An Efﬁcient Translation Scheme for Representing Nurse Rostering

Problems as Satisﬁability Problems

Stefaan Haspeslagh

, Tommy Messelis

, Greet Vanden Berghe

and Patrick De Causmaecker

1,2

CODeS (member of ITEC-IBBT-KU Leuven), Department of Computer Science, KU Leuven KULAK,

E. Sabbelaan 53, 8500 Kortrijk, Belgium

CODeS, Vakgroep IT, KAHO, Gebr. De Smetstraat 1, 9000 Gent, Belgium

Keywords:

Nurse Rostering, Satisﬁability Problems, Automated Translation, Counting Constraints.

Abstract:

In this paper we present efﬁcient translation schemes for converting nurse rostering problem instances into

satisﬁability problems (SAT). We deﬁne eight generic constraints types allowing the representation of a large

number of nurse rostering constraints commonly found in literature. For each of the generic constraint types,

we present efﬁcient translation schemes to SAT. Special attention is paid to the representation of counting

constraints. We developed a two way translation scheme for counting constraints using O(nlogn) variables

and O(n

) clauses. We translated the instances of the First international nurse rostering competition 2010 to

SAT and proved the infeasibility of the instances. The SAT translation was used for a hardness study of nurse

rostering problem instances based on SAT features.

1 INTRODUCTION

We present schemes for automatically translating

nurse rostering problem instances into satisﬁability

problem instances (SAT).

The SAT-translation scheme was originally de-

signed for a hardness study of nurse rostering prob-

lem instances (Bilgin et al., 2009). Some algorithms

perform well on some speciﬁc instances while those

same algorithms perform worse on others. (Leyton-

Brown et al., 2006) present an experimental approach

for predicting the running time of algorithms designed

to solve the winner determination problem for combi-

natorial auctions. The previous strategy was applied

to the more abstract problem of propositional satisﬁ-

ability (Nudelman et al., 2004) introducing a set of

91 features for the runtime prediction of several algo-

rithms that prove whether a certain problem instance

is satisﬁable or not. This work has led to the construc-

tion of SATzilla, a portfolio solver for SAT problems

(Xu et al., 2008). This portfolio was very success-

ful, winning several tracks of different SAT competi-

tions

. The aforementioned SAT-features set is used

to predict the runtime of a certain algorithm and the

value of the objective function obtained by the algo-

rithm.

More information about these competitions can be

found at http://www.satcompetition.org

A second potential application of the SAT transla-

tion scheme is applying SAT (or MaxSAT) solvers to

nurse rostering problem instances. As a ﬁrst attempt,

(Acharyya, 2008) translates a simple nurse rostering

problem with a small set of constraints into SAT and

applies GSAT to solve it. The approach is limited

to small instances as the number of clauses increases

rapidly with the size of the instances. It is thus impor-

tant to develop efﬁcient encodings of the constraints.

Modeling optimisation problems involving con-

straints on sequences of decision variables to mixed

integer programs (and SAT) can be very complex

e et al., 2011). Special attention is given to the

translation of counting constraints into SAT clauses.

We present an efﬁcient translation procedure generat-

ing O(n

) clauses and O(nlogn) variables. (Bailleux

and Boufkhad, 2003) developed a translation scheme

with similar complexity. (Sinz, 2005) provides a

O(n.k) and (As

ın et al., 2009) a O(nlog

(k)) scheme

to translate the constraint x

+...+x

≤ k. All contri-

butions however, only provide a one way translation,

preserving arc consistency. As for the hardness stud-

ies, we do not solely focus on solving nurse rostering

problems using SAT, a two way translation was re-

quired.

(Cadoli and Schaerf, 2005) present the automated

translation of problem speciﬁcations expressed in NP-

SEC into SAT. NP-SEC in a logic based language ca-

303

Haspeslagh S., Messelis T., Vanden Berghe G. and De Causmaecker P..

An Efﬁcient Translation Scheme for Representing Nurse Rostering Problems as Satisﬁability Problems.

DOI: 10.5220/0004259103030310

In Proceedings of the 5th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2013), pages 303-310

ISBN: 978-989-8565-39-6

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

pable of expressing all problems belonging to com-

plexity class NP. The authors tested the system on a

few classical problems such as graph colouring and

job-shop scheduling, NP-complete problems with a

rather simple problem deﬁnition. The authors high-

light the beneﬁts of an automated approach. Although

the performance of SAT solvers on the generated in-

stances is often inferior to manual encodings, the au-

thors stem that the system is a valuable tool for devel-

oping fast prototypes for new problems, or variations

of known ones for which no speciﬁc solver is avail-

able.

In Section 2, an abstract respresentation of

nurse rostering problems using numberings is de-

scribed. Section 3 elaborates on the formal deﬁni-

tion of generic constraints types. In Section 4, each

generic constraint type is translated to SAT. Section 5

demonstrates the applicability of the SAT translation

schemes for a hardness study of nurse rostering prob-

lem instances. Section 6 concludes and gives direc-

tions for further research.

2 ABSTRACT REPRESENTATION

OF NURSE ROSTERING

PROBLEMS USING

NUMBERINGS

The nurse rostering problem under study is the prob-

lem presented in the First international nurse rostering

competition (Haspeslagh et al., 2012). The authors

used numberings, originally designed for the efﬁcient

evaluation of constraint violations (Burke et al., 2001)

as an abstract representation of constraints. After we

elaborate on the deﬁnition of numberings, we give an

example numbering for some sample constraints. For

a full description of the competition’s constraint, we

refer to the original paper.

2.1 Numberings

We start with some elementary notation.

Deﬁnition 1. A time unit is an elementary interval of

time in which a nurse can be assigned a shift.

In our case, shift types determine those intervals.

So, each shift type has a corresponding time unit. In

the case of the nurse rostering problems considered

within this paper, the number of time units equalled

the number of shift types times the number of days

in the planning period. Thus, supposing we have a

planning horizon of D days and for each day there are

Sh shift types, we have a set T of D ∗ Sh time units.

A solution to a nurse rostering problem is then an as-

signment of nurses to shifts on speciﬁc time units.

Numberings on the time units are deﬁned as fol-

lows:

Deﬁnition 2. A numbering N

is a mapping of the

set of time units onto a set of numbers extended

with the symbol U i.e. N

: T → {−M,−M +

1,...,0,1,...,M − 1,M,U } where i = 1,...,I and I

is the total number of numberings. M is a positive

integer and U (undeﬁned) is a symbol introduced to

represent the time units that are not mapped onto a

number.

The mapping does not need to be into or onto, nor

does it need to preserve sequence. An event is a time

unit for which a nurse has a shift type assigned. The

idea of the evaluation method (Burke et al., 2001) is

to go through the set of events for which the time units

do not have U (undeﬁned) as value. We call these the

’numbered events’. More formally:

Deﬁnition 3. A personal schedule S

for person p is

a mapping S

: T → {working,free}.

Deﬁnition 4. For a given personal schedule S

event is a time unit e for which S

(e) = working.

Denote by T

the set of all time units for which

maps to working. T

is thus the set of time units

induced by S

, or in other words: the set of time units

for which nurse p is assigned to work. Denote by T

the set of time units for which the numbering N

deﬁned.

Deﬁnition 5. The event set T

:= T

∩ T

is the

set of time units induced by the schedule S

, for which

the numbering N

is deﬁned (6= U).

Deﬁnition 6. Two events e

and e

j+1

are consecutive

with respect to numbering N

if N

j+1

) − N

) = 1

We can now express the following four groups of

constraints:

• total number of assignments (total): for a num-

bering N

, this constraint type limits the number

of events e for which N

(e) 6= U.

• total number of assignments of a certain type (per-

Type): for a numbering N

, this constraint type

limits the number of events corresponding to a

speciﬁc number j (N

(e) = j).

• consecutive assignments (consecutiveness): for a

numbering N

, this constraint limits the length a a

sequence of consecutive events in T

• gaps between consecutive sequences of assign-

ments (between): for a numbering N

this con-

straint type limits the maximum gap (e.g. free

time) between two non-consecutive events e

and

j+1

(the gap equals N

j+1

) − N

)).

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Table 1: Example numbering.

Day Mo Tu We Th Fr

ST E L E L E L E L E L

T 0 1 2 3 4 5 6 7 8 9

N 0 0 1 1 2 2 3 3 4 4

E ∗ ∗ ∗ ∗

cons 1 1 1 2 2 3 0 0 1 1

last 1 1 1 2 2 3 3 3 4 4

For each constraint type, a maximum and min-

imum bound can be set: max total and min total,

max consecutive and min consecutive, max between

and min between and max pert and min pert.

Four counting variables are introduced. total rep-

resents the total number of events for the number-

ing. consecutive represents the number of consecutive

events. If an interruption in a sequence is found, the

counter is reset to 0. pert keeps track of the number

of events per value in the numbering and last keeps

track of the number of the last evaluated event.

With each numbering N

a set of four of the above

counters is associated. During the evaluation, by com-

paring the counter with one of the abovementioned

constraint types, constraint violations are detected.

An example for a simpliﬁed nurse rostering prob-

lem with two shift types (ST) is given in Ta-

ble 1. We want to evaluate two constraints: the em-

ployee should work at most two consecutive days

(max consecutive = 2) and should have at least two

consecutive days off (min between = 2). E shows

the assignments to one employee. At time unit 5, a

ﬁrst violation is detected as the value of the counter

cons exceeds max consecutive. At time unit 8, a sec-

ond violation is detected as (current number − last >

min between).

3 FORMAL DEFINITIONS OF

GENERIC CONSTRAINTS

In this study, we only consider monotonically ascend-

ing (Deﬁnition 7) numberings.

Deﬁnition 7. A numbering N

is monotonically as-

cending if, for every two time units j

and j

for which

is deﬁned (6= U), j

< j

⇒ N

( j

) ≤ N

( j

The constraints on the numberings can be ex-

pressed using event sets. An important concept in

the expression of constraints is the notion of event

sequences, in particular sequences that increase at a

steady pace.

Deﬁnition 8. The event set E of a numbering N

and

a personal schedule S

is the set T

With every event e ∈ E, a unique number (∈ N

) is

associated. We refer to Section 2.1 for detailed infor-

mation on the deﬁnition of S

and T

Deﬁnition 9. An event sequence r is any sequence of

events e

from E, ( j = 0 ...m), conserving the order

of the time units corresponding to the events.

Deﬁnition 10. A sequence r is contiguously ascend-

ing if ∀ j ∈ {1, . . . , m} : e

− e

j−1

= 1.

We can now give formal deﬁnitions for the con-

straints in Table 2. Note that the subject of these

constraints is depending on the numbering. All con-

straints of the competition’s instances can be de-

scribed using monotonically ascending numberings

and are of one of the presented constraint types.

Table 2: Formal deﬁnition of eight generic constraints.

Constraint Value Deﬁnition

max consecutive max

There is no contiguously ascending event se-

quence of length max

+ 1

min consecutive min

There is no contiguously ascending event se-

quence of length l: min

> l ≥ 1 which is not

part of another ascending event sequence of

length at least min

max between max

For any two consecutive events e

and e

j+1

) − N

) ≤ max

min between min

For any two consecutive events e

and e

j+1

) − N

) ≤ 1 or N

j+1

) − N

) ≥

min

max total max

The event set E contains at most max

events

min total min

The event set E contains at least min

events

max perType(i) max

The event set E contains at most max

events

corrensponding to a number k

min perType(i) min

If the event set E contains an event correspond-

ing to a number k, then it contains at least min

events corresponding to k

4 TRANSLATION OF GENERIC

CONSTRAINTS TO SAT

We present a scheme to translate each of the eight

generic constraints from Section 3 to Conjunctive

Normal Form SAT clauses (Cook, 1971). A prepro-

cessing step is performed before the translation of the

constraint types.

4.1 Preprocessing

We introduce boolean decision variables v

p, j

indicat-

ing whether nurse p is working time unit j. Since the

expression of the sequence constraints is basically the

same for all employees, we denote v

p, j

as v

, in order

to simplify the notation.

For some numberings, consecutive time units are

assigned the same number, e.g. the example number-

ing N in Table 1 does not distinguish between dif-

ferent shifts on the same day. Hence it is natural to

introduce a variable t

indicating for each sequence of

consecutive time units with equal numbers whether

an employee is working on a time unit within that se-

quence. Generally, for such a sequence of time units

starting at k and ending at l:

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Table 3: Preprocessing for ’between’ constraints.

Time unit 1 2 3 4 - - 5 6

Numbering 1 1 2 2 - - 5 5

Variable v

- - v

Preprocessing t

⇔

j=k

or in cnf: ¬t

∨ (

j=k

) and

j=k

∨ ¬v

)

This results in 2 + l − k clauses. For n numbers

in the numbering we generate at most dn/2e variables

and at most 1+dn/2e clauses. The preprocessing step

thus results in O(n) variables and O(n) clauses.

4.1.1 Preprocessing for between Constraints

Translating ’between’ constraints requires a slightly

different preprocessing step. Where a numbering is

interrupted, one or more implicit variables with value

’False’ should be placed. An example is given in

Table 3, both variables t

≡ False and t

≡ False.

Without the implicit variables, some event sequences

would lack. Suppose we want a gap of at most 1

between two consecutive events and an event occurs

for t

= True or v

= True) and t

= True or

= True). Between the two events, there is a gap of

2. By omitting the implicit variables, the two event

sequences detecting the violation would be missing.

The preprocessing step results in at most O(n) ex-

tra variables and as many extra clauses as variables

added.

4.2 Translation of ’Consecutive’ and

’between’ Constraints

4.2.1 Maximum Number of Consecutive Events

and Free Time Units between Two Events

As stated in Table 2 there should not be a contigu-

ously ascending event sequence of length max

+ 1.

For every contiguously ascending sequence cas (cas

is the index in the original sequence of the i

variable

within cas) we have the following clauses:

max

j=0

cas

, in cnf:

max

i=0

(¬t

cas

)

Analogously, for a maximum gap between two events,

we obtain:

max

j=0

¬t

cas

, in cnf:

max

i=0

cas

)

In general, both constraints for a monotonically as-

cending numbering with n numbers generates at most

(n − max

), respectively (n − max

) clauses and no

extra variables.

4.2.2 Minimum Number of Consecutive Events

and Free Time Units between Two Events

Following Table 2, there should not be a contiguously

ascending event sequence of length l (min

> l > 1)

which is not part of another contiguously ascending

event sequence of length at least min

. This implies

that in any consecutive sequence of variables t

length l (min

+1 ≥ l ≥ 3) the middle variables cannot

be true without one of the border variables. For ev-

ery contiguously ascending sequence cas of length l

(3 ≤ l ≤ min

+1) this results in the following clauses:

¬t

cas

∧ ¬t

cas

l−1

∧

l−2

i=1

cas

)

in cnf: t

cas

∨t

cas

l−1

∨

l−2

i=1

¬t

cas

Analogously, for a minimum gap between two events,

we obtain:

cas

∧t

cas

l−1

∧

l−2

i=1

¬t

cas

)

in cnf: ¬t

cas

∨ ¬t

cas

l−1

∨

l−2

i=1

cas

In general, both constraints for a monotonically as-

cending numbering consisting of n numbers, generate

at most (n − min

) clauses and no extra variables.

4.3 Translation of Counting Constraints

We developed an efﬁcient general procedure for trans-

lating counting constraints into CNF clauses. First we

elaborate on the procedure in general. Then we show

how to translate total and pert constraint types using

this procedure.

4.3.1 General Procedure

This procedure uses an iterative process of introduc-

ing variables and clauses. We start by some elemen-

tary deﬁnitions.

Deﬁnition 11. V is the set of variables v

for which

we want to count the number of variables that are as-

signed true.

Deﬁnition 12. U

α,β

is the set of indices (of time units)

k between α and β for which v

in V is assigned true.

Deﬁnition 13. F

α,β

is the set of indices (of time units)

k between α and β for which v

in V is assigned false.

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We then introduce the variables u

, respectively f

denoting whether there are at least i elements in the set

1,n

, respectively F

1,n

, with n the number of elements

in V :

⇔ |U

1,n

| ≥ i, f

⇔ |F

1,n

| ≥ i for i ∈ {1,...,n} (1)

More generally, we introduce the variables u

x,y,z

( f

x,y,z

) denoting whether there are at least z events in

the set U

x,y

x,y,z

⇔ |U

x,y

| ≥ z and f

x,y,z

⇔ |F

x,y

| ≥ z (2)

In the remainder of this section, we only consider the

ﬁrst equivalence in (2). As u

= u

1,n,i

, for representing

equivalence (1), we show we only need to translate:

1,n,i

⇒ |U

1,n

| ≥ i (3)

By contradiction, the other direction of the equiva-

lence is equivalent with

¬u

1,n,i

⇒ ¬(|U

1,n

| ≥ i) (4)

If no i variables may be assigned true, at least n −i+1

variables should be assigned f alse:

(|U

1,n

| < i) ⇔ (|F

1,n

| ≥ n − i + 1) ⇔ f

n−i+1

(5)

Using (5), implication (4) becomes:

n−i+1

⇒ |F

1,n

| ≥ n − i + 1 (6)

This can be translated in the same way as we translate

(3).

4.3.2 Translation of u

1,n,i

⇒ |U

1,n

| ≥ i

First we prove the following equivalence:

1,n,i

⇔ (u

∨ u

+1,n,l

) for k + l = i + 1.

Lemma 1. u

1,n,i

⇒ (u

∨u

+1,n,l

) for k + l = i +1

In words, when U

1,n

contains at least i elements, then

contains at least k elements or U

+1,n

contains at

least l elements. We will prove this by contradiction.

Proof. Let k

be the number of elements in U

and

the number of elements U

+1,n

, then i

= k

+ l

the number of elements in U

1,n

Suppose that the right part of the implication does not

hold, so we have k

< k and l

< l

( k

< k ) and ( l

< l )

⇔ ( k

≤ k − 1 ) and ( l

≤ l − 1 )

⇒ ( k

+ l

) ≤ ( k + l − 1 − 1 )

⇔ i

≤ ( i − 1 )

⇔ i

< i

which is per deﬁnition ¬u

1,n,i

and thus contradicts to

the left part of the implication.

Using the above lemma we can split up the event

sets and rewrite the deﬁnition of u

1,n,i

as:

1,n,i

⇒ (u

∨ u

+1,n,l

) for all k ≥ 0,l ≥ 0

s.t. k + l = i + 1,i ∈ {1,...,n}

This is equivalent to:

1,n,i

⇒

k,l

∨ u

+1,n,l

) for all k ≥ 0,l ≥ 0

s.t. k + l = i + 1,i ∈ {1,...,n}

This implication is then formulated as a CNF formula:

k,l

(¬u

1,n,i

∨ u

+1,n,l

) for all k ≥ 0,l ≥ 0

s.t. k + l = i + 1

Given the limitations we can ﬁnd (i + 2) pairs of val-

ues for k and l. Hence we need a total of (i + 2)

clauses to represent this implication as a cnf formula.

This procedure essentially expresses the variable u

1,n,i

in terms of lower level variables u

and u

+1,n,l

We need 2(i + 2) lower level variables to represent

the clauses. We must note here that we do not always

need all of these lower level variables, indeed the vari-

ables u

and u

+1,n,z

with z > (

+1) will trivially

be false, since there can not be more than (

+ 1) in-

dices in the sets U

or U

+1,n

. In general, a variable

x,y,z

with z > (y−x +1) will always be trivially false.

This brings the actual number of lower level to at most

+ 1).

Generally, we need min((n + 2),2(i + 2)) lower

level variables. In the worst case, when i equals n,

this is (n + 2).

Whereas the original deﬁnition of u

used sets of

size n, we are now left with sets of size

. We repeat

this recursively until the sets are of size 1. The vari-

ables u

x,x,0

and u

x,x,1

then correspond to ¬v

and v

respectively.

The ‘left’ variables u

with different k will in

their turn all use the same lower level variables u

and u

+1,

. This is similar for the ‘right’ variables.

The second iteration will thus introduce 2.(n/2 + 2)

lower level variables. The total process will ulti-

mately introduce the following number of lower level

variables, for 2

= n:

(n + 2) + 2(n/2 + 2) + ... + 2

k−1

(n/2

k−1

+ 2)

= (n + 2) + n + 4 + ... + n + 2n

which equals

(n + n + ··· + n) + 2(2

+ 2

+ ··· + 2

k−1

)

with twice k = log

n terms

= nlog

n + Σ

i=1

)

= nlog

n + 2n − 2

= O(nlogn + n)

= O(nlogn)

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The ﬁrst iteration introduces (i + 2) lower level

clauses. In the worst case i equals n, thus (n + 2)

clauses are introduced. In a formula, the set of k

and l variables will produce the following number of

clauses

bn/2c

∑

i=0

(i + 1) +

dn/2e

∑

i=0

(i + 1)

which is O(n

4.3.3 Translation of Total and perType

Constraints

For an employee p and numbering N

, the total con-

straints impose restrictions on the minimum and max-

imum number of decision variables v

p, j

(for which N

is deﬁned) that can be assigned true in a given per-

sonal schedule S

. Deﬁnition 11 to 13 can (assuming

V containts n elements) be rewritten:

V = {v

p,i

|i ∈ T

}

α,β

= {k|v

p,k

∧ k ∈ T

∧ (α ≤ k ≤ β)}

α,β

= {k|¬v

p,k

∧ k ∈ T

∧ (α ≤ k ≤ β)}

A min

total = min

constraint can be translated to

SAT by using the general procedure from the previ-

ous section to express:

min

⇔ |U

1,n

| ≥ min

A max total = max

constraint means at least n −

max

variables should be assigned false. Analogously,

this can be expressed by translating

n−max

⇔ |F

1,n

| ≥ n − max

The translation of pertType constraints is similar.

Only difference is that the set of variables V is lim-

ited to those corresponding to a speciﬁc value j.

V = {v

p,i

|i ∈ T

∧ N

(i) = j}

5 APPLICATION: HARDNESS

STUDY FOR NURSE

ROSTERING PROBLEM

INSTANCES

Empirical hardness can be understood as the com-

plexity of a problem instance, when solved with a

particular solution method, measured by some per-

formance criteria. This hardness is a combination of

the intrinsic hardness of the instance and the quality

of the procedure on this speciﬁc instance. This im-

plies that empirical hardness is inherently linked to

the algorithm that is used. Empirical hardness models

typically map a set of instance features onto a perfor-

mance criterion such as the running time of an algo-

rithm.

Instance features are often derived from expert

knowledge. However, experts may be biased and

good experts in the problem domain may not always

be available. Therefore, by translating the problem in-

stances under study to SAT, we examine the feasibil-

ity of a more general approach based on well known

SAT-features (Nudelman et al., 2004).

For the hardness studies presented in this paper,

we follow the methodology presented by (Leyton-

Brown et al., 2006). Six steps are to be followed:

1. Identiﬁcation of the problem instance distribution.

2. Selection of one or more algorithms and perfor-

mance criteria.

3. Selection of a set of inexpensive, distribution in-

dependent features.

4. Sampling the instance distribution for generating

a training set.

5. Elimination of redundant and uninformative fea-

tures.

6. Learning models that map the feature space onto

the performance criteria.

As hardness studies are not the scope of this paper,

we refer to the aforementioned paper for more details

on the method used and the technical report (Mes-

selis et al., 2012) for speciﬁc and detailed information

on the above steps in the light of the nurse rostering

problem. We present the results of two performed ex-

periments to demonstrate the applicability of the SAT

translation schemes presented in this paper.

The ﬁrst experiment focusses on the applicabil-

ity of the SAT features proposed in (Nudelman et al.,

2004). We designed a problem instance distribution

consisting of “smaller” problem instances

The second experiment demonstrates the applica-

bility of hardness models to real world like optimi-

sation problems. Testbed were randomisations of the

instances found in the First international nurse roster-

ing competition (Haspeslagh et al., 2012).

5.1 Applicability of Hardness Studies

using SAT Features for NRP

As stated before, a problem distribution of “smaller”

instances is generated. This allows the use of IBM

CPLEX to solve the instances. Besides an optimal

Smaller denotes a small planning horizon, a limited

number of employees available and a limited number of

shift to be covered.

ICAART2013-InternationalConferenceonAgentsandArtificialIntelligence

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Title Suppressed Due to Excessive Length 11

Fig. 1 Optimal objective value versus predicted optimal value.

NOTE: NRP model testset is a fake

NRP model - test set

0 10203040506070

real optimal value

predicted optimal value

(a) based on the NRP feature set

SAT model - testset

0 10203040506070

real optimal value

predicted optimal value

(b) based on the SAT feature set

There appears to be some loss in accuracy when compared to the models

for the quality of the optimal solution. This can be explained by the

fact that the metaheuristic is not always able to solve an instance to

optimality. For many instances of the distribution, the metaheuristic

does ﬁnd the optimal solution. For a much smaller number of instances,

the metaheuristic can only approximate the solution. This data is thus

the same data as for the complete search method, augmented with

some noise for a small set of instances. This noise produced by the

metaheuristic is hard to model. Nevertheless, these models are still very

accurate and yield good predictive power, as the plots show.

Fig. 2 Metaheuristic obtained objective value versus predicted metaheuristic value.

NRP model - testset

0 10203040506070

real approximate quality

predicted approximate quality

(a) based on the NRP feature set

SAT model - testset

0 10203040506070

real approximate quality

predicted approximate quality

(b) based on the SAT feature set

– Quality gap

The regression models for the quality gap are not as accurate as the

models for solution quality. The correlation coeﬃcients are R

= 0.40

Figure 1: Prediction of the optimal solution quality.