#### New Competitive Semi-online Scheduling Algorithms for Small Number of Identical Machines

##### Debasis Dwibedy, Rakesh Mohanty

Design and analysis of constant competitive deterministic semi-online algorithms for the multi-processor scheduling problem with small number of identical machines have gained significant research interest in the last two decades. In the semi-online scheduling problem for makespan minimization, we are given a sequence of independent jobs one by one in order and upon arrival, each job must be allocated to a machine with prior knowledge of some Extra Piece of Information (EPI) about the future jobs. Researchers have designed multiple variants of semi-online scheduling algorithms with constant competitive ratios by considering one or more EPI. In this paper, we propose four new variants of competitive deterministic semi-online algorithms for smaller number of identical machines by considering two EPI such as Decr and Sum. We obtain improved upper bound and lower bound results on the competitive ratio for our proposed algorithms, which are comparable to the best known results in the literature. In two identical machines setting with known Sum, we show a tight bound of 1.33 on the competitive ratio by considering a sequence of equal size jobs. In the same setting we achieve a lower bound of 1.04 and an upper bound of 1.16 by considering Sum and a sequence of jobs arriving in order of decreasing sizes. For three identical machines setting with known Decr and Sum, we show a lower bound of 1.11 on the competitive ratio. In this setting, we obtain an upper bound of 1.5 for scheduling a sequence of equal size jobs and achieves an upper bound of 1.2 by considering a sequence of decreasing size jobs. Further we develop an improved competitive algorithm with an upper bound of 1.11 on the competitive ratio.

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