Minimizing average and weighted-average completion time

Minimizing the average completion time (P||${\displaystyle \sum C_{i}}$) 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||${\displaystyle \sum C_{i}}$.

• There can be many SPT schedules; finding the SPT schedule with the smallest finish time (also called OMFT – optimal mean finish time) is NP-hard.

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.^[2]

Sahni^[2] presents an exponential-time algorithm and a polynomial-time approximation scheme for solving both these NP-hard problems on identical machines:

• Optimal average-completion-time;
• Weighted-average-completion-time.

Minimizing the maximum completion time (makespan)

Minimizing the maximum completion time (P||${\displaystyle C_{\max }}$) is NP-hard even for identical machines, by reduction from the partition problem. Many exact and approximation algorithms are known.

Graham proved that:

• Any list scheduling algorithm (an algorithm that processes the jobs in an arbitrary fixed order, and schedules each job to the first available machine) is a ${\displaystyle 2-1/m}$ approximation for identical machines.^[3] The bound is tight for any m. This algorithm runs in time O(n).
• The specific list-scheduling algorithm called Longest Processing Time First (LPT), which sorts the jobs by descending length, is a ${\displaystyle 4/3-1/3m}$ approximation for identical machines.^[4]: sec.5 It is also called greedy number partitioning.

Coffman, Garey and Johnson presented a different algorithm called multifit algorithm, using techniques from bin packing, which has an approximation factor of 13/11≈1.182.

Huang and Lu^[5] presented a simple polynomial-time algorithm that attains an 11/9≈1.222 approximation in time O(m log m + n), through the more general problem of maximin-share allocation of chores.

Sahni^[2] presented a PTAS that attains (1+ε)OPT in time ${\displaystyle O(n\cdot (n^{2}/\epsilon )^{m-1})}$. It is an FPTAS if m is fixed. For m=2, the run-time improves to ${\displaystyle O(n^{2}/\epsilon )}$. The algorithm uses a technique called interval partitioning.

Hochbaum and Shmoys^[6] presented several approximation algorithms for any number of identical machines (even when the number of machines is not fixed):

• For any r >0, an algorithm with approximation ratio at most (6/5+2^−r ) in time ${\displaystyle O(n(r+\log {n}))}$.
• For any r >0, an algorithm with approximation ratio at most (7/6+2^−r ) in time ${\displaystyle O(n(rm^{4}+\log {n}))}$.
• For any ε>0, an algorithm with approximation ratio at most (1+ε) in time ${\displaystyle O((n/\varepsilon )^{(1/\varepsilon ^{2})})}$ . This is a PTAS. Note that, when the number of machines is a part of the input, the problem is strongly NP-hard, so no FPTAS is possible.

Leung^[7] improved the run-time of this algorithm to ${\displaystyle O\left((n/\varepsilon )^{(1/\varepsilon )\log {(1/\varepsilon )}}\right)}$.

Maximizing the minimum completion time

Maximizing the minimum completion time (P||${\displaystyle C_{\min }}$) is applicable when the "jobs" are actually spare parts that are required to keep the machines running, and they have different life-times. The goal is to keep machines running for as long as possible.^[8] The LPT algorithm attains at least ${\displaystyle {\frac {3m-1}{4m-2}}}$ of the optimum.

Woeginger^[9] presented a PTAS that attains an approximation factor of ${\displaystyle 1-{\varepsilon }}$ in time ${\displaystyle O(c_{\varepsilon }n\log {k})}$, where ${\displaystyle c_{\varepsilon }}$ a huge constant that is exponential in the required approximation factor ε. The algorithm uses Lenstra's algorithm for integer linear programming.

General objective functions

Alon, Azar, Woeginger and Yadid^[10] consider a more general objective function. Given a positive real function f, which depends only on the completion times C_i, they consider the objectives of minimizing ${\displaystyle \sum _{i=1}^{m}f(C_{i})}$, minimizing ${\displaystyle \max _{i=1}^{m}f(C_{i})}$, maximizing ${\displaystyle \sum _{i=1}^{m}f(C_{i})}$, and maximizing ${\displaystyle \min _{i=1}^{m}f(C_{i})}$. They prove that, if f is non-negative, convex, and satisfies a strong continuity assumption that they call "F*", then both minimization problems have a PTAS. Similarly, if f is non-negative, concave, and satisfies F*, then both maximization problems have a PTAS. In both cases, the run-time of the PTAS is O(n), but with constants that are exponential in 1/ε.