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

Authors Lin Chen, Dániel Marx, Deshi Ye, Guochuan Zhang

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Lin Chen
Dániel Marx
Deshi Ye
Guochuan Zhang

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Lin Chen, Dániel Marx, Deshi Ye, and Guochuan Zhang. Parameterized and Approximation Results for Scheduling with a Low Rank Processing Time Matrix. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.STACS.2017.22

Abstract

We study approximation and parameterized algorithms for R||C_max, focusing on the problem when the rank of the matrix formed by job processing times is small. Bhaskara et al. initiated the study of approximation algorithms with respect to the rank, showing that R||C_max 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||C_max is APX-hard even if the rank is 3, resolving an open problem. We then show that R||C_max is FPT parameterized by the rank and the largest job processing time p_max. This generalizes the parameterized results on P||C_max and R||C_max with few different types of machines. 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.

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Keywords
  • APX-hardness
  • Parameterized algorithm
  • Scheduling
  • Exponential Time Hypothesis

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