Günter Rote and Gerhard J. Woeginger:

Time complexity and linear-time approximation of the ancient two machine flow shop

Journal of Scheduling 1 (1998), 149–155. doi:10.1002/(SICI)1099-1425(1998100)1:3<149::AID-JOS10>3.0.CO;2-4

Abstract

We consider the scheduling problems F2|.|Cmax and F2|no-wait|Cmax, i.e. makespan minimization in a two machine flow shop, with and without no wait in process. For both problems solution algorithms based on sorting with O(n log n) running time are known, where n denotes the number of jobs [Johnson 1954, Gilmore and Gomory 1964].

We prove that no asymptotically faster algorithms can solve these problems. This is done by establishing Ω(n log n) lower bounds in the algebraic computation tree model of computation. Moreover, we develop for every ε>0 approximation algorithms with linear running time O(n log(1/ε)) that deliver feasible schedules whose makespan is at most 1+ε times the optimum makespan.

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