Minimizing Total Completion Time on Parallel Machines with Deadline Constraints

• Authors: Joseph Y;.-T. Leung;Michael Pinedo
• Venue: SIAM Journal on Computing
• Year: 2003

Consider n independent jobs and m identical machines in parallel. Job j has a processing time pj and a deadline $\bar{d}_j$. It must complete its processing before or at its deadline. All jobs are available for processing at time t=0 and preemptions are allowed. A set of jobs is said to be feasible if there exists a schedule that meets all the deadlines; such a schedule is called a feasible schedule. Given a feasible set of jobs, our goal is to find a schedule that minimizes the total completion time $\sum C_j$. In the classical $\alpha \mid \beta \mid \gamma$ scheduling notation this problem is referred to as $P \mid prmt, \bar{d}_j \mid \sum C_j$. Lawler (Recent Results in the Theory of Machine Scheduling, in Mathematical Programming: The State of the Art, A. Bachem, M. Grötschel, and B. Korte, eds., Springer, Berlin, 1982, pp. 202--234) raised the question of whether or not the problem is NP-hard. In this paper we present a polynomial-time algorithm for every $m \ge 2$, and we show that the more general problem with m unrelated machines, i.e., $R \mid prmt, \bar{d}_j \mid \sum C_j$, is strongly NP-hard.