Minimum Makespan Scheduling with Low Rank Processing Times

We investigate approximation algorithms for the classical minimum makespan scheduling problem, focusing on instances where the rank of the matrix describing the processing times of the jobs is bounded. A bounded rank matrix arises naturally when the processing time of a job on machine depends upon a bounded set of resources. A bounded rank matrix also shows up when jobs have varying degrees of parallelizability, and the machines have multiple cores. We are interested in studying the tractability of the problem as a function of the (positive) rank of the processingtime matrix. At one extreme is the case of unit rank, also known as related machines, which admits a PTAS [7], and at the other extreme is the full rank case (unrelated machines), which is NP-hard to approximate within a factor better than 3/2 [8]. Our main technical contribution is in showing that the approximability of the problem is not smooth with the rank of the matrix. From the inapproximability side, we show that the problem becomes APX-hard, even for rank four matrices. For rank seven matrices, we prove that it is hard to approximate to a factor 3/2, matching the inapproximability result for general unrelated machines. From the algorithmic side, we obtain a quasi-polynomial approximation scheme (i.e., a (1 + ε) approximation in time n) for the rank two case. This implies that the problem is not APX-hard in this case, unless NP has quasi-polynomial algorithms. Our algorithm is a subtle dynamic program which runs in polynomial time in some interesting special cases. The classification of the three dimensional problem remains open.