Minimizing the average completion time

Minimizing the average completion time can be done in polynomial time:

• The SPT algorithm (Shortest Processing Time First), sorts the jobs by their length, shortest first, and then assigns them to the processor with the earliest end time so far. It runs in time O(n log n), and minimizes the average completion time on identical machines,^[1] P||${\displaystyle \sum C_{i}}$.
• Horowitz and Sahni^[1] present an exact algorithm, with run time O(n log m n), for minimizing the average completion time on uniform machines, Q||${\displaystyle \sum C_{i}}$.
• Bruno, Coffman and Sethi^[2] present an algorithm, running in time ${\displaystyle O(\max(mn^{2},n^{3}))}$, for minimizing the average completion time on unrelated machines, R||${\displaystyle \sum C_{i}}$.

Minimizing the weighted-average completion time

Minimizing the weighted average completion time is NP-hard even on identical machines, by reduction from the knapsack problem.^[1] It is NP-hard even if the number of machines is fixed and at least 2, by reduction from the partition problem.^[3]

Sahni^[3] presents an exponential-time algorithm and a polynomial-time approximation algorithm for identical machines.

Horowitz and Sahni^[1] presented:

• Exact dynamic programming algorithms for minimizing the weighted-average completion time on uniform machines. These algorithms run in exponential time.
• Polynomial-time approximation schemes, which for any ε>0, attain at most (1+ε)OPT. For minimizing the weighted average completion time on two uniform machines, the run-time is ${\displaystyle O(10^{l}n^{2})}$ = ${\displaystyle O(n^{2}/\epsilon )}$, so it is an FPTAS. They claim that their algorithms can be easily extended for any number of uniform machines, but do not analyze the run-time in this case. They do not present an algorithm for weighted-average completion time on unrelated machines.

Minimizing the maximum completion time (makespan)

Minimizing the maximum completion time is NP-hard even for identical machines, by reduction from the partition problem.

A constant-factor approximation is attained by the Longest-processing-time-first algorithm (LPT).

Horowitz and Sahni^[1] presented:

• Exact dynamic programming algorithms for minimizing the maximum completion time on both uniform and unrelated machines. These algorithms run in exponential time (recall that these problems are all NP-hard).
• Polynomial-time approximation schemes, which for any ε>0, attain at most (1+ε)OPT. For minimizing the maximum completion time on two uniform machines, their algorithm runs in time ${\displaystyle O(10^{2l}n)}$, where ${\displaystyle l}$ is the smallest integer for which ${\displaystyle \epsilon \geq 2\cdot 10^{-l}}$. Therefore, the run-time is in ${\displaystyle O(n/\epsilon ^{2})}$, so it is an FPTAS. For minimizing the maximum completion time on two unrelated machines, the run-time is ${\displaystyle O(10^{l}n^{2})}$ = ${\displaystyle O(n^{2}/\epsilon )}$. They claim that their algorithms can be easily extended for any number of uniform machines, but do not analyze the run-time in this case.

Hochbaum and Shmoys^[4] presented several approximation algorithms for any number of identical machines. Later,^[5] they developed a PTAS for uniform machines.

Epstein and Sgall^[6] generalized the PTAS for uniform machines to handle more general objective functions. Let C_i (for i between 1 and m) be the makespan of machine i in a given schedule. Instead of minimizing the objective function max(C_i), one can minimize the objective function max(f(C_i)), where f is any fixed function. Similarly, one can minimize the objective function sum(f(C_i)).

Monotonicity and Truthfulness

In some settings, the machine speed is the machine's private information, and we want to incentivize machines to reveal their true speed, that is, we want a truthful mechanism. An important consideration for attaining truthfulness is monotonicity.^[7] It means that, if a machine reports a higher speed, and all other inputs remain the same, then the total processing time allocated to the machine weakly increases. For this problem:

• Auletta, De Prisco, Penna and Persiano^[8] presented a 4-approximation monotone algorithm, which runs in polytime when the number of machines is fixed.
• Ambrosio and Auletta^[9] proved that the Longest Processing Time algorithm is monotone whenever the machine speeds are powers of some c ≥ 2, but not when c ≤ 1.78. In contrast, List scheduling is not monotone for c > 2.
• Andelman, Azar and Sorani^[10] presented a 5-approximation monotone algorithm, which runs in polytime even when the number of machines is variable.
• Kovacz^[11] presented a 3-approximation monotone algorithm.