Missouri Estimation of Distribution Algorithms Laboratory
Analysis of Estimation of Distribution Algorithms and Genetic Algorithms on NK Land-
scapes
Martin Pelikan
MEDAL Report No. 2008001
January 2008
Abstract
This study analyzes performance of several genetic and evolutionary algorithms on randomly generated NK fitness
landscapes with various values of n and k. A large number of NK problem instances are first generated for each
n and k, and the global optimum of each instance is obtained using the branch-and-bound algorithm. Next, the
hierarchical Bayesian optimization algorithm (hBOA), the univariate marginal distribution algorithm (UMDA), and
the simple genetic algorithm (GA) with uniform and two-point crossover operators are applied to all generated
instances. Performance of all algorithms is then analyzed and compared, and the results are discussed.
Keywords
NK fitness landscape, hierarchical BOA, genetic algorithm, branch and bound, performance analysis, scalability,
local search, crossover.
Missouri Estimation of Distribution Algorithms Laboratory (MEDAL)
Department of Mathematics and Computer Science
University of Missouri-St. Louis
One University Blvd., St. Louis, MO 63121
E-mail: medal@cs.umsl.edu
WWW: http : //medal . cs . umsl . edu/
Analysis of Estimation of Distribution Algorithms and Genetic
Algorithms on NK Landscapes
Martin Pelikan
Missouri Estimation of Distribution Algorithms Laboratory (MEDAL)
Dept. of Math and Computer Science, 320 CCB
University of Missouri at St. Louis
One University Blvd., St. Louis, MO 63121
pelikan@cs . umsl . edu
Abstract
This study analyzes performance of several genetic and evolutionary algorithms on randomly
generated NK fitness landscapes with various values of n and k. A large number of NK prob-
lem instances are first generated for each n and &, and the global optimum of each instance is
obtained using the branch-and-bound algorithm. Next, the hierarchical Bayesian optimization
algorithm (hBOA), the univariate marginal distribution algorithm (UMDA), and the simple
genetic algorithm (GA) with uniform and two-point crossover operators are applied to all gen-
erated instances. Performance of all algorithms is then analyzed and compared, and the results
are discussed.
Keywords: NK fitness landscape, hierarchical BOA, genetic algorithm, branch and bound, per-
formance analysis, scalability, local search, crossover.
1 Introduction
NK fitness landscapes (Kauffman, 1989; Kauffman, 1993) were introduced by Kauffman as tunable
models of rugged fitness landscape. An NK landscape is a function defined on binary strings of
fixed length and is characterized by two parameters: (1) n for the overall number of bits and (2) k
for the neighborhood size. For each bit, k neighbors are specified and a function is given that
determines the fitness contribution of the bit and its neighbors. Usually, the function for each bit
is given as a lookup table of size 2 /c+1 (one value for each combination of the bit and its neighbors),
and both the neighbors as well as the subfunction lookup tables are initialized randomly in some
way.
NK landscapes are NP-complete for A: > 1, although some variants of NK landscapes are
polynomially solvable and there exist approximation algorithms for other cases (Wright, Thompson,
& Zhang, 2000; Gao & Culberson, 2002; Choi, Jung, & Kim, 2005). Nonetheless, NK landscapes
remain a challenge for any optimization algorithm and they are also interesting from the perspective
of complexity theory and computational biology; that is why since their inception NK landscapes
have attracted researchers in all these areas (Kauffman, 1993; Altenberg, 1997; Wright, Thompson,
& Zhang, 2000; Gao & Culberson, 2002; Aguirre & Tanaka, 2003; Choi, Jung, & Kim, 2005).
This paper presents an in-depth empirical performance analysis of various genetic and evolu-
tionary algorithms on NK landscapes with varying n and k. For each value of n and &;, a large
number of problem instances are first generated. Then, the branch-and-bound algorithm is applied
1
to each of these instances to provide a guaranteed global optimum of this instance. Although the
application of branch and bound limits the size of problems that we can study, one of the primary
goals was to ensure that we are able to verify the global optimum of each instance for every algo-
rithm considered in this study. Several genetic and evolutionary algorithms are then applied to all
generated problem instances and their performance is analyzed and compared. More specifically,
we consider the hierarchical Bayesian optimization algorithm (hBOA), the univariate marginal dis-
tribution algorithm (UMDA), and the simple genetic algorithm (GA) with bit-flip mutation, and
uniform or two-point crossover operator. Additionally, GA without any crossover is considered.
The results provide insight into the difficulty of NK landscapes with respect to the parameters n
and k and performance differences between all compared algorithms. Several interesting avenues
for future work are outlined.
The paper starts by describing NK landscapes and the branch-and-bound algorithm used to
verify the global optima of generated NK landscapes in section 2. Section 3 outlines compared
algorithms. Section 4 presents experimental results. Section 5 discusses future work. Finally,
section 6 summarizes and concludes the paper.
2 NK Landscapes
This section describes NK landscapes and a method to generate random problem instances of NK
landscapes. Additionally, the section describes the branch-and-bound algorithm, which was used
to obtain global optima of all NK problem instances considered in this paper. Branch and bound is
a complete algorithm and it is thus guaranteed to find the true global optimum; this was especially
useful for scalability experiments and performance analyses of different evolutionary algorithms.
Nonetheless, branch and bound requires exponential time and thus the size of instances it can solve
in practical time is severely limited.
2.1 Problem Definition
An NK fitness landscape (Kauffman, 1989; Kauffman, 1993) is fully defined by the following com-
ponents:
• The number of bits, n.
• The number of neighbors per bit, k.
• A set of k neighbors II(JQ) for the i-th bit, JQ, for every i E {0, . . . , n — 1}.
• A subfunction fi defining a real value for each combination of values of X{ and II (JQ) for
every i E {0, . . . , n — 1}. Typically, each subfunction is defined as a lookup table with 2 k+1
values.
The objective function f nk to maximize is then defined as
n-l
fnk(Xo,Xi, . . . , X n -i) — y jfi(Xi,IL(Xj)).
2=0
The difficulty of optimizing NK landscapes depends on all of the four components defining an
NK problem instance. One useful approach to analyzing complexity of NK landscapes is to focus on
the influence of k on problem complexity. For k = 0, NK landscapes are simple unimodal functions
similar to onemax or binint, which can be solved in linear time and should be easy for practically
any genetic and evolutionary algorithm. The global optimum of NK landscapes can be obtained in
polynomial time (Wright, Thompson, & Zhang, 2000) even for k = 1; on the other hand, for k > 1,
the problem of finding the global optimum of unrestricted NK landscapes is NP-complete (Wright
et al., 2000). The problem becomes polynomially solvable with dynamic programming even for
k > 1 if the neighbors are restricted to only adjacent string positions (using circular strings) (Wright
et al., 2000) or if the subfunctions are generated according to some distributions (Gao & Culberson,
2002). For unrestricted NK landscapes with k > 1, a polynomial-time approximation algorithm
exists with the approximation threshold 1 — l/2 fc+1 (Wright et al., 2000).
2.2 Generating Random NK Problem Instances
Typically, both the neighbors as well as the lookup tables defining the subfunctions are generated
randomly. In this paper, for each string position JQ, we first generate a random set of k neighbors
where each string position except for Xi is selected with equal probability. Then, the lookup table
defining fi is generated using the uniform distribution over [0, 1).
Consequently, the studied class of NK landscapes is NP-complete for any k > 1. Since the case
for k = 1 is extremely simple to solve, we only considered k > 1; specifically, we considered k = 2
to 6 with step 1. To study scalability of various evolutionary algorithms, for each &;, we considered
a range of values of n with the minimum value of n = 20 and the maximum value bounded mainly
by the available computational resources and the scope of the empirical analysis.
2.3 Branch and Bound
The basic idea of branch and bound is to recursively explore all possible binary strings of n bits
using a recursion tree where each level corresponds to one of the bits and the subtrees below each
level correspond to the different values of the bit corresponding to this level. To make the algorithm
more efficient, some subtrees are cut if they can be proven to not lead to any solution that is better
than the best-so-far solution found. While this cannot eliminate the exponential complexity, which
can be expected due to the NP-completeness for NK landscapes with k > 1, it significantly improves
the performance of the algorithm and allows it to solve much larger problem instances than if a
complete recursion tree had to be explored. The branch-and-bound procedure is illustrated in
figure 1.
Before running the branch-and-bound algorithm, we first use a simple hill climber based on
bit-flip mutation with several random restarts to locate high-quality local optima. The best of the
discovered optima is then used as the best-so-far solution when the branch-and-bound algorithm is
started. In the branch-and-bound approach used in this paper, the bits are assigned sequentially
from Xq to X n (there are two subtrees of each node at level i, each corresponding to one value of
Xi), although reordering the bits might improve performance under some conditions.
When processing a node at level i, the best value we can obtain by setting the remaining n — i
bits is given by
max fnk(Xi =xi,...,X n = x n )
x ii ...,x n e{o i i} n - i
where bits xo to Xi-\ are assumed to be fixed to the values defined by the path from the root
of the recursion tree to the current node. If a solution has been found already that has a higher
fitness than this maximum possible value, the processing below the currently processed node does
not have to continue and the remaining unexplored parts of the recursion tree can be explored with
the exception of those parts that have already been cut.
cut
Figure 1: Branch and bound traverses the recursion tree where each each level sets the value of
one bit and each leaf thus corresponds to one instance of all n bits. Subtrees that lead to solutions
that cannot improve the best-so-far solution are cut to improve efficiency.
We also tried another variant of the branch-and-bound algorithm, in which the best value of
the objective function is computed incrementally for subsets containing only the first i bits with
i = 2 to ?2. While this approach has been very efficient in solving instances of the Sherrington-
Kirkpatrick spin glass model (Hartwig, Daske, & Kobe, 1984), for NK landscapes, the algorithm
described earlier performed more efficiently.
The aforedescribed branch-and-bound algorithm is complete and it is thus guaranteed to find
the global optimum of any problem instance. Nonetheless, the complexity of branch and bound can
be expected to grow exponentially fast and solving large NK instances becomes intractable with
this algorithm. For example, for k = 2, the proposed branch-and-bound algorithm was fast enough
to solve ten thousand unique instances of n < 52; for k = 6, the algorithm was fast enough to
deal with instances of size n < 36. While the evolutionary algorithms presented in the next section
should be capable of reliably solving larger instances, their convergence to the global optimum
cannot be guaranteed; nonetheless, section 5 discusses how to extend this study to deal with larger
NK problem instances, which are intractable with the branch-and-bound algorithm.
3 Compared Algorithms
This section outlines the optimization algorithms discussed in this paper: (1) the hierarchical
Bayesian optimization algorithm (hBOA) (Pelikan & Goldberg, 2001; Pelikan, 2005), (2) the uni-
variate marginal distribution algorithm (UMDA) (Miihlenbein & Paafi, 1996), and (3) the genetic
algorithm (GA) (Holland, 1975; Goldberg, 1989). Additionally, the section describes the deter-
ministic hill climber (DHC) (Pelikan & Goldberg, 2003), which is incorporated into all compared
algorithms to improve their performance. In all compared algorithms, candidate solutions are repre-
sented by binary strings of n bits and a niching technique called restricted tournament replacement
(RTR) (Harik, 1995) is used for effective diversity maintenance.
3.1 Genetic Algorithm
The genetic algorithm (GA) (Holland, 1975; Goldberg, 1989) evolves a population of candidate
solutions typically represented by fixed-length binary strings. The first population is generated at
random. Each iteration starts by selecting promising solutions from the current population. We
use binary tournament selection. New solutions are created by applying variation operators to the
population of selected solutions. Specifically, crossover is used to exchange bits and pieces between
pairs of candidate solutions and mutation is used to perturb the resulting solutions. Here we use
uniform or two-point crossover, and bit-flip mutation (Goldberg, 1989). To ensure effective diver-
sity maintenance, the new candidate solutions are incorporated into the original population using
restricted tournament replacement (RTR) (Harik, 1995). The run is terminated when termination
criteria are met.
3.2 Univariate Marginal Distribution Algorithm (UMDA)
The univariate marginal distribution algorithm (UMDA) (Miihlenbein & Paafi, 1996) also evolves
a population of candidate solutions represented by fixed-length binary strings with the initial pop-
ulation generated at random. Each iteration starts by selecting a population of promising solutions
using any common selection method of genetic and evolutionary algorithms; we use binary tour-
nament selection. Then, the probability vector is learned that stores the proportion of Is in each
position of the selected population. Each bit of a new candidate solution is then set to 1 with
the probability equal to the proportion of Is in this position; otherwise, the bit is set to 0. Con-
sequently, the variation operator of UMDA preserves the proportions of Is in each position while
decorrelating different string positions. The new candidate solutions are incorporated into the
original population using RTR. The run is terminated when termination criteria are met.
UMDA is an estimation of distribution algorithm (EDA) (Baluja, 1994; Miihlenbein & Paafi,
1996; Larrahaga & Lozano, 2002; Pelikan, Goldberg, & Lobo, 2002). EDAs — also called probabilis-
tic model-building genetic algorithms (PMBGAs) (Pelikan, Goldberg, & Lobo, 2002) and iterated
density estimation algorithms (IDEAs) (Bosman & Thierens, 2000) — replace standard variation
operators of genetic algorithms such as crossover and mutation by building a probabilistic model of
promising solutions and sampling the built model to generate new candidate solutions. The only
difference between the GA and UMDA is in the way the selected solutions are processed to generate
new solutions.
3.3 Hierarchical BOA (hBOA)
The hierarchical Bayesian optimization algorithm (hBOA) (Pelikan & Goldberg, 2001; Pelikan,
2005) is also an EDA and the basic procedure of hBOA is similar to that of the UMDA variant
described earlier. However, to model promising solutions and sample new solutions, Bayesian
networks with local structures (Chickering, Heckerman, & Meek, 1997; Friedman & Goldszmidt,
1999) are used instead of the simple probability vector of UMDA. Similarly as in the considered
GA and UMDA variants, the new candidate solutions are incorporated into the original population
using RTR and the run is terminated when termination criteria are met.
3.4 Deterministic Hill Climber (DHC)
The deterministic hill climber (DHC) is incorporated into GA, UMDA and hBOA to improve their
performance. DHC takes a candidate solution represented by an n-bit binary string on input.
Then, it performs one-bit changes on the solution that lead to the maximum improvement of
solution quality. DHC is terminated when no single-bit flip improves solution quality and the
solution is thus locally optimal. Here, DHC is used to improve every solution in the population
before the evaluation is performed.
4 Experiments
This section describes experiments and presents experimental results. First, problem instances and
experimental setup are discussed. Next, the analysis of hBOA, UMDA and several GA variants is
presented. Finally, all algorithms are compared and the results of the comparisons are discussed.
4.1 Problem Instances
NK instances for k = 2 to k = 6 with step 1 were studied. The only restriction on problem size was
the efficiency of the branch-and-bound algorithm, the complexity of which grew very fast with n.
For k = 2, we considered n = 20 to n = 52 with step 2; for k = 3, we considered n = 20 to n = 48
with step 2; for k = 4, we considered n = 20 to n = 40 with step 2; for k = 5, we considered n = 20
to n = 38 with step 2; finally, for k = 6, we considered n = 20 to n = 32 with step 2.
For each combination of n and fc, we generated 10,000 random problem instances and for each
instance we used the branch-and-bound algorithm to locate the global optimum. Then, we applied
hBOA, UMDA and several GA variants to each of these instances and collected empirical results,
which were subsequently analyzed. That means that overall 600,000 unique problem instances were
generated and all of them were tested with every algorithm included in this study.
4.2 Compared Algorithms
The following list summarizes the algorithms included in this study:
(i) Hierarchical BOA (hBOA).
(ii) Univariate marginal distribution algorithm (UMDA).
(hi) Genetic algorithm with uniform crossover and bit-flip mutation.
(iv) Genetic algorithm with two-point crossover and bit-flip mutation.
(v) Genetic algorithm with bit-flip mutation and no crossover.
(vi) Hill climbing (results omitted due to inferior performance and infeasible computation).
4.3 Experimental Setup
To select promising solutions, binary tournament selection is used. New solutions (offspring) are
incorporated into the old population using RTR with window size w = min{n, N/5} as suggested
in Pelikan (2005). In hBOA, Bayesian networks with decision trees (Chickering et al., 1997; Fried-
man & Goldszmidt, 1999; Pelikan, 2005) are used and the models are evaluated using the Bayesian-
Dirichlet metric with likelihood equivalence (Heckerman et al., 1994; Chickering et al., 1997) and
a penalty for model complexity (Friedman & Goldszmidt, 1999; Pelikan, 2005). All GA variants
use bit-flip mutation with the probability of flipping each bit p m = 1/n. Two common crossover
operators are considered in a GA: two-point and uniform crossover. For both crossover operators,
the probability of applying crossover is set to 0.6. To emphasize the importance of using crossover,
the results for GA without any crossover are also included, where only bit-flip mutation is used.
A stochastic hill climber with bit-flip mutation has also been considered in the initial stage, but
the performance of this algorithm was far inferior compared to any other algorithm included in
the comparison and it was intractable to solve most problem instances included in the comparison;
that is why the results for this algorithm are omitted.
For each problem instance and each algorithm, an adequate population size is approximated
with the bisection method (Sastry, 2001; Pelikan, 2005), which estimates the minimum population
size required for reliable convergence to the optimum. Here, the bisection method finds an adequate
population size for the algorithms to find the optimum in 10 out of 10 independent runs. Each run
is terminated when the global optimum has been found. The results for each problem instance com-
prise of the following statistics: (1) the population size, (2) the number of iterations (generations),
(3) the number of evaluations, and (4) the number of flips of DHC. For each value of n and fc, all
observed statistics were averaged over the 10,000 random instances. Since for each instance, 10
successful runs were performed, for each n and k and each algorithm the results are averaged over
100,000 successful runs. Overall, for each algorithm, the results correspond to 6,000,000 successful
runs on a total of 600,000 unique problem instances.
4.4 Performance Analysis
Figure 2 shows the average performance statistics for hBOA on NK problem instances for k = 2
to k = 6. As expected, performance of hBOA gets worse with increasing k. More specifically, the
population size, the number of iterations, the number of evaluations, and the number of DHC flips
appear all to grow exponentially with k. For a fixed &;, the time complexity appears to grow with n
slightly faster than polynomially regardless of whether it is measured by the number of evaluations
or the number of flips.
Figure 3 shows the average performance statistics for UMDA. Similarly as with hBOA, time
complexity of UMDA grows exponentially fast with k and its growth with n for a fixed k appears
to be slightly faster than polynomial.
Figures 4, 5 and 6 show the average performance statistics for all three GA variants. Similarly
as with hBOA and UMDA, time complexity of all GA variants grows exponentially fast with k and
its growth with n for a fixed k is slightly faster than polynomial.
4.5 Comparison of All Algorithms
To compare performance of algorithms A and £>, for each problem instance, we can compute the
ratio of the number of evaluations required by A and the number of evaluations required by B;
analogically, we can compute the ratio of the number of flips required by A and the number of
flips required by B. Then, the ratios can be averaged over all instances with specific n and k. If
A performs better than £>, the computed ratios should be smaller than 1; if A performs the same
as £>, the ratios should be about 1; finally, if the A performs worse than I?, the ratios should be
greater than 1.
A comparison based on the aforementioned ratio was computed for each pair of algorithms
studied in this work. To make the results easier to read, the superior algorithm was typically used
as the second algorithm in the comparison (in the denominator of the ratios), so that the ratios
should be expected to be greater than 1.
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12
Figure 7 compares performance of GA with two-point crossover and that of hBOA. Figure 8
compares performance of GA with uniform crossover and that of hBOA. Figure 12 compares perfor-
mance of GA with two-point crossover and that of GA with uniform crossover. Finally, figures 13
and 14 compare performance of GAs with and without crossover.
One of the important trends to observe in the results of the comparisons is the change in the
two ratios with problem size. In most cases, when one algorithm outperforms another one, the
differences become more significant as the problem size increases. In some cases, although one
algorithm outperforms another one on small problems, because of the observed dynamics with
problem size, we can expect the situation to reverse for large problems.
The comparisons based on the number of evaluations and the number of flips can be summarized
as follows:
hBOA. While for small values of &;, hBOA is outperformed by other algorithms included in the
comparison, as k increases, the situation changes rapidly. More specifically, for larger &;,
hBOA outperforms all other algorithms and its relative performance with respect to other
algorithms improves with increasing problem size. The larger the fc, the more favorably hBOA
compares to other algorithms.
GA with uniform crossover. GA with uniform crossover performs better than GA with two-
point crossover and UMDA regardless of k and its relative performance with respect to these
algorithms improves with problem size. However, as mentioned above, for larger values of
&;, GA with uniform crossover is outperformed by hBOA and the factor by which hBOA
outperforms GA with uniform crossover grows with problem size.
GA with two-point crossover. GA with two-point crossover performs worse than hBOA and
GA with uniform crossover for larger values of &;, but it still outperforms UMDA with respect
to the number of flips, which is the most important performance measure.
UMDA. UMDA performs worst of all recombination-based algorithms included in the comparison
except for a few cases with small values of k.
Crossover versus mutation. Crossover has proven to outperform pure mutation, which is clear
from all the results. First of all, for the most difficult instances, hBOA — which is a pure
selectorecombinative evolutionary algorithm with no explicit mutation — outperforms other
algorithms with increasing n. Second, eliminating crossover from GA significantly decreases
its efficiency and the mutation-based approaches perform worst of all compared algorithms.
Specifically, GA with no crossover is outperformed by all other variants of GA, and the
stochastic hill climbing is not even capable of solving many problem instances in practical
time.
5 Future Work
There are several interesting ways of extending the work presented in this paper. First of all, the
problem instances generated in this work can be used for analyzing performance of other optimiza-
tion algorithms and comparing different optimization algorithms on a broad class of problems with
tunable difficulty. Second, the class of problems considered in this study can be extended sub-
stantially using genetic and evolutionary algorithms with adequate settings for solving instances
unsolvable with branch and bound. Although the global optimum would no longer be guaranteed,
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better the performance of uniform crossover compared to two-point crossover.
16
30 35 40
Problem size
30 35 40
Problem size
Figure 13: Comparison of GA with uniform crossover and GA with no crossover (mutation only)
with respect to the number of evaluations and the number of flips. The comparison is visualized by
the average ratio of the number of evaluations (number of flips) required by GA without crossover
and the number of evaluations (number of flips) required by GA with uniform crossover. The
greater the ratio, the better the performance of uniform crossover compared to mutation only.
4.5
<3.5
I 3^
> 2.5h
CD
O
<
1.5-
20
25
-B-k=6
-e-k=5
-«f-k=4
^-k=3
-A-k=2
f
>#// ■
id^
30
35 40
Problem size
45
50
35 40
Problem size
Figure 14: Comparison of GA with two-point crossover and GA with no crossover (mutation only)
with respect to the number of evaluations and the number of flips. The comparison is visualized by
the average ratio of the number of evaluations (number of flips) required by GA without crossover
and the number of evaluations (number of flips) required by GA with two-point crossover. The
greater the ratio, the better the performance of two-point crossover compared to mutation only.
17
methods can be devised that still guarantee that the global optimum is found reliably. Finally,
other probability distributions for generating NK problem instances can be considered to provide
further insights into the difficulty of various classes of NK landscapes and the benefits and costs of
using alternative optimization strategies in each of these classes.
6 Summary and Conclusions
This paper presented an in-depth empirical performance study of several genetic and evolutionary
algorithms on NK landscapes with various values of n and k. Specifically, the algorithms considered
in this work included the hierarchical Bayesian optimization algorithm (hBOA), the univariate
marginal distribution algorithm (UMDA), and the simple genetic algorithm (GA) with bit-flip
mutation, and two-point or uniform crossover. Additionally, GA with bit-flip mutation but no
crossover was considered. For each value of n and fc, a large number of NK instances were generated
and solved with the branch-and-bound algorithm, which is a complete algorithm that is guaranteed
to find the global optimum. Performance of all algorithms was analyzed and compared, and the
results were discussed.
The main contributions of this work are summarized in what follows. First of all, NK landscapes
represent an important class of test problems and despite that there has been practically no work
on using advanced estimation of distribution algorithms (EDAs) on NK landscapes. This work
provides many experimental results on one advanced and one simple EDA, and it shows that
advanced EDAs can significantly outperform other genetic and evolutionary algorithms on NK
landscapes for larger values of k. Second, most studies concerned with NK landscapes do not verify
the global optimum of the considered problem instances and it is thus often difficult to interpret the
results and evaluate their importance. In this study, the global optimum of each instance is verified
with the complete branch-and-bound algorithm. Third, while the difficulty of NK landscapes can
be expected to vary substantially from instance to instance, most studies presented in the past
used only a limited sample of problem instances; here we provide an in-depth study where about
600,000 unique problem instances are considered. Finally, the results in this paper are not based
on only one evolutionary algorithm; instead, we consider several qualitatively different evolutionary
algorithms, providing insight into the comparison of genetic algorithms and EDAs, as well as into
the comparison of the mutation-based and recombination-based evolutionary algorithms.
Acknowledgments
This project was sponsored by the National Science Foundation under CAREER grant ECS-
0547013, by the Air Force Office of Scientific Research, Air Force Materiel Command, USAF,
under grant FA9550-06-1-0096, and by the University of Missouri in St. Louis through the High
Performance Computing Collaboratory sponsored by Information Technology Services, and the
Research Award and Research Board programs.
The U.S. Government is authorized to reproduce and distribute reprints for government pur-
poses notwithstanding any copyright notation thereon. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the authors and do not necessarily reflect
the views of the National Science Foundation, the Air Force Office of Scientific Research, or the
U.S. Government. Some experiments were done using the hBOA software developed by Martin
Pelikan and David E. Goldberg at the University of Illinois at Urbana-Champaign and most exper-
iments were performed on the Beowulf cluster maintained by ITS at the University of Missouri in
18
St. Louis.
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