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Paintings, Polygons and Plant
Propagation
Misha Paauw and Daan van den Berg(B
)
Institute for Informatics, University of Amsterdam, Science Park 904,
1098XH Amsterdam, The Netherlands
mishapaauw@gmail.com,d.vandenberg@uva.nl
Abstract. It is possible to approximate artistic images from a limited
number of stacked semi-transparent colored polygons. To match the tar-
get image as closely as possible, the locations of the vertices, the drawing
order of the polygons and the RGBA color values must be optimized for
the entire set at once. Because of the vast combinatorial space, the rel-
atively simple constraints and the well-defined objective function, these
optimization problems appear to be well suited for nature-inspired opti-
mization algorithms.
In this pioneering study, we start off with sets of randomized poly-
gons and try to find optimal arrangements for several well-known paint-
ings using three iterative optimization algorithms: stochastic hillclimb-
ing, simulated annealing and the plant propagation algorithm. We discuss
the performance of the algorithms, relate the found objective values to
the polygonal invariants and supply a challenge to the community.
Keywords: Paintings ·Polygons ·Plant propagation algorithm ·
Simulated annealing ·Stochastic hillclimbing
1 Introduction
Since 2008, Roger Johansson has been using “Genetic Programming” to repro-
duce famous paintings by optimally arranging a limited set of semi-transparent
partially overlapping colored polygons [1]. The polygon constellation is rendered
to a bitmap image which is then compared to a target bitmap image, usually a
photograph of a famous painting. By iteratively making small mutations in the
polygon constellation, the target bitmap is being matched ever closer, resulting
in an ‘approximate famous painting’ of overlapping polygons.
Johansson’s program works as follows: a run gets initialized by creating a
black canvas with a single random 3-vertex polygon. At every iteration, one or
more randomly chosen mutations are applied: adding a new 3-vertex polygon to
the constellation, deleting a polygon from the constellation, changing a polygon’s
RGBA-color, changing a polygon’s place in the drawing index, adding a vertex
to an existing polygon, removing a vertex from a polygon and moving a vertex to
a new location. After one such mutation, the bitmap gets re-rendered from the
c
Springer Nature Switzerland AG 2019
A. Ek´art et al. (Eds.): EvoMUSART 2019, LNCS 11453, pp. 84–97, 2019.
https://doi.org/10.1007/978-3-030-16667-0_6
Paintings, Polygons and Plant Propagation 85
constellation, compared to the target bitmap by calculating the total squared
error over each pixel’s RGB color channel. If it improves, it is retained, otherwise
the change is reverted. We suspect the program is biased towards increasing
numbers of polygons and vertices, but Johansson limits the total numbers of
vertices per run to 1500, the total number of polygons to 250 and the number
of vertices per polygon from 3 to 10. Finally, although a polygon’s RGBA-color
can take any value in the program, its alpha channel (opaqueness) is restricted
to between 30 and 601.
Although the related FAQ honestly enough debates whether his algorithm is
correctly dubbed ‘genetic programming’ or might actually be better considered
a stochastic hillclimber (we believe it is), the optimization algorist cannot be
unsensitive to the unique properties of the problem at hand. The lack of hard
constraints, the vastness of the state space and the interdependency of param-
eters make it a very interesting case for testing (new) optimization algorithms.
But exploring ground in this direction might also reveal some interesting prop-
erties about the artworks themselves. Are some artworks easier approximated
than others? Which algorithms are more suitable for the problem? Are there
any invariants to be found across different artworks?
Today, we present a first set of results. Deviating slightly from Johansson’s
original program, we freeze the numbers of vertices and polygons for each run,
but allow the other variables to mutate, producing some first structural insights.
We consistently use three well-known algorithms with default parameter settings
in order to open up the lanes of scientific discussion, and hopefully create an entry
point for other research teams to follow suit.
2 Related Work
Nature-inspired algorithms come in a baffling variety, ranging in their metaphoric
inheritance from monkeys, lions and elephants to the inner workings of DNA [3]
(for a wide overview, see [4]). There has been some abrasive debate recently
about the novelty and applicability of many of these algorithms [5], which is a
sometimes painful but necessary process, as the community appears to be in tran-
sition from being explorative artists to rigourous scientists. Some nature-inspired
algorithms however, such as the Cavity Method [6,7] or simulated annealing [8],
have a firm foothold in classical physics that stretches far beyond the metaphor
alone, challenging the very relationship between physics and informatics. Others
are just practically useful and although many of the industrial designs generated
by nature-inspired or evolutionary algorithms will never transcend their digital
existence, some heroes in the field make the effort of actually building them
out [9–11]. We need more of that.
The relatively recently minted plant propagation algorithm has also demon-
strated its practical worth, being deployed for the Traveling Salesman Prob-
lem [12,13], the University Course Timetabling Problem [14] and parametric
1Details were extracted from Johansson’s source code [2] wherever necessary.
86 M. Paauw and D. van den Berg
optimization in a chemical plant [12]. It also shows good performance on bench-
mark functions for continuous optimization [12].
The algorithm revolves around the idea that a given strawberry plant off-
shoots many runners in close proximity if it is in a good spot, and few runners
far away if it is in a bad sp ot. Applying these principles to a population of can-
didate solutions (“individuals”) provides a good balance between exploitation
and exploration of the state space. The algorithm is relatively easily imple-
mented, and does not require the procedures to repair a candidate solution from
crossovers that accidentally violated its hard constraints. A possible drawback
however, is that desirable features among individuals are less likely to be com-
bined in a single individual, but these details still await experimental and theo-
retical verification.
Recent history has seen various initiatives on the intersection between nature-
inspired algorithms and art, ranging from evolutionary image transition between
bitmaps [15] to artistic emergent patterns based on the feeding behavior of sand-
bubbler crabs [16] and non-photorealistic rendering of images based on digital
ant colonies [17]. One of the most remarkable applications is the interactive
online evolutionary platform ‘DarwinTunes’ [18], in which evaluation by public
choice provided the selection pressure on a population of musical phrases that
‘mate’ and mutate, resulting in surprisingly catchy melodies.
3 Paintings from Polygons
For this study, we used seven 240 ×180-pixel target bitmaps in portrait or
landscape orientation of seven famous paintings (Fig.1): Mona Lisa (1503) by
Leonardo da Vinci, The Starry Night (1889) by Vincent van Gogh, The Kiss
(1908) by Gustav Klimt, Composition with Red, Yellow and Blue (1930) by Piet
Mondriaan, The Persistance of Memory (1931) by Salvador Dali, Convergence
(1952) by Jackson Pollock, and the only known portrait of Leipzig-based com-
poser Johann Sebastian Bach (1746) by Elias Gottlieb Haussman. Together, they
span a wide range of ages, countries and artistic styles which makes them suit-
able for a pioneering study. Target bitmaps come from the public domain, are
slightly cropped or rescaled if necessary, and are comprised of 8-bit RGB-pixels.
Every polygon in the pre-rendering constellation has four byte sized RGBA-
values: a red, green, blue and an alpha channel for opaqueness ranging from 0
to 255. The total number of vertices v∈{20,100,300,500,700,1000}is fixed
for each run, as is the number of polygons p=v
4. All polygons have at least 3
vertices and, at most v
4+ 3 vertices as the exact distribution of vertices over the
polygons can vary during a run. Vertices in a polygon have coordinate values in
the range from 0 to the maximum of the respective dimension, which is either
180 or 240. Finally, every polygon has a drawing index: in rendering the bitmap
from a constellation, the polygons are drawn one by one, starting from drawing
index 0, so a higher indexed polygon can overlap a lower indexed polygon, but
not the other way around. The polygon constellation is rendered to an RGB-
bitmap by the Python Image Library [19]. This library, and all the programmatic
Paintings, Polygons and Plant Propagation 87
Fig. 1. The paintings used in this study, from a diverse range of countries, eras and
artistic styles. From top left, clockwise: Mona Lisa (1503, Leonardo da Vinci), Com-
position with Red, Yellow, and Blue (1930, Piet Mondriaan), The Kiss (1908, Gustav
Klimt), Portrait of J.S. Bach (1746, Elias Gottlieb Hausmann), The Persistance of
Memory (1931, Salvador Dali), Convergence (1952, Jackson Pollock), and The Starry
Night (1889, Vincent van Gogh).
resources we use, including the algorithms, are programmed in Python 3.6.5 and
are publicly available [20].
After rendering, the proximity of the rendered bitmap to the target bitmap,
which is the mean squared error (MSE) per RGB-channel, is calculated:
180·240·3
i=1
(Renderedi−Target
i)2
180 ·240 (1)
in which Renderediis a Red, Green or Blue channel in a pixel of the rendered
bitmap’s pixel and Target
iis the corresponding channel in the target bitmap’s
pixel. It follows that the best possible MSE for a rendered bitmap is 0, and is
only reached when each pixel of the rendered bitmap is identical to the target
bitmap. The worst possible fitness is 2552·3 = 195075, which corresponds to the
situation of a target bitmap containing only ‘extreme pixels’ with each of the
three RGB-color registers at either 0 or 255, and the rendered bitmap having
the exact opposite for each corresponding color register.
Although it is a rare luxury to know the exact maximal and minimal objective
values (in this case the MSE) for any optimization problem, they’re hardly usable
due to the sheer vastness of the state space |S|of possible polygon constellations.
It takes some special tricks to enable computers to perform even basic operations
on numbers of this magnitude, but nonetheless, some useful bounds can be given
by
88 M. Paauw and D. van den Berg
S=α·(240 ·180)v·(2564)v
4·(v
4)! (2)
in which (240·180)vis the combination space for vertex placement, (2564)v
4rep-
resents all possible polygon colorings and ( v
4)! is the number of possible drawing
orders. The variable αreflects the number of ways the vertices can be distributed
over the polygons. In our case, 3v
4vertices are priorly allocated to assert that
every polygon has at least three vertices, after which the remaining v
4vertices are
randomly distributed over the v
4polygons. This means that for these specifics,
α=P(v
4), in which P is the integer partition function, and |S|can be calculated
exactly from the number of vertices v.
The state space size of the target bitmaps is also known. It is, neglecting
symmetry and rotation, given by
2563(180·240) ≈7.93 ·10312,107 (3)
which reflects the three RGB-values for each pixel in the bitmap. From Eq. 2,
we can derive that a constellation of 39,328 vertices could render to a total of
≈1.81·10312,109 different bitmap images. These numbers are incomprehensibly
large, but nonetheless establish an exact numerical lower bound on the number
of vertices we require to render every possible bitmap image of 180 ×240 pixels.
A feasible upper bound can be practically inferred: if we assign a single square
polygon with four vertices to each pixel, we can create every possible bitmap of
aforementioned dimensions with 180 ·240 ·4 = 172,800 vertices.
Upholding these bounds and assumptions, we would need somewhere between
39,328 en 172,800 vertices to enable the rendering of every possible target bitmap
and thereby guarantee the theoretical reachability of a perfect MSE of 0. These
numbers are nowhere near practically calculable though, and therefore it can
not be said whether the perfect MSE is reachable for vertex numbers lower than
172,800, or even what the best reachable value is. But even if a perfect MSE
was reachable, it is unclear what a complete algorithm for this task would look
like, let alone whether it can do the job within any reasonable amount of time.
What we can do however, is compare the performance of three good heuristic
algorithms, which we will describe in the next section.
4 Three Algorithms for Optimal Polygon Arrangement
We use a stochastic hillclimber, simulated annealing, and the newly developed
plant propagation algorithm to optimize parameter values for the polygon con-
stellations of the seven paintings (Fig.2). The mutation operators are identical
for all algorithms, as is the initialization procedure:
4.1 Random Initialization
Both the stochastic hillclimber and the simulated annealing start off with a sin-
gle randomly initialized polygon constellation, whereas the plant propagation
Paintings, Polygons and Plant Propagation 89
algorithm has a population of M= 30 randomly initialized constellations (or
individuals). Initialization first creates a black canvas with the dimensions of
the target bitmap. Then, vvertices and p=v
4polygons are created, each with
the necessary minimum of three vertices, and the remaining v
4vertices randomly
distributed over the polygons. All vertices in all polygons are randomly placed
on the canvas with 0 ≤x<Xmaxand 0 ≤y<Ymax. Finally, all polygons are
assigned the four values for RBGA by randomly choosing values between 0 and
255. The only variables which are not actively initialized are the drawing indices
i, but the shrewd reader might have already figured out that this is unneces-
sary, as all other properties are randomly assigned throughout the initialization
process.
4.2 Mutations
All three algorithms hinge critically on the concept of a mutation. To maximize
both generality and diversity, we define four types of mutation, all of which are
available in each of the three algorithms:
1. Move Vertex randomly selects a polygon, and from that polygon a random
single vertex. It assigns a new randomly chosen location from 0 ≤x<xMax
and 0 ≤y<yMax.
2. Transfer Vertex: randomly selects two different polygons p1andp2, in
which p1 is not a triangle. A random vertex is deleted from p1 and inserted
in p2 on a random place between two other vertices of p2. Because of this
property, the shape of p2 does not change, though it might change in later
iterations because of operation 1. Note that this operation enables diversifi-
cation of the polygon types but it keeps the numbers of polygons and vertices
constant, facilitating quantitative comparison of the end results.
3. Change Color: randomly chooses either the red, green, blue or alpha channel
from a randomly chosen polygon and assigns it a new value 0 ≤q≤255.
4. Change Drawing Index: randomly selects a polygon and assigns it a new
index in the drawing order. If the new index is lower, all subsequent polygons
get their index increased by one (“moved to the right”). If the new index is
higher, all previous polygons get their index decreased by one (“moved to the
left”).
It follows that for any constellation at any point, there are vpossible move
vertex operators, v
4possible transfer vertex operators, 3v
4possible change color
operators, and v
4possible change drawing index operators. As such, the total
number of possible mutation operators for a polygon constellation of vvertices
is 9v
4.
4.3 Stochastic Hillclimber and Simulated Annealing
A run of the stochastic hillclimber works as follows: first, it initializes a random
constellation of vvertices and v
4polygons as described in Sect. 4.1. Then, it
90 M. Paauw and D. van den Berg
renders the polygon constellation to a bitmap and calculates its MSE respective
to the target bitmap. Each iteration, it selects one of the four mutation types
with probability 1
4, and randomly selects appropriate operands for the mutation
type with equal probability. Then it re-renders the bitmap; if the MSE increases
(which is undesirable), the change is reverted. If not, it proceeds to the next
iteration unless the prespecified maximum number of iterations is reached, in
which case the algorithm halts.
Simulated annealing [8] works in the exact same way as the stochastic hill-
climber, accepting random improvements, but has one important added feature:
whenever a random mutation increases the error on the rendered bitmap, there
is still a chance the mutation gets accepted. This chance is equal to
e
−ΔMS E
T(4)
in which ΔMSE is the increase in error and Tis the ‘temperature’, a variable
depending on the iteration number i. There are many ways of lowering the
temperature, but we use the cooling scheme by Geman and Geman [21]:
T=c
ln(i+1).(5)
This scheme is the only one proven to be optimal [22], meaning that it is guaran-
teed to find the global minimum of the MSE as igoes to infinity, given that the
constant cis “the highest possible energy barrier to be traversed”. To understand
what this means, and correctly implement it, we need to examine the roots of
simulated annealing.
In condensed matter physics, e
−E
kT is known as “Boltzmann’s factor”. It
reflects the chance that a system is in a higher state of energy E, relative to
the temperature. In metallurgic annealing, this translates to the chance of dis-
located atoms moving to vacant sites in the lattice. If such a move occurs, the
system crosses a ‘barrier’ of higher energy which corresponds to the distance the
atom traverses. But whenever the atom reaches a vacant site, the system drops
to a lower energetic state, from which it is unlikelier to escape. More importantly:
it removes a weakness from the lattice and therefore the process of annealing sig-
nificantly improves the quality of the metal lattice. The translation from energy
state to the MSE in combinatorial optimization shows simulated annealing truly
fills the gap between physics and informatics: for the algorithm to escape from
a local minimum, the MSE should be allowed to increase, to eventually find a
better minimum.
So what’s left is to quantify “the highest possible energy barrier to be tra-
versed” for the algorithm to guarantee its optimality. As we have calculated
the upper bound for MSE in Sect. 2,wecansetc= 195075 for our 180 ×240-
paintings, thereby guaranteeing the optimal solution (eventually). The savvy
reader might feel some suspicion along the claim of optimality, especially consid-
ering the open issue on P ?
= NP, but there is no real contradiction here. The snag
is in igoing to infinity; we simply do not have that much time and if we would,
we might just as well run a brute-force algorithm, or even just do stochastic
sampling to find the global minimum. In infinite time, everything is easy.
Paintings, Polygons and Plant Propagation 91
Fig. 2. Typical runs for the stochastic hillclimber (top), the plant propagation algo-
rithm (middle) and simulated annealing (bottom). Polyptychs of rendered bitmaps
illustrate the visual improvement from left to right, while the in-between graphs show
the corresponding decrease in MSE throughout the iterative improvement process of
the respective algorithm.
92 M. Paauw and D. van den Berg
4.4 Plant Propagation
The plant propagation algorithm (PPA) is a relatively new member of the opti-
mization family and the only population-based algorithm we use. First, fit-
ness values are assigned. Normalized to (0,1) within the current population,
this ‘relative normalized fitness assignment’ might be beneficial for these kinds
of optimization problems in which the absolute values of the MSE are (usu-
ally)(practically) unknown. Each individual will then produce between 1 and
nMax = 5 offspring, proportional to its fitness, which will be mutated inversely
proportional to the normalized (relative) fitness. So fitter individuals produce
more offspring with small mutations, and unfitter individuals produce fewer off-
spring with large mutations. The notions of ‘large’ and ‘small’ mutations how-
ever, needs some careful consideration.
In the seminal work on PPA, mutations are done on real-valued dimensions
of continuous benchmark functions [12], which has the obvious advantage that
the size of the mutation can be scaled directly proportional to the (real-valued)
input dimension. An adaptation to a non-numerical domain has also been made
when PPA was deployed to solve the Traveling Salesman Problem [13]. In this
case, the size of the mutation is reflected in the number of successive 2-opts
which are applied to the TSP-tour. It does not, however take into account the
multiple k-opts might overlap, or how the effect of the mutation might differ
from edge to edge, as some edges are longer than others.
So where does it leave us? In this experiment, a largely mutated offspring
could either mean ‘having many mutations’ or ‘having large mutations’. Consult-
ing the authors, we tried to stick as closely to the idea of (inverse) proportionality
and find middle ground in the developments so far. In the benchmark paper, all
dimensions are mutated by about 50% in the worst-case fitness. In the TSP-
paper, the largest mutations vary between 7% and 27% of the edges, depending
on the instance size. We chose to keep the range of mutations maximal, while
assigning the number of mutations inversely proportional to the fitness, resulting
in the following PPA-implementation:
1. Initialization: create a population of M= 30 individuals (randomly initial-
ized polygon constellations).
2. Assign Fitness Values: first, calculate the MSEifor all individuals iin
the population, and normalize the fitness for each individual:
fi=MSEmax −MSEi
MSEmax −MSEmin
(6)
Ni=1
2(tanh(4 ·(1 −fi)−2) + 1) (7)
in which MSEmax and MSEmin are the maximum and minimum MSE in
the population, fiis an individual’s relative fitness and Niis an individual’s
normalized relative fitness.
3. Sort Population on fitness, keep the fittest Mindividuals and discard the
rest.
Paintings, Polygons and Plant Propagation 93
4. Create Offspring: each individual iin the population creates nrnew indi-
viduals as
nr=nmax ·Ni·r1(8)
with mrmutations as
mr=9v
4·1
nmax
·(1 −Ni)·r2(9)
in which 9v
4is the number of possible mutation operations for a constellation
of vvertices and r1and r2are random numbers from (0,1).
5. Return to step 2 unless the predetermined maximum number of function
evaluations is exceeded, in which case the algorithm terminates. In the exper-
iments, this number is identical to the number of iterations for the other two
algorithms, that both perform exactly one evaluation each iteration.
5 Experiments and Results
We ran the stochastic hillclimber and simulated annealing for 106itera-
tions and completed five runs for each painting and each number of v∈
{20,100,300,500,700,1000}. Simulated annealing was parametrized as described
in Sect. 4.2 and we continually retained the best candidate solution so far, as the
MSE occasionally increases during a run. The plant propagation algorithm was
parametrized as stated in Sect. 4.3 and also completed five runs, also until 106
function evaluations had occurred. After completing a set of five runs, we aver-
aged the last (and therefore best) values (Fig. 3).
For almost all paintings with all numbers of v, the stochastic hillclimber is by
far the superior algorithm. It outperformed the plant propagation algorithm on
48 out of 49 vertex-painting combinations with an average MSE improvement of
766 and a maximum improvement of 3177, on the Jackson Pollock with 1000 ver-
tices. Only for the Mona Lisa with 20 vertices, PPA outperformed the hillclimber
by 37 MSE. Plant propagation was the second best algorithm, better than simu-
lated annealing on all painting-vertex combinations by an average improvement
of 17666 MSE. The smallest improvement of 4532 MSE was achieved on Bach
with 20 vertices and a largest improvement of 46228 MSE on the Mondriaan with
20 vertices. A somewhat surprising observation is that for both the stochastic
hillclimber and plant propagation, end results improved with increasing vertex
numbers, but only up to about v= 500, after which the quality of the rendered
bitmaps largely leveled out. This pattern also largely applies to the end results
of simulated annealing, with the exception of Bach, which actually got worse
with increasing numbers of v. This surprising and somewhat counterintuitive
phenomenon might be explained from the fact the algorithm does not perform
very well, and the fact that Hausmann’s painting is largely black which is the
canvas’ default color. When initialized with more vertices (and therefore more
polygons), a larger area of the black canvas is covered with colored polygons,
increasing the default MSE-distance from Bach to black and back.
94 M. Paauw and D. van den Berg
Fig. 3. Best results of five runs for the three algorithms on all seven paintings for all
numbers of vertices. Remarkably enough, results often do not improve significantly with
numbers of vertices over 500. Note that results for the hillclimber and plant propagation
are depicted on a logarithmic scale.
6 Discussion, Future Work, and Potential Applications
In this work, we set out to compare three different algorithms to optimize a con-
stellation of semi-transparent, partially overlapping polygons to approximate a
set of famous paintings. It is rather surprising that in this exploration, the sim-
plest algorithm performs best. As the hillclimber is extremely susceptible to
getting trapped in local minima, this might indicate the projection of the objec-
tive function from the state space to the MSE might be relatively convex when
neighbourized with mutation types such as ours. Another explanation might be
the presence of many high-quality solutions despite the vastness of the state
space.
Plant propagation is not very vulnerable to local minima, but it does not
perform very well considering its track record on previous problems. This might
be due to the mutation operators, or the algorithm’s parametrization, but also
to the vastness of the state space; even if the algorithm does avoid local minima,
their attractive basins might be so far apart that the maximum distance our
mutations are likely to traverse are still (far) too small.
The rather poor performance of simulated annealing can be easily explained
from its high temperature. The logarithmic Geman&Geman-scheme in itself
cools down rather slowly, even over a run of a million iterations. But our choice of
c-value might also be too high. The canvas is not comprised of ‘extreme colors’,
and neither are the target bitmaps, so the maximum energy barrier (increase in
MSE) might in practice be much lower than our upper bound of 195075. An easy
way forward might be to introduce an ‘artificial Boltzmann constant’, effectively
facilitating faster temperature decrease. This method however, does raise some
theoretical objections as the principle of ‘cooling down slowly’ is a central dogma
in both simulated and metallurgic annealing.
Paintings, Polygons and Plant Propagation 95
An interesting observation can be made in Fig. 3. Some paintings appear to
be harder to approximate with a limited number of polygons than others, leading
to a ‘polygonicity ranking’ which is consistently shared between the hillclimber
and plant propagation. The essentially grid based work of Mondriaan seems
to be more suitable for polygonization than Van Gogh’s Starry Starry Night,
which consists of more rounded shapes, or Jackson Pollock’s work which has no
apparent structure at all. To quantitatively assert these observations, it would
be interesting to investigate whether a predictive data analytic could be found
to a priori estimate the approximability of a painting. This however, requires a
lot more theoretical foundation and experimental data.
Fig. 4. Best results for various paintings and various algorithms. From top left, clock-
wise: 1: Mona Lisa, SA, v= 1000. 2: Mondriaan, HC, v= 1000. 3: Klimt, PPA,
v= 1000. 4: Bach, HC v= 1000. 5: Dali, HC, v= 1000. 6: Jackson Pollock, SA,
v= 1000. 7: Starry Night, PPA, v= 1000. This Mondriaan is the best approximation
in the entire experiment; when looking closely, even details like irregularities in the
painting’s canvas and shades from the art gallery’s photograph can be made out.
7 Challenge
We hereby challenge all our colleagues in the field of combinatorial optimization
to come up with better results than ours (Fig. 4). Students are encouraged to
get involved too. Better results include, but are not limited to:
1. Better instance results: finding better MSE-values for values of vand pidenti-
cal to ours, regardless of the method involved. If better MSE-values are found
for smaller vand p, than ours, they are considered stronger.
96 M. Paauw and D. van den Berg
2. Better algorithmic results: finding algorithms that either significantly improve
MSE-results, or achieve similar results in fewer function evaluations.
3. Better parameter settings: finding parameter settings for our algorithms that
that either significantly improve MSE-results, or achieve similar results in
fewer function evaluations. We expect this is well possible.
4. Finding multiple minima: it is unclear whether any minimum value of MSE
can have multiple different polygon constellations, apart from symmetric
results.
For resources, as well as improved results, refer to our online page [20]. We
intend to keep a high score list.
Acknowledgements. We would like to thank Abdellah Salhi (University of Essex)
and Eric Fraga (University College London) for their unrelenting willingness to discuss
and explain the plant propagation algorithm. A big thanks also goes to Arnoud Visser
(University of Amsterdam) for providing some much-needed computing power towards
the end of the project, and to Jelle van Assema for helping with the big numbers.
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