The robust coloring problem

Abstract

Some problems can be modeled as graph coloring ones for which the criterion of minimizing the number of used colors is replaced by another criterion maintaining the number of colors as a constraint. Some examples of these problem types are introduced; it would be the case, for instance, of the problem of scheduling the courses at a university with a fixed number of time slots––the colors––and with the objective of minimizing the probability to include an edge to the graph with its endpoints equally colored. Based on this example, the new coloring problem introduced in this paper will be denoted as the Robust coloring problem, RCP for short. It is proved that this optimization problem is NP-hard and, consequently, only small-size problems could be solved with exact algorithms based on mathematical programming models; otherwise, for large size problems, some heuristics are needed in order to obtain appropriate solutions. A genetic algorithm which solves the RCP is outlined.

Full text

Discrete Optimization
The robust coloring problem
Javier Y
aa~
nnez
a,*
, Javier Ram
ıırez
b
a
Department of Statistics and O.R., Facultad de Matematicas, Universidad Complutense de Madrid, 28040 Madrid, Spain
b
Department of Systems, Universidad Aut
oonoma Metropolitana-Azcapotzalco, 02200 M
eexico D.F., Mexico
Received 28 March 2001; accepted 28 March 2002
Abstract
Some problems can be modeled as graph coloring ones for which the criterion of minimizing the number of
used colors is replaced by another criterion maintaining the number of colors as a constraint. Some examples of
these problem types are introduced; it would be the case, for instance, of the problem of scheduling the courses
at a university with a fixed number of time slots––the colors––and with the objective of minimizing the probabil-
ity to include an edge to the graph with its endpoints equally colored. Based on this example, the new color-
ing problem introduced in this paper will be denoted as the Robust coloring problem, RCP for short. It is proved
that this optimization problem is NP-hard and, consequently, only small-size problems could be solved with
exact algorithms based on mathematical programming models; otherwise, for large size problems, some heu-
ristics are needed in order to obtain appropriate solutions. A genetic algorithm which solves the RCP is
outlined.
Ó2002 Elsevier Science B.V. All rights reserved.
Keywords: Graph theory; Graph coloring; Timetabling
1. Introduction
Some scheduling problems can be stated as
minimal coloring problems. In these problems, the
objective is to minimize the number of colors in
such a way that any pair of items to be scheduled
and that cannot share the same resource must have
a different color. This problem is stated, for in-
stance, when we want to schedule the courses at a
university in such a way that two courses taught by
the same individual cannot be scheduled at the
same time; the courses are represented by the
nodes of a graph and every pair of incompatible
courses are connected by an edge; the coloring
of this graph provides a feasible schedule of
the courses and a minimal coloring computes the
minimal number of time slots needed as the chro-
matic number of the graph.
In some circumstances, however, this scheme
seems to be very restrictive in the sense that it does
not include any other scheduling problems which
can be modeled as coloring problems.
European Journal of Operational Research 148 (2003) 546–558
www.elsevier.com/locate/dsw
*
Corresponding author. Tel.: +34-91-394-4522; fax: +34-91-
394-4606.
E-mail addresses: jayage@mat.ucm.es (J. Ya
´n˜ez), jar-
aro@correo.azc.uam.mx (J. Ram

ıırez).
0377-2217/03/$ - see front matter Ó2002 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0377-2217(02)00362-4

This is the case if the resource size to be shared is
fixed. In the courses scheduling of a university
problem, for instance, the number of available time
slots could appear as a constraint instead of the
objective function to minimize. Any c-coloring
function, where cdenotes the fixed resource size,
gives a feasible schedule and the problem would
be to choose the best one following some other
criterion.
In this way, the minimal coloring problem can
be a very restrictive model for some schedul-
ing problems and a new graph coloring prob-
lem, denoted as the Robust coloring problem
(RCP) in Ram
ıırez [7], will be analyzed in this
paper.
The major innovation of this problem is that
the coloring function takes into account the
overall topology of the graph: the endpoints of the
edges cannot be equally colored––as in the classi-
cal coloring problem––and, also, the complemen-
tary edges are valued in such a way that their
equally colored endpoints increase the objective
function. Taking into account the classification of
Pardalos et al. [6], this new problem cannot be
considered as a internal or external generalization
of the graph coloring problem.
Some examples of the new graph coloring
problem are introduced in Section 2. In Section 3
the RCP is formalized and its computational
complexity is analyzed; it will be concluded that
the RCP can be classified as an NP––hard prob-
lem. The solution methods are analyzed in Section
4: an exact algorithm can be based on the binary
programming model introduced in Section 4.1; an
approximated genetic algorithm is introduced in
Section 4.2. Computational experiences are in-
cluded in Section 5.
2. Some introductory examples
Given three classical minimal coloring prob-
lems, the courses scheduling problem, the cluster
problem and the map coloring problem, their ob-
jective function––number of time slots, number of
cluster and number of colors––will be introduced
as a constraint and three new coloring problems
will be defined.
2.1. The examination timetabling problem
The following instance of the examination
timetabling problem previously introduced in
Section 1 is outlined.
Example 2.1. The examinations of six courses of a
particular program at a university must be sched-
uled. Two courses sharing at least one student
cannot be scheduled the same day. Taking into
account the course incompatibilities, the following
6-nodes graph G¼ðV;EÞis constructed from V,
the courses set, and the couple fi;jgwill be in-
cluded in the edge set Ewhen the courses iand j
share at least one student. The adjacency matrix is
the following:
BG¼
0
10
110
1000
00110
010110
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
The graph is depicted in Fig. 1.
In order to minimize the duration of examina-
tions, a minimal coloring problem can be stated
and the chromatic number of the incompatibility-
graph is computed as vðGÞ¼3. The following
coloring function Cis in this sense optimal:
Fig. 1. Incompatibility graph of Example 2.1.
547J. Y
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ıırez / European Journal of Operational Research 148 (2003) 546–558

Cð1Þ¼1;Cð2Þ¼2;Cð3Þ¼3;
Cð4Þ¼2;Cð5Þ¼1;Cð6Þ¼3
Moreover, the stability of the scheduled solu-
tion deduced from the coloring function Cwould
be desirable in the sense that it remains valid under
some changes in the associated incompatibility
graph. In Example 2.1, for instance, after the ex-
amination schedule has been published, one stu-
dent could choose a new course in such a way that
a new edge is included in the set E, the initial
coloring becoming invalid.
Let Edenote the complementary edge set:
fi;jg2E() f i;jg 62 E
and G¼ðV;EÞwill be the complementary graph
of G.
The validity of a coloring function Cunder such
changes of graph G, which can be defined as the
robustness of the coloring, could be an important
criterion which must be considered.
For Example 2.1, if the probabilities that any
complementary edge 
ee 2Ewill be added to Eare
known, the robustness of the coloring can be
measured as the probability of such coloring re-
maining valid after one random complementary
edge has been added to the edge set. Let us sup-
pose that the probability of the complementary
edge fi;jgwill be proportional to the number of
students registered in courses iand j. Let n¼50 be
the total number of students which must choose
two out of the six courses and let also nibe the
number of students registered in course i:
n1¼5;n2¼30;n3¼10;
n4¼30;n5¼20;n6¼5
The normalized probabilities are depicted in
Table 1.
Given the above coloring function C, the
probability that remain valid, assuming statistical
independence, can be computed as:
ð1pr15Þð1pr24 Þð1pr36 Þ
¼ð10:0506Þð10:4557Þð10:0253Þ
¼0:5037
However, if the coloring function were C0, defined
by:
C0ð1Þ¼1;C0ð2Þ¼2;C0ð3Þ¼3;
C0ð4Þ¼3;C0ð5Þ¼2;C0ð6Þ¼1
the probability would be:
ð1pr16Þð1pr25 Þð1pr34 Þ
¼ð10:0127Þð10:3038Þð10:1519Þ
¼0:5829
Moreover, if the number of days to schedule the
exams were not critical, say for instance 4, this
probability will increase choosing the coloring
function:
C00ð1Þ¼2;C00 ð2Þ¼4;C00ð3Þ¼1;
C00ð4Þ¼3;C00 ð5Þ¼2;C00ð6Þ¼1
which has a probability of remaining valid:
ð1pr15Þð1pr36 Þ¼ð10:0506Þð10:0253Þ
¼0:9254
2.2. Cluster analysis
A basic problem in cluster analysis is how to
partition the entities of a given set into a preas-
signed number of homogeneous subsets (clusters).
The homogeneity of the clusters can be expressed
as a function of a dissimilarity measure between
entities. Let Vbe the set of entities and let Ebe the
set of edges defined by every pair of entities i;j2V
whose dissimilarity dði;jÞis less than a>0,a
given threshold. A c-coloring of this graph de-
fines the ccolor classes as clusters. Hansen and
Delattre (see Hansen et al. [3]) have shown that
any of those c-coloring define a minimum-diame-
ter partition of the entities set into cclusters. A
numerical example of this problem is introduced.
Table 1
Normalized probabilities of Example 2.1
fi;jg2En
injprij
{1,5} 100 0.0506
{1,6} 25 0.0127
{2,4} 900 0.4557
{2,5} 600 0.3038
{3,4} 300 0.1519
{3,6} 50 0.0253
J. Y
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nnez, J. Ram
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ıırez / European Journal of Operational Research 148 (2003) 546–558548

Example 2.2. Let V¼f1;2;3;4;5gbe the set of
entities and let dbe the dissimilarity distance de-
fined by the matrix:
D¼
0:010:020:050:04
0:040:030:04
0:060:07
0:03

0
B
B
B
B
@
1
C
C
C
C
A
For different values of the threshold a, different
graphs Gaand coloring functions are obtained:
For a¼0:05, the graph Gahas two edges, i.e.
E¼ff3;4g;f3;5gg. This graph is depicted in Fig.
2.
Taking into account that vðG0:05Þ¼2, the 2-
coloring function C2
1defined by
C2
1ð1Þ¼1;C2
1ð2Þ¼1;C2
1ð3Þ¼2;
C2
1ð4Þ¼1;C2
1ð5Þ¼1
defines two clusters:
VC2
1ð1Þ¼f1;2;4;5g;VC2
1ð2Þ¼f3g
Given another valid 2-coloring function C2
2,
defined by
C2
2ð1Þ¼1;C2
2ð2Þ¼1;C2
2ð3Þ¼1;
C2
2ð4Þ¼2;C2
2ð5Þ¼2
defines another partition in clusters:
VC2
2ð1Þ¼f1;2;3g;VC2
2ð2Þ¼f4;5g
Decreasing the threshold from a¼0:05 to 0.04,
one new edge is included, f1;4g. The new graph
G0:04 is depicted in Fig. 3.
The chromatic number is also vðG0:04 Þ¼2. The
coloring function C2
1is not valid but C2
2remains
valid. This coloring function is in this sense pref-
erable to C2
1.
Decreasing the threshold again to the value
a¼0:03, neither of the two coloring functions
remains valid because the chromatic number
vðG0:03Þ¼3.
Allowing three colors, the 3-coloring C3
1defined
by:
C3
1ð1Þ¼1;C3
1ð2Þ¼2;C3
1ð3Þ¼1;
C3
1ð4Þ¼2;C3
1ð5Þ¼3
whose color classes are:
VC3
1ð1Þ¼f1;3g;VC3
1ð2Þ¼f2;4g;VC3
1ð3Þ¼f3g
defines three clusters which remain valid until the
threshold value a¼0:03. The graph G0:03 is de-
picted in Fig. 4.
In the cluster analysis, the minimal coloring
seeks the minimum number of clusters. However,
in some real situations, the number of cluster can
be fixed, let cbe this number, and the problem is
how these cclusters can be filled in such a way that
they remain valid under lower values for the dis-
similarity measure threshold. Moreover, the limit
Fig. 2. Graph G0:05 of Example 2.2.
Fig. 3. Graph G0:04 of Example 2.2.
549J. Y
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ıırez / European Journal of Operational Research 148 (2003) 546–558

value aðcÞalso can be computed, verifying that a c-
coloring function Cc
exists, and such that it re-
mains valid for graph Gafor all aPaðcÞ. It is not
difficult to see that aðcÞis a decreasing function of
c. In the limit case c¼n,aðnÞ¼0.
In Example 2.2, að2Þ¼0:04;C2
¼C2
2and að3Þ¼
0:03;C3
¼C3
2, where
C3
2ð1Þ¼1;C3
2ð2Þ¼1;C3
2ð3Þ¼2;
C3
2ð4Þ¼3;C3
2ð5Þ¼3
2.3. Coloring geographical maps
The classical four-color problem faced with the
minimal coloring of a planar map.
Example 2.3. Let Gbe the planar graph associated
to the geographical map of eight regions depicted
in Fig. 5.
Let C4
1and C4
2be two 4-coloring functions of G:
and the corresponding color classes:
VC4
1ð1Þ¼f4g;VC4
1ð2Þ¼f2;3;6;7g;
VC4
1ð3Þ¼f1;8g;VC4
1ð4Þ¼f5g
and
VC4
2ð1Þ¼f4;7g;VC4
2ð2Þ¼f2;6g;
VC4
2ð3Þ¼f1;5g;VC4
2ð4Þ¼f3;8g
We can notice that the color classes of C4
2are more
homogeneous than the color classes of C4
1. All of
them have two regions. It is in this sense that we
prefer coloring function C4
2to coloring function
C4
1.
All of these three problems have two types of
constraints: a hard one which defines the valid
coloring and a soft type constraint, which takes
into account other criteria, for instance, the exis-
tence of additional edges of the graph. In these
problems, the number of units of some used re-
source is considered as a constraint of the prob-
lem, instead of the objective function which must
be minimized in the classical coloring problem.
3. The problem
In the previously proposed examples, the col-
oring functions of graph Gdoes not search the
minimum number of colors, which is fixed as c.
Fig. 4. Graph G0:03 of Example 2.2.
Fig. 5. Geographical map of Example 2.3.
12345678
C4
132214223
C4
232413214
J. Y
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nnez, J. Ram
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ıırez / European Journal of Operational Research 148 (2003) 546–558550

Obviously, this parameter must verify that
cPvðGÞ. As was pointed out previously, and dif-
fering to the minimal coloring graph problem, the
complementary edges are also considered.
In Example 2.1 the objective is to obtain a valid
coloring function with no more than ccolors in
such a way that the probability of an added edge
to the incompatibility graph with the two end-
points equally colored is minimized.
In Example 2.2 a coloring function is searched
with the property of remaining valid when some
edges were added after a decreasing of the
threshold dissimilarity value.
In Example 2.3, the coloring function with the
property of maintaining as closely as possible the
cardinal of the color classes is preferred.
There is, however, a unified coloring graph
problem which generalizes all of these examples.
Taking into account that this graph coloring
problem looks like a valid coloring function under
possible changes of the graph, this problem will be
denoted as the RCP for short, which can be stated
formally in the following.
Given the graph G¼ðV;EÞwith jVj¼nand
jEj¼m, let c>0 be an integer number, then, a c-
coloring is a mapping
Cc:V!f1;2;...;cg
verifying
CcðiÞ 6¼ CcðjÞ8fi;jg2E
Ac-coloring exists if and only if the number of
colors allowed is equal or greater than the chro-
matic number of the graph G, i.e., cPvðGÞ.
Remark 3.1. From now on, any c-coloring will be
presumed strict in the sense that all color classes
are not empty:
i2f1;...;ng=CcðiÞ
f¼rg6¼; 8r2f1;...;cg
The kernel of this new graph problem is the
valuation of any valid c-coloring. In order to ac-
complish this objective, the following concept is
introduced.
Definition 3.1. Given graph Gand the penalty
matrix defined on the complementary edge set
fpij P0;fi;jg2Eg
the rigidity level of the c-coloring Cc, denoted by
RðCcÞ, is defined as the sum of the penalties of
complementary edges whose extremes are equally
colored:
RðCcÞ X
fi;jg2E;CcðiÞ¼CcðjÞ
pij
In Example 2.1, if any complementary edge
pij ¼lnð1prij Þ8fi;jg2E
then, taking into account these penalties, the ri-
gidity level of the c-coloring Cis
RðCcÞ X
fi;jg2E;CcðiÞ¼CcðjÞ
lnð1prijÞ
¼ln Y
fi;jg2E;CcðiÞ¼CcðjÞ
ð1
0
@prijÞ1
A
In this way, the rigidity level is equal to minus the
logarithm of the probability that the coloring re-
mains valid. The most robust coloring minimizes
the probability that an added edge to the incom-
patibility graph makes such coloring invalid.
In Example 2.2, however, the rigidity level has a
more complex interpretation. It depends on the
finite and ordered dissimilarity values of the ob-
jects.
Let fd1;d2;...;dkgbe the dissimilarity values
set defined by EE. Consequently,
dij 2fd1;d2;...;dkg8fi;jg2Ea
verifying
d1<d2<<dk
Given the threshold level a, the penalty of any
complementary edge fi;jg2Eawhose endpoints
are equally colored must be greater than the sum
of all complementary edge penalties with dissimi-
larity lower than dij, i.e.
pij >X
fi0;j0g2Ea=di0j0<dij
pi0j0
This property assures that any c-coloring which
remains valid for lower values of awill have a ri-
gidity level lower than another c-coloring which
551J. Y
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does not remain valid for the same lower values of
a. Let m¼jEjbe the cardinal of the complemen-
tary edges set, the following penalties verify the
above property:
pij ¼ðmÞs1if dij ¼ds8fi;jg2E
s2f1;...;kg
In Example 2.2, given the threshold a¼0:05,
the dissimilarity set is
f0:01;0:02;0:03;0:04;0:05g
in such a way that k¼5 and
d1¼0:01;d2¼0:02;d3¼0:03;
d4¼0:04;d5¼0:05
Taking into account that m¼8, one penalty set
for this example is depicted in Table 2.
In Example 2.3, setting
pij ¼18fi;jg2E
the rigidity level of a c-coloring will be decreased
as the number of regions equally colored tends to
be uniform with respect to all colors.
The rigidity level RðCcÞof a c-coloring Cc
measures its robustness in the sense that the com-
plementary edges whose endpoints are equally
colored are penalized and, consequently, this c-
coloring would not be valid if they were added to
the graph.
Given a number of colors c, for a lower rigidity
level the c-coloring will be more robust. Of course,
as cincreases, a more robust c-coloring can be
obtained; in the limit case c¼n, the most robust n-
coloring will be
CnðiÞ¼i8i2V
which has a rigidity level RðCnÞ¼0.
The RCP, can be stated in the following way:
Given a graph G¼ðV;EÞ, an integer number c
and the penalty set fpij ;fi;jg2Eg. The RCP looks
like those c-coloring Cc
Rwith the least rigidity
level:
RðCc
RÞ¼min
CcRðCcÞ
Taking into account that the adjacency matrix
Bof a graph is symmetrical, any instance of the
RCP can be characterized by ðn;c;HÞ, where nis
the number of nodes of the graph, cis the num-
ber of valid colors, and His the nnmatrix
which stores the adjacency matrix of the graph
in the lower triangle matrix and the penalties of
its complementary edges in the upper triangle
matrix:
hij ¼pij if i<j
bji if i>j

where
bij ¼1iffi;jg2E
0 if fi;jg 62 E
i<j
and pij is the penalty of complementary edge
fi;jg 62 Eor 0 if fi;jg2E.
The parameters ðn;c;HÞof Example 2.1 are:
n¼6c¼4
H¼
000 0 0:0519 0:0128
1000:6083 0:3621 0
1100:1648 0 0:0256
100 0 0 0
001 1 0 0
010 1 1 0
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
where penalties hij have been obtained from the
Table 3:
Table 2
Complementary edge penalties of Example 2.2
fi;jg2Ed
ij dspij
{1,2} 0.01 d11
{1,3} 0.02 d28
{1,4} 0.05 d54096
{1,5} 0.04 d4512
{2,3} 0.04 d4512
{2,4} 0.03 d364
{2,5} 0.04 d4512
{4,5} 0.03 d364
Table 3
Complementary edge penalties of Example 2.1
fi;jg2Eprij pij
{1,5} 0.0506 0.0519
{1,6} 0.0127 0.0128
{2,4} 0.4557 0.6083
{2,5} 0.3038 0.3621
{3,4} 0.1519 0.1648
{3,6} 0.0253 0.0256
J. Y
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nnez, J. Ram

ıırez / European Journal of Operational Research 148 (2003) 546–558552

The most robust 4-coloring is:
C4
Rð1Þ¼2;C4
Rð2Þ¼4;C4
Rð3Þ¼1;C4
Rð4Þ¼3;
C4
Rð5Þ¼2;C4
Rð6Þ¼1
Its rigidity level is
RðC4
RÞ¼0:0519
in such a way that the probability that it will
remain valid under a random change in the edge
set is:
1
expð0:0519Þ¼0:9494
Given the threshold level a¼0:05, Example 2.2
is characterized by:
n¼5c¼3H¼
0 1 8 4096 512
0 0 512 64 512
00000
001064
00100
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
The most robust 3-coloring is:
C3
Rð1Þ¼1;C3
Rð2Þ¼1;
C3
Rð3Þ¼2;C3
Rð4Þ¼3;C3
Rð5Þ¼3
Its rigidity level is RðC3
RÞ¼p12 þp45 ¼65 which
can be interpreted in the following way: The 3-
coloring C3
Rwill remain valid for any threshold
value a>d3¼0:03, where the exponent 3 is cho-
sen from the inequality chain
64 ¼82665 ¼RðC3
RÞ<512 ¼83
The critical threshold að3Þ¼aðC3
RÞ¼0:03
Example 2.3 is characterized by:
n¼8c¼4H¼
00001111
10100111
10001111
11100010
01010100
00010010
00001000
00011110
0
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
A
The most robust 4-coloring is C4
2. Its rigidity level is
RðC4
2Þ¼p47 þp26 þp15 þp38 ¼4
In general, when all penalties are equal to a
constant p>0, then the most robust c-coloring
computes the c-coloring with the most homo-
geneous cardinal of color classes.
Proposition 3.1. Given graph G¼ðV;EÞwith a c-
coloring Cc
1,if the penalties of the complementary
edges verify
pij ¼p>08fi;jg2E
and two different color classes verifying
n1¼jVC1ðkÞj <jVC1ðlÞj ¼ n2
exist with n2>n1þ1, then the resulting c-coloring
Cc
2after changing, if it is possible, one node from the
color class VC1ðlÞto the color class VC1ðkÞis more
robust than Cc
1.
Proof. The cardinals of color classes VC2ðkÞand
VC2ðlÞare n1þ1andn21 respectively. The ri-
gidity level added by these classes is lower than
that added by color classes VC1ðkÞand VC1ðlÞ:
n1
2

þn2
2

>n1þ1
2

þn21
2

developing the binomial coefficients:
1
2ðn2
1n1þn2
2n2Þ>1
2ðn2
1þn1þn2
23n2þ2Þ
which is equivalent to
n1n2>n13n2þ2
and, it is also equivalent to
n2>n1þ1
and the proposition is thus proven.
Consequently, the equitable coloring problem
(see de Werra [9] and Lih [5]) which seeks a c-
coloring whose color classes are nearly as equal in
size as possible, can be viewed as a particular case
of the RCP.
3.1. Computational complexity analysis
When we are confronted with a new problem, in
our case the RCP, a natural question to ask is: Can
it be solved with a polynomial time algorithm? If
the answer is positive, any instance of this problem
553J. Y
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can be solved in a reasonable amount of time;
otherwise, the computational complexity theory
can help us to decide if this problem is computa-
tionally difficult in the sense that only small size
instances can be solved. See Garey and Johnson [2]
for further details. The complexity analysis of the
RCP begins with its associated decision problem:
DRCP:
INSTANCE: A graph G¼ðV;EÞ, one positive
integer c6jVj, a family of non-negative penal-
ties defined over Eand an upper bound 
rr,
QUESTION: Does a c-coloring Ccof Gexist
such that RðCcÞ¼Pi<j;CcðiÞ¼CcðjÞhij 6
rr?
Proposition 3.2. The DRCP is NP––complete.
Proof. The DRCP 2NP, because a non-deter-
ministic algorithm only needs to check in polyno-
mial time that a c-coloring verifies all constraints.
Let DMCP be the decision problem associated
with the minimal coloring problem:
DMCP:
INSTANCE: A graph G¼ðV;EÞand one pos-
itive integer k6jVj,
QUESTION: Is Gk-colorable, i.e., does a k-col-
oring of Gexist?
We transform the DMCP to DRCP. Given an
instance I2DDMCP, let hij ¼08fi;jg2E, be the
trivial penalty matrix and let also c¼kbe the
number of valid colors; then, considering 
rr ¼0,an
associated instance fðIÞ2DDRCP has been defined
in such a way that
I2YDMCP () fðIÞ2YDRCP
This mapping fcan be constructed in polyno-
mial time. 
4. The solution methods
As a consequence of the above computational
complexity analysis, the optimization problem
RCP is NP-hard and only small size instances can
be solved exactly; otherwise, some heuristics must
be used. In Section 4.1 a binary programming
model for the RCP will be introduced. In Section
4.2 a genetic algorithm to solve the RCP will be
outlined.
4.1. Mathematical programming model
Let G¼ðV;EÞbe the graph, with n¼jVjand
m¼jEj. Let cbe the number of valid colors. The
penalties of complementary edges are defined by
fpij=fi;jg2Eg.
In order to solve the RCP exactly the following
binary programming model is introduced.
Let xik 2f0;1gbe the decision variable defined
by
xik ¼1CcðiÞ¼k
0 otherwise
i2f1;...;ngk2f1;...;cg
identifying the function
Cc:V!f1;...;cg
as the c-coloring of the graph G.
Introducing the auxiliary variables:
yij ¼
1 if there exists k2f1;...;cg
such that xik ¼xjk
0 otherwise
8
<
:8fi;jg2E
the RCP can be stated as:
Min X
fi;jg2E
pijyij
subject to
X
c
k¼1
xik ¼18i2f1;...;ng
xik þxjk 618fi;jg2Eand 8k2f1;...;cg
xik þxjk 16yij 8fi;jg2Eand 8k2f1;...;cg
The first and second constraint groups assure that
all nodes have been colored with an unique color
and that two adjacent nodes cannot be equally
colored. The third constraint group assigns the
values yij ¼1 for any endpoints of a complemen-
tary edge equally colored.
Taking into account that 
mm ¼j
Ej¼ðnðn1Þ=
2Þmthe number of binary variables is nc þm
and the number of constraints is nþcm þcm¼
nþcðnðn1Þ=2Þ.
J. Y
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The binary programming model of Example
2.1, with c¼4, is the following:
Minf0:0519y15 þ0:0128y16 þ0:6083y24 þ0:3621y25
þ0:1648y34 þ0:0256y36g
subject to
x11 þx12 þx13 þx14 ¼1

x61 þx62 þx63 þx64 ¼1
x11 þx21 61x14 þx24 61
x11 þx31 61x14 þx34 61

x51 þx61 61x54 þx64 61
x11 þx51 16y15 x14 þx54 16y15

x31 þx61 16y36 x34 þx64 16y36
The non-null binary variables identifying the most
robust 4-coloring C4
Rare:
x12 ¼1;x24 ¼1;x31 ¼1;x43 ¼1;
x52 ¼1;x61 ¼1
The non-null auxiliary variables identifying the
equally colored complementary edges are:
y15 ¼1;y36 ¼1
and the non-null terms of the objective function
are:
p15y15 þp36 y36 ¼0:0519 þ0:0256 ¼0:0775
¼RðC4
RÞ
4.2. A genetic algorithm for the RCP
Genetic algorithm (GA), originally developed
by Holland [4], proved efficient in solving several
combinatorial optimization problems. The GAs
(see Chelouah and Siarry [1]) use natural pro-
cesses, such as selection, crossover and mutation to
manage a population or set of valid solutions for
the problem. The selection operator determines the
individuals to be chosen for mating; crossover
between two selected individuals produces two
new ones, the sons, which will replace their par-
ents; the mutation alters some characteristics of
some (very few) individuals from the population.
The representation of any individual of the
population will be a vector of arranged nodes r.
From this vector r,ac-coloring function will be
greedily constructed: for any node, the least fea-
sible color is computed taking into account the
colors of the previous adjacent nodes. This pro-
cess does not guarantee the feasibility of the col-
oring, eventually more than ccolors would be
needed, in this case, this invalid coloring must be
penalized, let PI>0 be such a constant which must
be high enough for that any invalid coloring is
avoided.
Let mbe the population size. This parameter is
chosen based on the trade off between a small
value––a low computational time consuming––and
a high value––a broad coverage of the solution
space. Empirical results from many authors sug-
gest that population sizes as small as 30 are quite
adequate (see Reeves [8]).
Another parameter of the GA is the maximum
number of iterations itM. This parameter can be
adjusted from empirical results and in order to
decrease the time consumption.
At any iteration it 2f0;1;...;itMg, the popu-
lation is identified by ðrit
1;rit
2;...;rit
mÞ:
Each individual rit
iof the population at itera-
tion it induces––through the greedy procedure al-
ready outlined––a coloring function Cit
ivalued as
b
RRit
i¼RðCit
iÞþyIðCit
iÞPI
where
RðCÞ¼ X
fi;jg2E;CðiÞ¼CðjÞ
pij
yIðCÞ¼ 1ifCis invalid
0 otherwise

Since the RCP is stated as a minimization
problem, to each individual will be assigned the
inverse value:
Fit
i¼1
b
RRit
i
8i2f1;...;mg
555J. Y
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The overall population at iteration it is also valued
with
TF it ¼X
m
i¼1
Fit
i
As is proved by the fundamental schema theo-
rem (see Reeves [8]) as it increases the expected
total value EðTF itÞalso increases. Moreover, the
optimization solution proposed by the algorithm
will be the best obtained individual of any iteration
rand this solution is updated through the evo-
lution of the population.
In order to improve the quality of the popula-
tions at generation it, those individuals rit
ibetter
adapted will be selected, with i2f1;...;mg; i.e.,
with a higher value Fit
iand, consequently, a lower
value b
RRit
i.
One way to accomplish this objective is through
a Monte-Carlo method. The probability of select-
ing the individual rit
iwill be proportional to its
relative value with respect to the total value of its
generation:
ProbfSelect individual rit
ig¼ Fit
i
TF it 8i2f1;...;mg
Generating mrandom numbers, with the above
probabilities mindividuals will be selected; repe-
titions are allowed.
Given a randomly generated couple from the
population, with probability pcthese two individ-
uals will be crossed and they will be replaced by
their offspring.
Given the two individuals rand r0of a selected
couple, the crossover operator works in the fol-
lowing way:
1. Let ube an integer uniform random number in
the set f1;2;...;ng.
2. The first uelements of first (second) offspring
are ðrð1Þ;...;rðuÞÞ (ðr0ð1Þ;...;r0ðuÞÞ).
3. The last nuelements of first (second) off-
spring are ðr0ðuþ1Þ;...;r0ðnÞÞ (ðrðuþ1Þ;...;
rðnÞÞ) avoiding the repetitions and maintaining
the relative order of r0(r).
For instance, let n¼10 be the number of
groups and let rand r0be two parents:
r¼ð1;2;3;4;5;6;7;8;9;10Þ
r0¼ð7;3;9;1;4;6;2;8;10;5Þ
If u¼3, then the two offsprings are:
r¼ð1;2;3;4;6;8;10;5;7;9Þ
r0¼ð7;3;9;4;5;6;8;10;1;2Þ
Applying the operator selection and crossover
to the population ðrit
1;rit
2;...;rit
mÞa new popula-
tion ðritþ1
1;ritþ1
2;...;ritþ1
mÞis obtained.
With probability pm, each of the individuals of
this new population is selected so that two ele-
ments rðkÞand rðk0Þof it are permuted. The ele-
ments kand k0are integer uniform random
numbers in the set f1;2;...;ng.
The family of genetic algorithms is character-
ized by the parameter vector ðm;itM;pc;pmÞ.
5. Computational experiences
A set of random graphs Gn;pwas generated,
where nis the number of nodes and pis the graph
density (the probability that an edge exists between
two arbitrary nodes). The parameter nvaries from
10 to 15. The graph density was fixed as p¼0:5.
The parameter cvaries from 4 to 6. Each com-
plementary edge penalty pij , with fi;jg2E, has
also been randomly generated with the uniform
distribution in the interval [0,1].
The associated RCP have been solved exactly
modeling them as binary programming models
and using CPLEX 6.5. They have also been solved
approximately with a genetic algorithm with the
following parameter setting ðm¼20;itM¼20;
pc¼0:6;pm¼0:1Þand setting also the penalty
constant PI¼104. A personal computer Pentium
III was used.
The computational results are shown in Table
4. For any random graph (row), is identified the
input file, the number of nodes n, the number of
valid colors c, and the minimum rigidity obtained
level RðCc
RÞand the CPU time in seconds obtained
by the exact and approximated algorithms are
identified. The symbol ðÞ indicates those cases
where the optimum solution was attained.
J. Y
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Given the explosive growth of the number of
variables and constraints when the dimension of
the graph increases, the binary programming
model can only solve small or medium size prob-
lems. As a consequence, for high size instances for
the RCP only heuristics should be applied.
In order to check the validity of the proposed
GA, some other graphs have been randomly gen-
erated. In these graphs, and without any loss of a
generality, the penalty of complementary edges
was fixed as 1. Table 5 shows the obtained results
for the parameter setting:
ðm¼20;itM¼50ð30Þ;pc¼0:6;pm¼0:1Þ
In all cases, except for instance of file aleato12,
for which there is no feasible 4-coloring, the ob-
tained coloring function is feasible. We can see
that the results obtained by genetic algorithms are
very good when compared to the exact method
(Table 4). Moreover, very high size instances for
the RCP can be solved with a very low CPU time
consumption (Table 5).
6. Conclusions
The RCP takes into account the complementary
edges penalizing them if their extremes were
equally colored and they were added to the graph.
In this way, other criteria can be considered for
classical coloring problems.
For the examination problem, we can look for a
robust coloring function in the sense that it re-
mains valid under some changes of the graph to-
pology. In this way, the name of the new proposed
problem is justified.
For the cluster problem, the most robust
c-coloring will compute cclusters, the color
classes, in such a way that they remain valid for
lower values for the dissimilarity measure thresh-
old.
Also, with a constant penalty for every com-
plementary edges, the most robust c-coloring
function will compute the most homogeneous
color classes in the sense that their cardinals are
very similar.
The RCP can be considered as a coloring
graph extension problem and, consequently, it is
an NP-hard. This fact has been shown in the
computational experiences section. As the graph
size increases, only approximate solutions can be
Table 4
Comparative computational results
Input file ncBinary programming GA
RðCc
RÞCPU time RðCc
RÞCPU time
aleato10 10 4 3.548 1.01 (*) 3.548 0.06
aleato10 10 5 2.177 4.04 2.903 0.06
aleato11 11 4 3.445 1.50 (*) 3.445 0.06
aleato11 11 5 2.023 8.80 2.962 0.06
aleato12 12 4 (#) (#) (#) (#)
aleato12 12 5 2.904 5.92 2.915 0.06
aleato13 13 5 3.587 21.69 (*) 3.587 0.06
aleato14 14 5 5.269 27.25 5.490 0.06
aleato15 15 5 5.637 130.01 (*) 5.637 0.06
aleato15 15 6 3.559 1053.43 4.671 0.06
Note: the symbol # indicates that this problem has no feasible coloring function.
Table 5
High size computational experiences with GAs
Input
file
ncitMCPU
time
RðCc
RÞ
al0050 50 18 50 1.54 46
al0100 100 35 50 5.50 97
al0250 250 70 50 54.32 334
al0250 250 80 50 46.47 280
al0250 250 90 50 39.38 238
al0500 500 200 50 166.14 411
al1000 1000 300 30 656.42 1235
557J. Y
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obtained. However, all approximate algorithms
solving the minimal coloring problem can be easily
adapted and very good solutions can be obtained
in a few seconds of CPU time.
As a consequence, some other interesting
problems, as in the case of scheduling problems,
can be stated as RCPs so that the graph-based
heuristics could be applied to solve them.
Acknowledgements
Research supported by DGICYT (National
grant number PB95-0407) and UCM (grant num-
ber PR52/00-8920).
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