Scheduling jobs on identical and uniform processors revisited

• Authors: Klaus Jansen;Christina Robenek
• Venue: WAOA'11 Proceedings of the 9th international conference on Approximation and Online Algorithms
• Year: 2011

We study the problem of scheduling jobs on uniform processors with the objective to minimize the makespan. In scheduling theory this problem is known as Q||C max . We present an EPTAS for scheduling on uniform machines avoiding the use of an MILP or ILP solver. Instead of solving (M)ILPs we solve the LP-relaxation and use structural information about the "closest" ILP solution. For a given LP-solution x we consider the distance to the closest ILP solution y in the infinity norm, i.e. ||x−y||∞. We call this distance max -gap(Aδ, where Aδ is the constraint matrix of the considered (I)LP. For identical machines and δ=Θ(ε) the matrix Aδ has integral entries in {0,…,(1+δ)/δ} and O(1/δlog(1/δ)) rows representing job sizes and 2O(1/δ log2(1/δ)) columns representing configurations of jobs, so that the column sums are bounded by (1+δ)/δ. The running-time of our algorithm is 2O(1/ε log(1/ε)\log(C(Aδ)) + O(n log n) where C(Aδ) denotes an upper bound for max -gap(Aδ. Furthermore, we can generalize the algorithm for uniform machines and obtain a running-time of 2O(1/ε log(1/ε)\log(C(Ãδ)) + poly(n), where Ãδ is the constraint matrix for a sub-problem considered in this case. In both cases we show that C(Aδ), C(Ãδ) ≤ 2O(1/ε2 log2(1/ε)). Consequently, our algorithm has running-time at most 2O(1/ε3 log3(1/ε)) + O(n log n) for identical machines and 2O(1/ε2 log3(1/ε)) + poly(n) for uniform machines, the same as in [11]. But, to our best knowledge, no instance is known to take on the value 2O(1/ε log2(1/ε)) for max -gap (Aδ) or max -gap(Ãδ). If C(Ãδ), C(Aδ) ≤ poly(1/ε), the running-time of the algorithm would be 2O(1/ε log2(1/ε)) + poly(n) and thus improve the result in [11].