Optimal graph design using a knowledge-driven multi-objective evolutionary graph algorithm

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Abstract—Designing appropriate graphs is a problem
frequently occurring in several common applications ranging
from designing communication and transportation networks
to discovering new drugs. More often than not the graphs to
be designed need to satisfy multiple, sometimes conflicting,
objectives e.g. total length, cost, complexity or other shape
and property limitations. In this paper we present our
approach to solving the multi-objective graph design
problem and obtaining a set of multiple equivalent
compromising solutions. Our method uses multi-objective
evolutionary graphs, a graph-specific
optimization method that combines evolutionary algorithms
with graph theory and local search techniques exploiting
domain-specific knowledge. In the experimental section we
present results obtained for the problem of designing
molecules satisfying multiple pharmaceutically relevant
objectives. The results suggest that the proposed method can
provide a variety of valid solutions.

Index Terms—Optimal graph design, de novo drug
design, multi-objective optimization,
evolutionary algorithms MEGA.
meta-heuristic
multi-objective
I. INTRODUCTION
PTIMAL Graph Design (OGD) deals with the
discovery of graph structures that satisfy
certain predefined criteria while conforming to a
set of specifications concerning the building
blocks and the rules available for the
construction of the graph. The problem is
characterized by its combinatorial nature and the
potentially large size of the search space defined
by the number of feasible graph structures. As is
the case in most real life problems, OGD is
typically multi-objective, i.e. the graphs need to
satisfy more than one criterion and thus, it is
essentially a multi-objective combinatorial
optimization problem. OGD includes a wide
variety of problems including communication
network design [1], route planning [2] and
molecular design [3] among others.
The aim of this paper is to introduce a newer
development of our recently proposed multi-

Manuscript received August 17, 2009.
C. A. Nicolaou is with the Cyprus Institute, Nicosia, Cyprus, the
University of Cyprus, Nicosia, Cyprus and Noesis Chemoinformatics,
Nicosia, Cyprus. (phone: +357-22208644; fax: +357-22208625; e-mail:
c.nicolaou@cyi.ac.cy).
C. Kannas, is with Noesis Chemoinformatics and the University of
Cyprus, Nicosia, Cyprus (e-mail: ckannas@noesisinformatics.com).
C. S. Pattichis is with the Computer Science Department, University
of Cyprus, Nicosia, Cyprus (e-mail: pattichi@ucy.ac.cy).
objective optimization algorithm specifically
designed to evolve graphs using a pool of
existing building blocks and exploiting problem
domain specific knowledge. The implementation
of the algorithm, as well as the results presented,
focus on the specific problem of molecular/drug
design, also known as de novo drug design
(DND). The rest of the paper is organized as
follows. The next two sections of the paper
briefly describe fundamental graph theory
elements and their application to molecular
graphs and, multi-objective
Section IV of the paper introduces the proposed
algorithm while section V describes the general
DND problem and presents some experimental
results from the application of the algorithm.
The final section summarizes our conclusions
and outlines some directions for future research.
optimization.
II. BACKGROUND
A graph G = (V, E) consists of a set of vertices
V(G) and a set of edges E(G). In the case of
labeled graphs both vertices and edges have
identifiers, i.e. each vertex and edge has a label
drawn from a predefined set of vertex and edge
labels. Graphs can be directed or undirected. In
directed graphs edges are ordered pairs of the
vertices they connect where, in undirected, edges
simply list the pair of vertices they connect. Two
vertices VI, VJ of graph G are connected, or
adjacent, if there is an edge EIJ = (VI, VJ) ∈E(G).
If there is a path P = (E1, E2, …, En) between
every pair of vertices in a graph G, then G is a
connected graph. A graph S = (Vs, Es) is a
subgraph of G = (V, E) if and only if VS∈ V and
ES∈ E. If ES contains all edges in E connecting
the vertices in VS then S is an induced subgraph
of G. A common induced subgraph between G1
and G2 is a graph CS that is an induced subgraph
of both G1 and G2. The largest induced subgraph
between G1 and G2 is known as the Maximum
Common Induced Subgraph (MCIS). The largest
contiguous common substructure is known as
the Maximum Common Substructure (MCS).
Optimal Graph Design Using A Knowledge-driven
Multi-objective Evolutionary Graph Algorithm
Christos A. Nicolaou, Christos Kannas, and Constantinos S. Pattichis, Member, IEEE
O
Proceedings of the 9th International Conference on Information Technology
and Applications in Biomedicine, ITAB 2009
November 5-7, 2009
Larnaca, Cyprus

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Chemical structures are typically represented
as labeled, undirected graphs where atoms
correspond to vertices and chemical bonds are
represented by edges. In this context, molecular
fragments, or substructures,
subgraphs of molecular graphs. Scaffolds are
molecular fragments defined in association to
well-defined sets of compounds. A scaffold
derived from a compound set is a substructure
characteristic of the compounds in the set. Often,
a compound set scaffold is thought of as the
MCS of that set. More informally scaffolds can
be thought of as the common, distinguishing
“core” of a compound
substructures are scaffolds positively correlated
with favorable behavior, i.e. scaffolds present
preferentially in compounds with a desired
biological profile. For a more thorough review
of chemical substructure mining see [4].
are induced
set. Privileged
III. MULTI-OBJECTIVE OPTIMIZATION
A large class of optimization problems need to
simultaneously achieve
conflicting objectives. In contrast to single-
objective problems where optimization methods
explore the feasible search space to find the
single best solution, in multi-objective settings
no best solution can be found that outperforms
all others in every criterion [5]. Instead, multiple
“best” solutions exist representing the range of
possible compromises of the objectives [1].
These solutions, known as non-dominated, have
no other solutions that are better than them in all
of the objectives considered. The set of non-
dominated solutions is also known as the Pareto-
front or the tradeoff surface. Fig. 1 illustrates the
concept of non-dominated solutions and the
Pareto-front in a bi-objective problem.
Problems that require the accommodation of
multiple objectives are widely known as multi-
objective problems
optimization problems [6]. Traditionally MOPs
have been simplified, either by ignoring all
objectives but one, or, by aggregating them.
Multi-objective optimization (MOOP) methods
follow a different approach that is founded on
compromises and tradeoffs among the various
objectives to be met. The aim of MOOP methods
is to discover a set of satisfactory compromise
solutions and, through them, the globally
multiple, often
(MOP) or “vector”
optimal solution(s) by optimizing numerous
dependent properties [1]. A major benefit of
MOOP methods is
corresponding to one objective can be avoided
by consideration of
simultaneously, thereby
objective dead-ends.
that local optima
all the objectives
single escaping

Fig. 1. A MOP with two minimization objectives and a set of
solutions (circles). Non-dominated solutions are labeled ‘0’.
In recent years, Evolutionary Algorithms
(EAs) have been used extensively for MOPs
with several multi-objective optimization EAs
(MOEA) cited in the literature [3], [7], [8].
MOEAs are particularly attractive since their
population-based approach
simultaneous search of multiple search space
regions and thus the identification of numerous
Pareto-solutions in a single run. Additionally,
since EAs impose no constraints on the
morphology of the search space, they are
suitable for complex, multimodal search spaces
with various local optima such as the ones
typically found in MOPs [9].
enables the
IV. METHOD
Recently, we proposed the Multi-objective
Evolutionary Graph Algorithm (MEGA) that
combines evolutionary techniques with graph
data structures to directly manipulate graphs and
perform a global search for promising solutions
[3]. MEGA has been designed to enable the use
of problem-specific knowledge and local search
techniques, to improve performance and
scalability. This section briefly describes the
algorithm and elaborates on some new features
implemented. A more detailed description is
found in [3].
MEGA operates on two population sets, the
normal, working population and the secondary
population or the Pareto-archive. The former
population consists of the individuals subjected
to objective performance calculation and
Proceedings of the 9th International Conference on Information Technology
and Applications in Biomedicine, ITAB 2009
November 5-7, 2009
Larnaca, Cyprus

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obtained through evolution in a s
The latter supports a form of el
preserving promising solutions fo
evolution and, ensuring that th
approximation will contain the
found.
MEGA requires the supply
molecular building blocks, the
objectives to be used for scoring
a set of attributes controlling
crossover methods and probab
selection method, hard filters
elimination, etc. Optionally, a set
used as the initial working popu
supplied as well. The supplied
create graph-based chromosomes
list of building block objects
additional internal data structure
algorithm execution. At this sta
archive of solutions intended
secondary population is also creat
The first phase of the algorith
objectives on the working popula
list of scores for each individu
scores may be used for the
solutions with values outside the
by the corresponding active hard
next step, the two populations
secondary, are merged and the i
of scores is subjected to a
procedure to set the rank of each
combined population forms the
population. The algorithm then
calculate an efficiency score for
using a novel methodology that o
parameter and objective space. Th
employs an elaborate niching m
performs diversity analysis of t
based on the genotype, i.e. th
graph structure, and subsequen
efficiency score that takes into ac
Fig
single iteration.
litism aimed at
ound throughout
e final Pareto-
best solutions
of a set of
e implemented
the graphs and
mutation and
bilities, parent
s for solution
of graphs to be
ulation may be
data is used to
s, to construct a
and to initiate
es required for
ge the external
to store the
ted.
hm applies the
ation to obtain a
ual. The list of
elimination of
e range allowed
d filters. In the
, working and
individuals’ list
Pareto-ranking
individual. The
e new working
n proceeds to
each individual
operates both in
he methodology
mechanism that
the population,
he chromosome
ntly assigns an
ccount both the
Pareto-rank and the diver
efficiency score calculati
fine tuned by user supplie
the parameter space, the
balance between the two
for efficiency calculation
on the phenotype, i.e. the
also been
assessment comparisons. T
each individual is then use
archive. The current Pare
with a subset of the wo
favors individuals with hig
low domination rank an
graph diversity. Note that
selected is limited by a use
Following the update
MEGA checks for the term
satisfied the process termi
is not the case the proces
parent subset population
population set using the “r
the efficiency scores of th
The parents are then subj
crossover according to the
by the user. The new w
formed by merging th
population and the newly
crossover children. The p
shown in Fig. 2.
MEGA incorporates he
exploitation of existin
knowledge. The heuristics
the weights associated wi
provided and result in fa
increased weight. Additio
the Pareto-front approxim
self-adapt certain attrib
optimization process.
identifies underrepresented
front and adjusts the bu
implemente
g. 2: A diagram of the MEGA algorithm

rsity analysis [3]. The
ion technique can be
ed parameters to favor
objective space or to
o. Traditional methods
operating exclusively
e objective space, have
ed for performance
The efficiency score of
ed to update the Pareto-
eto-archive is replaced
orking population that
gh efficiency score, i.e.
nd high chromosome
t the size of the subset
er-supplied parameter.
of the Pareto-archive
mination conditions; if
inates. However, if this
ss moves to select the
n from the combined
roulette” method [1] on
he candidate solutions.
jected to mutation and
e probabilities indicated
working population is
he original working
produced mutants and
process then iterates as
euristics to enable the
ng problem specific
s involve the usage of
ith the building blocks
avoring those with an
onally, the progress of
mation is monitored to
butes controlling the
Specifically, MEGA
d regions in the Pareto-
uilding blocks weights
Proceedings of the 9th International Conference on Information Technology
and Applications in Biomedicine, ITAB 2009
November 5-7, 2009
Larnaca, Cyprus

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favoring the generation of solutions from certain
regions of the space.
V. APPLICATION
The validation of MEGA was performed
through application on the multi-objective DND
problem. In the next sections DND is described
and details about the application are provided.
A. De Novo Design
De-novo drug (or ligand) design is an attempt
to generate ligands from scratch based only on
information about the pharmaceutical target site
or known ligands [3]. Effectively, DND methods
face the task of exploring a chemical search
space estimated to be in the order of 1060 [10].
Such space cannot possibly be fully enumerated
and so powerful search methods need to be
applied to detect the best possible solutions in a
limited amount of time.
DND algorithms proposed in the literature
typically use an evolutionary algorithm related
technique for searching and a set of molecular
fragments as building blocks. The use of a
predefined collection of molecular fragments is
sometimes combined with the identification of
reaction points and synthetic rules that, when
used, increase the synthetic feasibility potential
of the designed chemical structures. Most
methods are designed to accommodate a single
objective either predicted binding affinity to a
known protein target or similarity to a known
ligand. MOOP-based methodologies are limited
to [3] and [11] although the need for their greater
adoption is gaining support within the drug
discovery community [5] [9].
B. Experimental Design
The experiments performed involved the
design of selective Estrogen Receptors (ER), i.e.
ligands that bind to ER-β and not ER-α. All runs
were performed on a computer equipped with an
Intel Core 2 Duo 3 GHz processor and 2GB of
memory. An account of the tests performed and
the results obtained follows.
Datasets. Two datasets were used during the
MEGA tests performed. Dataset 1, a set of well-
known ER ligands, contains 7 compounds, 5
with increased selectivity to ER-β and 2 with
selectivity to ER-α. Dataset 2 is an ER-inhibitor
dataset obtained from Pubchem [12]. The dataset
consists of 86098 compounds tested on both ER-
α (Bioassay 629) and ER-β (Bioassay 633).
Building Blocks. A single collection of 51123
building blocks was used for all the tests
performed. The building blocks were obtained
via fragmentation of Dataset 2 described above
with the substructure mining tool provided by
[13]. Building block weights were assigned
according to the activity labels of the molecules
containing it.
C. Objectives
The application of the algorithm relied on the
encoding of two ligand-based objectives that
measured the average similarity of a query
molecule to known ligands. Similarity was
calculated using the tool Fuzzee [14]. The
specific method used operates on abstractions of
molecular graphs that replace atoms with
molecular features to produce the so-called
feature graphs. The actual similarity is calculated
in a pair-wise manner by first aligning the
feature graphs of two molecules, identifying
common features and then applying the
Tanimoto similarity measure [15]. The two
objectives encoded measured similarity of a
given query molecule to a set of ER-α selective
and ER-β selective ligands. The experiments
aimed at designing molecules selective to ER-β
and so the algorithm was set so as to maximize
average similarity to the ER-β ligand set and
minimize the average similarity to the ER-α
ligand set. A set of hard filters based on
chemical structure objectives was also applied in
order to remove potentially problematic designs
from further consideration.
D. Algorithmic Settings
The experiments performed aimed to measure
the performance of the MEGA, MOGA and
SPEA algorihms on the selectivity problem
described previously. Tests using MOGA and
SPEA served to assess the performance of
MEGA in comparison to commonly used
algorithms from the MOOP field. MOGA [8]
was selected since it is one of the earliest, most
commonly cited algorithms in the MOOP field.
SPEA [7] is a more recent method that
popularized the use of the Pareto-archive in
MOEA algorithms. The experimental settings
used population sizes 25, 50 and 100 and 100
generations. For each combination of input
Proceedings of the 9th International Conference on Information Technology
and Applications in Biomedicine, ITAB 2009
November 5-7, 2009
Larnaca, Cyprus

Page 5

parameter settings 5 runs were pe
different initial population sets. In
the initial population was selecte
defined data set. Runs were perf
roulette parent selection mechan
mutation and crossover. Mutati
was set at 0.25 and crossover at
of MEGA and SPEA the max
archive set was set to 1000
assessment of the results from
place through a post-process
included the calculation of
approximation set hypervolume
chromosome/structural diversity.
calculated by averaging the Eucl
of each solution to all other so
proposed set, using atom-pair des
the molecules involved. MEGA n
to balance between the diversit
and objective space.
E. Results
Experimental results indicate th
generate solution sets consisting
diverse solutions characterized
similarity to ER-β ligands over
Throughout our tests MEGA
compare favorably with those
MOGA and SPEA.
Fig. 3: Hypervolume performance for
SPEA (LSPEA), and MOGA for populat
correspond to better performance.
Fig. 3 presents the hypervolum
plots for MEGA, MOGA and
approximations produced by ME
consistently perform better th
MOGA. A similar trend has bee
all population sizes examined.
Fig. 4 presents the results o
calculations for the three algor
parameter and objective space. I
the plot that MEGA, with its
mechanism specifically designe
structural diversity, can produce
with increased diversity both in
erformed, using
n each test case
ed from a user-
formed with the
nism using both
ion probability
1.0. In the case
ximum Pareto-
0. Performance
each run took
ing step that
the Pareto-
e [7] and the
The latter was
lidean distances
olutions in the
scriptors [16] of
niching was set
y in parameter
hat MEGA can
g of structurally
by increased
ER-α ligands.
solution sets
obtained using

MEGA (eMEGA),
tion 25. Low values
me measure box
SPEA. Pareto-
EGA and SPEA
han those of
en observed for
f the diversity
rithms both in
It is clear from
unique niching
ed to promote
e solution sets
parameter and
objective space. The sets
and SPEA, while of comp
measured by hypervolu
diversity measures where
successful.
Fig. 4: Comparison of the
approximations produced by M
parameter (genotype) space an
space. Higher values correspond
Overall, the results of M
ability of the algorithm to
graph space given clear
solution scoring. In comp
used MOOP algorithms
outperforms MOGA, the
technique in DND, and co
SPEA since it produces
sets of similar quality but g
Time requirements for
runs were sufficiently rea
of MEGA with population
took approximately 40 min
VI. CONC
The research presented
work on MEGA, a new
objective algorithms that
specifically for the OGD
graphs for solution repre
available knowledge th
privileged fragment bu
algorithm uses a unique
developed to preserve chro
elitism in the form of a P
loss of promising solution
from the application of M
and challenging problem
selective ligands indicat
produced solutions havin
similarity to ER-β ligands
that its results comp
established MOOP method

s proposed by MEGA
parable performance as
ume differ in both
MEGA is clearly more

diversity of the Pareto-
MEGA, SPEA and MOGA in
nd in objective (phenotype)
to greater diversity.
MEGA demonstrate the
o explore the chemical
objectives to use for
parison with commonly
MEGA consistently
most commonly used
ompares favorably with
Pareto-approximation
greater diversity.
the execution of the
asonable. A typical run
n 50 and 100 iterations
nutes.
CLUSION
d builds on our recent
class of hybrid multi-
t have been designed
problem. MEGA uses
esentation and exploits
hrough the use of
uilding blocks. The
e niching mechanism
omosome diversity and
Pareto archive, to avoid
ns. The results obtained
MEGA to the significant
m of designing ER-β
te that the algorithm
ng significantly higher
than ER-α ligands and
pare favorably with
dologies.
Proceedings of the 9th International Conference on Information Technology
and Applications in Biomedicine, ITAB 2009
November 5-7, 2009
Larnaca, Cyprus

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Our future work plans focus mainly on the
inclusion of additional knowledge into the
algorithm. We plan to add further domain
knowledge exploitation capabilities to MEGA in
the form of rules, and local search techniques.
An additional, ongoing direction of research,
investigates the parallelization of the algorithm
implementation to enable the application of the
system on large scale distributed systems.
ACKNOWLEDGMENT
The authors would like to thank Dr. J.
Apostolakis for providing access to the Chil
chemoinformatics software suite and Prof. E.
Mikros for supplying his expert opinion on
estrogen receptors and their inhibitors.
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Formulation, Discussion and
Ltd. Nicosia, Cyprus.
Proceedings of the 9th International Conference on Information Technology
and Applications in Biomedicine, ITAB 2009
November 5-7, 2009
Larnaca, Cyprus