Consider the following preemptive variant of the Load Balancing problem: There are m identical machines, and n jobs with processing times t1, t2, ..... tn respectively. Each machine can process at most one job at a time. Each job can run on more than one machines but must run on at most one machine at any time. Let T = max{max1<=j<=n tj ,1/ m Σ 1<=j<=n tj } : (a) Prove that the optimal makespan is at least T. (b) Give a linear-time algorithm which produces a preemptive schedule with makespan exactly T.

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Polynomial Time

The problems for which the deterministic polynomial-time algorithm exists belong to the complexity class (p) and it is central in the field of computational complexity theory. According to Cobham’s thesis, polynomial time is a synonym for “feasible”, “fast”, and “efficient”.