Tighter Bounds for Makespan Minimization on Unrelated Machines

We consider the problem of scheduling n jobs to minimize the makespan on m unrelated machines, where job j requires time pij if processed on machine i. A classic algorithm of Lenstra et al. [5] yields the best known approximation ratio of 2 for the problem. Improving this bound has been a prominent open problem for over two decades. In this paper we obtain a tighter bound for a wide subclass of instances which can be identified efficiently. Specifically, we define the feasibility factor of a given instance as the minimum fraction of machines on which each job can be processed. We show that there is a polynomial-time algorithm that, given values L and T , and an instance having a sufficiently large feasibility factor h ∈ (0, 1], either proves that no schedule of mean machine completion time L and makespan T exists, or else finds a schedule of makespan at most T + L/h which is smaller than 2T for a wide class of instances. For the restricted version of the problem, where for each job j and machine i, pij ∈ {pj ,∞}, we show that a simpler algorithm yields a better bound, thus improving for highly feasible instances the best known ratio of 33/17 + ǫ, for any fixed ǫ > 0, due to Svensson [8].