  One of the classic results in scheduling theory is the 2-approximation
algorithm by Lenstra, Shmoys, and Tardos for the problem of scheduling jobs to
minimize makespan on unrelated machines, i.e., job j requires time p_{ij} if
processed on machine i. More than two decades after its introduction it is
still the algorithm of choice even in the restricted model where processing
times are of the form p_{ij} in {p_j, \infty}. This problem, also known as the
restricted assignment problem, is NP-hard to approximate within a factor less
than 1.5 which is also the best known lower bound for the general version.
  Our main result is a polynomial time algorithm that estimates the optimal
makespan of the restricted assignment problem within a factor 33/17 + \epsilon
\approx 1.9412 + \epsilon, where \epsilon > 0 is an arbitrarily small constant.
The result is obtained by upper bounding the integrality gap of a certain
strong linear program, known as configuration LP, that was previously
successfully used for the related Santa Claus problem. Similar to the strongest
analysis for that problem our proof is based on a local search algorithm that
will eventually find a schedule of the mentioned approximation guarantee, but
is not known to converge in polynomial time.
