Tighter approximation bounds for LPT scheduling in two special cases

Q||CmaxQ||Cmax 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||CmaxQ||Cmax are considered: case I, when m−1m−1 machine speeds are equal, and there is only one faster machine; and case II, when machine speeds are all powers of 2 (2-divisible machines). 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 ‘Largest Processing Time (LPT)’. We provide an analysis of this ratio Lpt/OptLpt/Opt for both special cases: For ‘one fast machine’, a tight bound of (3+1)/2≈1.3660 is given. For 2-divisible machines, we show that in the worst case 1.3673(3+1)/2 when LPT breaks ties arbitrarily.To our knowledge, the best previous lower and upper bounds were (4/3,3/2−1/2m] in case I [T. Gonzalez, O.H. Ibarra, S. Sahni, Bounds for LPT schedules on uniform processors, SIAM Journal on Computing 6 (1) (1977) 155–166], respectively [4/3−1/3m,3/2] in case II [R.L. Graham, Bounds on multiprocessing timing anomalies, SIAM Journal on Applied Mathematics 17 (1969) 416–429; A. Kovács, Fast monotone 3-approximation algorithm for scheduling related machines, in: Proc. 13th Europ. Symp. on Algs. (ESA), in: LNCS, vol. 3669, Springer, 2005, pp. 616–627]. Moreover, Gonzalez et al. conjectured the lower bound 4/3 to be tight in the ‘one fast machine’ case [T. Gonzalez, O.H. Ibarra, S. Sahni, Bounds for LPT schedules on uniform processors, SIAM Journal on Computing 6 (1) (1977) 155–166].