An EPTAS for Scheduling Jobs on Uniform Processors: Using an MILP Relaxation with a Constant Number of Integral Variables

Abstract

In this paper, we present an efficient polynomial time approximation scheme (EPTAS) for scheduling on uniform processors, i.e. finding a minimum length schedule for a set of n independent jobs on m processors with different speeds (a fundamental NP-hard scheduling problem). The previous best polynomial time approximation scheme (PTAS) by Hochbaum and Shmoys has a running time of (n/) O(1/ 2). Our algorithm, based on a new mixed integer linear programming (MILP) formulation with a constant number of integral variables and an interesting rounding method, finds a schedule whose length is within a relative error of the optimum, and has running time 2 O(1/ 2 log(1/) 3) poly(n).

An EPTAS for Machine Scheduling with Bag-Constraints

“…We can observe that this origin l function for every priority bag B l is injective, as every large job was assigned to exactly one unique machine. Further we can conclude that for a large job j ∈ B l and machine i = origin l (j) that in our current solution i cannot hold a small or a medium job from B l , as this would contradict either constraint (5) of the MILP (for a small job) or the definition of patterns (for a medium job). Machine i may however still hold either j or another large job, that was moved there after j was moved away.…”

“…The constraint (4) will ensure that the average area that is scheduled on top of a pattern does not exceed the optimal height T . To respect conflicts among priority bags we added constraint (5). Constraint (5) first will not allow to pack any small job of a bag B l on top of a pattern p that already holds large or medium jobs of B l .…”

“…Finally assume now that constraint (5) failed and let p be the pattern and B l be the bag of the failed constraint. As in the previous case we can follow that x p = 0.…”

“…For the classical makespan minimization problem it is known that the problem is strongly NP-hard and there are known PTAS's [4,10] and furthermore EPTAS's [5,6]. One known generalization of the machine scheduling problem is to change the model of machines.…”

“…The proof follows from lemma 8. We remark that with constraint (5) we ensured that at most x p jobs could be assigned to each pattern. Therefore we get that:…”