An EPTAS for scheduling jobs on uniform processors using an MILP relaxation with a constant number of integral variables

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/\epsilon)^{O(1/\epsilon^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 $\epsilon$ of the optimum, and has running time 2^{O(1/\epsilon^2 \log(1/\epsilon)^3)} + poly(n).

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