We consider the machine covering problem for selfish related
machines. For a constant number of machines, m, we show a
monotone polynomial time approximation scheme (PTAS) with running
time that is linear in the number of jobs. It uses a new
technique for reducing the number of jobs while remaining close
to the optimal solution. We also present an FPTAS for the classical
machine covering problem (the previous best result was a PTAS)
and use this to give a monotone FPTAS.
Additionally, we give a monotone approximation algorithm with
approximation ratio min(m,(2+\eps)s_1/s_m) where \eps>0 can
be chosen arbitrarily small and s_i is the (real) speed of
machine i. Finally we give improved results for two machines.
Our paper presents the first
results for this problem in the context of selfish machines.