Abstract:
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We show how to approximate in NC the problem of Scheduling Unrelated
Parallel Machines, for a fixed number of machines. We develop a
(2+epsilon)-approximate parallel algorithm for the problem. Our
approach shows how to relate the linear program obtained by relaxing
the integer programming formulation of the problem with a linear program
formulation that is positive and in the packing/covering form. The
relationship established enables us to transfer approximate fractional
solutions from the later formulation that is known to be approximable
in NC. Then, we show how to obtain an integer approximate
solution, i.e. a schedule, from the fractional one, using
the randomized rounding
technique. Finally, we show that the same technique can be applied to
the General Assignment Problem of fixed number of machines and
a given makespan T, thus yielding a schedule whose cost
is at most (2+epsilon) times the minimum cost and has makespan at
most 2T. |