The Doctor Rostering Problem - Asser Fahrenholz

Chapter 4. Solving the DRP 16
it in a short amount of time (Hopcroft et al. [13]). If the DRP problem is NP-complete
and an algorithm does exist, able to solve the problem deterministically in polynomial
time, then it would follow that all NP-complete problems are solvable deterministically
in polynomial time and so P = NP , a statement that has been investigated for the last
50 years without any definite results 1 . As such, it is widely believed that P = NP , since
no polynomial time algorithms has been devised to solve an NP-complete problem.
To prove that a problem is NP-complete, one can transform an already known NP-
complete problem (in polynomial time) to the problem for which a proof is sought.
Fortunately, a problem, very close to the DRP, has already been proven NP-complete. It
is stated in Garey and Johnson [10] as [SS19], referring to Even et al. [9] who transformed
the 3SAT (Karp [15]) problem, which again was transformed from the SAT problem
(Cook [6]). The chain of transformations is shown in figure 4.1.
Figure 4.1: The transformation chain from SAT to Timetable Design
Garey and Johnson [10] defines the instance as: A set H of ”work periods”, a set C
of ”craftsmen”, a set T of ”tasks”, a subset A(c) ⊆ H of ”available hours” for each
craftsman c ∈ C, a subset A(t) ⊆ H of available hours for each task t ∈ T , and, for each
pair (c, t) ∈ C × T , a number R(c, t) ∈ Z + 0
of ”required work periods”.
Applying this to the DRP: The set H defines the work periods in which shifts must lie,
the set C is the set of doctors and the set T is the shifts. A(c) defines when a doctor
is available, which in this project is defined by all the hard constraints except H4. A(t)
describes when a shift can be performed and R(c,t) defines how long a shift t takes
for doctor c to perform. In this project R(c, t) = 1 ∀ c, t. This instance describes a
timetable for completing all shifts in time. The Timetable Design problem can thus be
transformed to the DRP and hence, DRP is NP-complete. This means that no known
algorithm can deterministically find an optimal solution to the DRP in polynomiel time.
4.2 Exact methods
Given a set of constraints, as described in chapter 3, the goal of finding a solution to the
problem can be accomplished by using an optimal solver such as GAMS. As concluded
above, solving the DRP to optimality can take an exponential amount of time. Due to
1 One of the Millennium Problems : Claymath.org/millennium/