Parameterized and Approximation Results for Scheduling with a Low Rank Processing Time Matrix

We study approximation and parameterized algorithms for R||Cmax, focusing on the problem when the rank of the matrix formed by job processing times is small. Bhaskara et al. [2] initiated the study of approximation algorithms with respect to the rank, showing that R||Cmax admits a QPTAS (Quasi-polynomial time approximation scheme) when the rank is 2, and becomes APX-hard when the rank is 4. We continue this line of research. We prove that R||Cmax is APX-hard even if the rank is 3, resolving an open problem in [2]. We then show that R||Cmax is FPT parameterized by the rank and the largest job processing time pmax. This generalizes the parameterized results on P ||Cmax [17] and R||Cmax with few different types of machines [15]. We also provide nearly tight lower bounds under Exponential Time Hypothesis which suggests that the running time of the FPT algorithm is unlikely to be improved significantly. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.1 Combinatorics