Semi-online scheduling on two parallel processors with combined partial information

The classical formulation of the multiprocessor scheduling problem requires to assign a set of tasks to m processors so that the makespan is minimized. This problem is well known to be NP - hard. The online version of the multiprocessor scheduling problem has been studied widely and many results are available. The LS algorithm is optimal for two processors with competitive ratio 3/2. Various semi-online multiprocessor problems on two processors have been studied. The cases where the sum of the tasks (T) is known, or the largest size of the tasks (P_max) is known, or the optimal offline value of the instance (C*) is known have been studied. For each of those problems optimal algorithm with competitive ratio 4/3 has been proposed, which improved the 3/2 of the online version. In this paper, we study the semi-online version where we know both T and P_max in advance. And there also exists an optimal algorithm with competitive ratio 6/5. But the optimal algorithm is not good, because it was designed only focus on the worst cases. In this research, we propose algorithms with better competitive ratio in terms of the parameter r = 2p_max/T. For different values of r, we also show lower bounds. Both the lower bounds and competitive ratios of the algorithms are given as functions of parameter r. The algorithms are optimal in some cases.