  We consider the online makespan minimization problem on identical machines.
Chen and Vestjens (ORL 1997) show that the largest processing time first (LPT)
algorithm is 1.5-competitive. For the special case of two machines, Noga and
Seiden (TCS 2001) introduce the SLEEPY algorithm that achieves a competitive
ratio of $(5 - \sqrt{5})/2 \approx 1.382$, matching the lower bound by Chen and
Vestjens (ORL 1997). Furthermore, Noga and Seiden note that in many
applications one can kill a job and restart it later, and they leave an open
problem whether algorithms with restart can obtain better competitive ratios.
  We resolve this long-standing open problem on the positive end. Our algorithm
has a natural rule for killing a processing job: a newly-arrived job replaces
the smallest processing job if 1) the new job is larger than other pending
jobs, 2) the new job is much larger than the processing one, and 3) the
processed portion is small relative to the size of the new job. With
appropriate choice of parameters, we show that our algorithm improves the 1.5
competitive ratio for the general case, and the 1.382 competitive ratio for the
two-machine case.
