On the Complexity of Scheduling. Peter Brucker University of Osnabrueck Germany. 1.Scheduling Problems. In a scheduling problem one has to find time slots in which activities (or jobs) should be processed under given constraints. The main constraints are
University of Osnabrueck
In a scheduling problem one has to find time slots
in which activities (or jobs) should be processed
under given constraints. The main constraints are
resource constraints and precedence constraints
between activities. A quite general scheduling
problem is the Resource Constrained Project
Scheduling Problem (RCPSP) which can be
formulated as follows:
The objective is to determine starting times Sj
for all activities j in such a way that
The fact that activities j start at time Sj and
finish at time Sj + pj implies that the activities
j are not preempted. We may relax this
condition by allowing preemption (activity
If preemption is not allowed the vector S = (Sj)
defines a schedule. S is called feasible if all
resource and precedence constraints are
fulfilled. One has to find a feasible schedule
which minimizes the objective function
f( C1, ... , Cn). In project planning f( C1, ... , Cn) is
often replaced by the makespan Cmax which is
the maximum of all Cj - values.
Consider a project with n = 4 activities, r = 2
resources with capacities R1 = 5 and R2 = 7, a
precedence relation 2 3 and the following data:
A corresponding schedule with minimal makespan
Next machine scheduling problems will be
discussed in detail.
Here we will consider
To model the general shop scheduling problem as
RCPSP we consider
Classes of scheduling problems can be specified
in terms of the three-field classification a | b | g
P parallel identical machines
Q uniform machines
R unrelated machines
MPM multipurpose machines
O open-shopMachine Environment
To describe the machine environment the following symbols are used:
The above symbols are used if the number of machines is part of the input. If the number of machines is fixed to m we write Pm, Qm, Rm, MPMm, Jm, Fm, Om.
Two types of objective functions are most
Cmax and Lmaxsymbolizethe bottleneck
objective functions with fj(Cj) = Cj
(makespan)and fj(Cj) = Cj - dj (maximum
Common sum objective functions are:
3.1 Polynomial algorithms
3.2 Classes P and NP
3.3 NP- complete and NP- hard problems
1 | | S wjCj can be solved by scheduling the jobs
in an ordering of non-increasing wj/pj - values.
Complexity: O(n log n)
An algorithm for a scheduling problem with
computational effort O(S pj) is pseudo-polynomial.
Properties of polynomial reductions:
If any single NP- complete decision problem Q
could be solved in polynomial time then we
would have P = NP.
To prove that a decision problem P is NP -
complete it is sufficient to prove the
following two properties:
An optimization problem is NP- hard if its
decision version is NP- complete.
Cook  has shown that the satisfiability
problem from Boolean logic is NP- complete.
Using this result he used reduction to prove
that other combinatorial problems are NP-
complete as well.
In a follow-up paper Karp  derived NP-
completeness results for many other problems.
Complexity results for different classes of
scheduling problems can be found under
These results are based on
In these web-pages for several classes of machine
scheduling problems the following results are
Hereby the following elementary reductions
For single machine problems we have more specific elementary reductions.
Small sized problems can be solved by
To solve problems of larger size one has to
Multiprocessor task scheduling
Scheduling with controllable data
Shop problems with buffers
No-idle time scheduling
Scheduling with no-available constraints6. Other Types of Scheduling Problems
There are other classes of scheduling problems.
Some of them are discussed in this book.
Scheduling problems are also discussed in
connection with other areas:
P. Brucker (2007), Scheduling Algorithms, fifth edition, Springer, Heidelberg
M.R. Garey, D.S. Johnson (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman and Company, San Francisco.