Approximation algorithm for multiprocessor parallel job scheduling

P-k \fix\ C-max problem is a new scheduling problem based on the multiprocessor parallel job, and it is proved to be NP-hard problem when k greater than or equal to 3. This paper focuses on the case of k = 3. Some new observations and new techniques for P-3 \fix\ C-max problem are offered. The concept of semi-normal schedulings is introduced, and a very simple linear time algorithm Semi-normal Algorithm for constructing semi-normal schedulings is developed. With the method of the classical Graham List Scheduling, a thorough analysis of the optimal scheduling on a special instance is provided, which shows that the algorithm is an approximation algorithm of ratio of 9/8 for any instance of P-3 \fix\ C-max problem, and improves the previous best ratio of 7/6 by M. X. Goemans.