Homework 1

1. We proved in class that the LPT algorithm for P||C_max is a 4/3-approximation, assuming that the last job to start is the same as the last job to finish. Show that this assumption is not necessary, i.e. prove that LPT is always a 4/3-approximation algorithm for P||C_max
2. Show that the analysis of list scheduling for P||C_max is tight, by giving a class of inputs for which list scheduling produces a schedule whose length is (2-1/m)C_max. (The analysis of list scheduling is easily extended from 2 to 2-1/m).
3. Prove that if we have a r-relaxed decision procedure for a problem X, and we know that the possible objective function values for problem X are integers from a polynomial-sized range, then there exists a r-approximation algorithm for problem X.
4. Prove that there exists an optimal schedule for P2|prec,p_j=1|C_max that is HLF, that is, the jobs are processed highest level first.
5. Analyze list scheduling for P|r_j,prec|C_max. Give the best approximation ratio that you can.
6. Prove that LRPT-FM is optimal for Q|pmtn|C_max