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Abstract:
Q||C max denotes the problem of scheduling n jobs on m machines of different
speeds such that the makespan is minimized. In the paper two special cases of
Q||C max are considered: Case I, when m–1 machine speeds are equal, and there
is only one faster machine; and Case II, when machine speeds are all powers of
2. Case I has been widely studied in the literature, while Case II is
significant in an approach to design so called monotone algorithms for the
scheduling problem.
We deal with the worst case approximation ratio of the classic list scheduling
algorithm ’Longest Processing Time (LPT)’. We provide an analysis of this ratio
Lpt/Opt for both special cases: For one fast machine, a tight bound of is
given. When machine speeds are powers of 2 (2-divisible machines), we show that
in the worst case 41/30 <Lpt/Opt<42/30=1.4.
To our knowledge, the best previous lower bound for both problems was 4/3–ε,
whereas the best known upper bounds were 3/2–1/2m for Case I [6] resp. 3/2 for
Case II [10]. For both the lower and the upper bound, the analysis of Case II
is a refined version of that of Case I.
