Scheduling Jobs on Identical and Uniform Processors Revisited

Abstract

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 $\max\mbox{-gap}(A_\delta)$, where A _δ is the constraint matrix of the considered (I)LP. For identical machines and δ = Θ(ε) the matrix A _δ has integral entries in {0,…,(1 + δ)/δ} and O(1/δlog(1/δ)) rows representing job sizes and $2^{O(1/\delta\log^2(1/\delta))}$ columns representing configurations of jobs, so that the column sums are bounded by (1 + δ)/δ. The running-time of our algorithm is $2^{O(1/\varepsilon \log(1/\varepsilon )\log(C(A_\delta))}+O(n\log n)$ where C(A _δ ) denotes an upper bound for $\max\mbox{-gap}(A_\delta)$. Furthermore, we can generalize the algorithm for uniform machines and obtain a running-time of $2^{O(1/\varepsilon \log(1/\varepsilon )\log(C(\tilde{A}_\delta))}+poly(n)$, where $\tilde{A}_\delta$ is the constraint matrix for a sub-problem considered in this case. In both cases we show that $C(A_\delta),C(\tilde{A}_\delta)\le 2^{O(1/\varepsilon \log^2(1/\varepsilon ))}$. Consequently, our algorithm has running-time at most $2^{O(1/\varepsilon ^2 \log^3(1/\varepsilon ))}+O(n\log n)$ for identical machines and $2^{O(1/\varepsilon ^2 \log^3(1/\varepsilon ))}+poly(n)$ for uniform machines, the same as in [11]. But, to our best knowledge, no instance is known to take on the value $2^{O(1/\varepsilon \log^2(1/\varepsilon ))}$ for $\max\mbox{-gap}(A_\delta)$ or $\max\mbox{-gap}(\tilde{A}_\delta)$. If $C(\tilde{A}_\delta),C(A_\delta)\le poly(1/\varepsilon )$, the running-time of the algorithm would be $2^{O(1/\varepsilon \log^2(1/\varepsilon ))}+poly(n)$ and thus improve the result in [11].

Research supported by German Research Foundation (DFG) project JA 612/14-1, “Design and analysis of efficient polynomial approximation schemes for scheduling and related optimization problems”.