speeds to be part of the input. Due to strong NP-hardness of $Q|G =
2-clique\ graph|C_{max}$, the result establishes the approximation status of
$Q|G = k-block\ graph|C_{max}$. The PTAS might be of independent interest
because the problem is tightly related to the NUMERICAL k-DIMENSIONAL MATCHING
WITH TARGET SUMS problem. The second case that we analyze is when the number of
blocks is arbitrary, but the number of cut-vertices is bounded and jobs are of
unit time. In this case, we present an exact algorithm. In addition, we present
an FPTAS for graphs with bounded treewidth and a bounded number of unrelated
machines.
