8.1 Makespan Scheduling

Definition 8.1.1 Let P = ∑ i∈J Pi. Clearly 1 ≤ OPT ≤ P . Since 1 ≤ OPT ≤ P , if we have access to a (1 + )-relaxed decision procedure we can use it O(logP ) times to do a binary search on different values of T (between 1 and P ) to find a (1 + )approximation. Since logP is polynomial in the input size, if the relaxed decision procedure runs in polynomial time then we have a (1 + )-approximation for Makespan Scheduling that runs in polynomial time.