  Makespan minimization on unrelated machines is a classic problem in
approximation algorithms. No polynomial time $(2-\delta)$-approximation
algorithm is known for the problem for constant $\delta> 0$. This is true even
for certain special cases, most notably the restricted assignment problem where
each job has the same load on any machine but can be assigned to one from a
specified subset. Recently in a breakthrough result, Svensson [Svensson, 2011]
proved that the integrality gap of a certain configuration LP relaxation is
upper bounded by $1.95$ for the restricted assignment problem; however, the
rounding algorithm is not known to run in polynomial time.
  In this paper we consider the $(1,\varepsilon)$-restricted assignment problem
where each job is either heavy ($p_j = 1$) or light ($p_j = \varepsilon$), for
some parameter $\varepsilon > 0$. Our main result is a $(2-\delta)$-approximate
polynomial time algorithm for the $(1,\epsilon)$-restricted assignment problem
for a fixed constant $\delta> 0$. Even for this special case, the best
polynomial-time approximation factor known so far is 2. We obtain this result
by rounding the configuration LP relaxation for this problem. A simple
reduction from vertex cover shows that this special case remains NP-hard to
approximate to within a factor better than 7/6.
