In a classical scheduling problem, we are given a set of n jobs of unit
length along with precedence constraints and the goal is to find a schedule of
these jobs on m identical machines that minimizes the makespan. This problem
is well-known to be NP-hard for an unbounded number of machines. Using standard
3-field notation, it is known as P∣prec,pj=1∣Cmax.
We present an algorithm for this problem that runs in O(1.995n) time.
Before our work, even for m=3 machines the best known algorithms ran in
O∗(2n) time. In contrast, our algorithm works when the number of
machines m is unbounded. A crucial ingredient of our approach is an algorithm
with a runtime that is only single-exponential in the vertex cover of the
comparability graph of the precedence constraint graph. This heavily relies on
insights from a classical result by Dolev and Warmuth (Journal of Algorithms
1984) for precedence graphs without long chains.Comment: 26 pages, 7 figure