Enhancing Partition Crossover with Articulation Points Analysis

Enhancing Partition Crossover with
Articulation Points Analysis
Francisco Chicano, Gabriela Ochoa, Darrell Whitley, Renato Tinós
DENTIDAD
rca Universidad de Málaga
VERSIÓN VERTICAL EN POSITIVO
VERSIÓN VERTICAL EN NEGATIVO

Enhancing Partition Crossover with Articulation Points Analysis
GECCO 2018, 15-19 July, Kyoto, Japan 2
• Gray-Box (vs. Black-Box) Optimization
• Partition Crossover and Articulation Points
• Deterministic Recombination and Iterated Local Search
• Experiments
• Conclusions and Future Work
Outline

Gray-Box (vs. Black-Box) Optimization
x f(x)
x f(x)
For most of real problems we
know (almost) all the details

Gray-Box structure: MK Landscapes
Example (k=2):
f = + + +f(1)(x) f(2)(x) f(3)(x) f(4)(x)
x1 x2 x3 x4
ndscape [3] with bounded epistasis k is de-
tion f(x) that can be written as the sum
ns, each one depending at most on k input
is:
f(x) =
mX
i=1
f(i)
(x), (1)
unctions f(i)
depend only on k components
d Landscapes generalize NK-landscapes and
problem. We will consider in this paper that
ubfunctions is linear in n, that is m 2 O(n).
pes m = n and is a common assumption in
t m 2 O(n).
S IN THE HAMMING BALL
Equ
change
f(l)
th
this su
On the
we onl
change
acteriz
we can
3.1
Each subfunction is unknown
and depends on k variables
All compresible pseudo-Boolean
functions can be transformed into
this in polynomial time

Variable Interaction
f = + + +f(1)(x) f(2)(x) f(3)(x) f(4)(x)
x1 x2 x3 x4
xi and xj interact when they appear together in the same subfunction*
If xi and xj don’t interact: ∆ij = ∆i + ∆j
x4 x3
x1 x2
Variable Interaction Graph (VIG)

Let us suppose our function has the following VIG…
if(x1, x2, . . . , xn) = f(x1, x2, . . . , 1i, . . . , xn) f(x1, x2, . . . , 0i, . . . , xn)
f(x1, x2, x3, x4) = x1x2 + x2x3 + x2x4 + x3x4
x9
x20
x23
x22
x21
x8
x10
x1 x2
x3
x4
x5
x6
x7
x15
x14
x13
x12
x11
x16
x19
x18
x17
Partition Crossover (PX)

f(x1, x2, x3, x4) = x1x2 + x2x3 + x2x4 + x3x4
x9
x20
x23
x22
x21
x8
x10
x1 x2
x3
x4
x5
x6
x7
x15
x14
x13
x12
x11
x16
x19
x18
x17
0
1
0
0
1
00
0
1
0
1
0
1
0
0
1
1
0 1
0
0
0
1

f(x1, x2, x3, x4) = x1x2 + x2x3 + x2x4 + x3x4
x9
x20
x23
x22
x21
x8
x10
x1 x2
x3
x4
x5
x6
x7
x15
x14
x13
x12
x11
x16
x19
x18
x17
0
1
1
0
1
00
0
1
1
0
0
1
0
1
1 1
0 1
1
0
1
1
0
1
0
0
1
00
0
1
0
1
0
1
0
0
1
1
0 1
0
0
0
1

PX creates a graph with only the differing variables (recombination graph)
All the variables in a component are taken from the same parent
The contribution of each component to the fitness value of the offspring is
independent of each other
x23
x18
x9
x3
x5
x16
FOGA 2015: Tinós, Whitley, C.

PX creates a graph with only the differing variables (recombination graph)
All the variables in a component are taken from the same parent
The contribution of each component to the fitness value of the offspring is
independent of each other
x23
x18
x9
x3
x5
x16
If there are q
components, the best
offspring out of 2q
solutions is obtained
FOGA 2015: Tinós, Whitley, C.

Articulation Points in a Graph
Articulation point
Connected sub-component
a
C1
C2
C3
C4

Articulation Points in a Graph
both parents and applying Partition Crossover. This computation
is independently performed for each connected component and all
the contributions are added to give the overall contribution. If there
is no articulation point in the recombination graph or removing an
articulation point does not increase the objective value, the operator
works as the original PX. In the following sections we detail the
theoretical background of the operator.
x2
x1
x3 x4
x0
Figure 3: Example of articulation points. Nodes x3 and x4 are
articulation points of the graph.
Articulation Points

f(x1, x2, x3, x4) = x1x2 + x2x3 + x2x4 + x3x4
x9
x20
x23
x22
x21
x8
x10
x1 x2
x3
x4
x5
x6
x7
x15
x14
x13
x12
x11
x16
x19
x18
x17
0
1
1
0
1
00
0
1
1
0
0
1
0
1
1 1 1
1
1
0
1
1
0
1
0
0
1
00
0
1
0
1
0
1
0
0
1
1
0 1
0
0
0
1
Articulation Points Partition Crossover (APX)

x23
x18
x9
x3
x5
x16
x1
Original PX would find 2
components, and would
provide the best of 4 solutions

x23
x18
x9
x3
x5
x16
x1
APX identifies articulation points in the recombination graph
It implicitly considers all the solutions PX would consider if one or none articulation
point is removed from each connected component
APX will consider 2 and 3
components and will provide
the best of 32 solutions
APX can break one connected
component by flipping
variables in one of the parents

All the analysis can be done using Tarjan’s algorithm to find articulation points (DFS-
like algorithm) : time complexity is the same as the original PX
a1
C1
C2
C3 C4
a2

The number of implicitly studied solutions is:
C (z) = max
a2AP(C)
t 2{x, }
≠
´h2Fa⇤
h(t 1a) +
i=1
Ca
i ,a(t)Æ
¨
(5)
where Ca
i ,a(t) = max
⇣
Ca
i
(t 1a), Ca
i
(t 1C )
⌘
is the contribu-
tion of the connected sub-component Ca
i when articulation point a is
removed and Fa⇤ = F{a} [da
i=1FCa
i
is the set of subfunctions that de-
pend on a but not on any other variable in C. The number of solutions
that APX implicitly explores is:
E(x, ) = 2|CC(G)|
÷
C 2CC(G)
©
≠
´
1 eC +
’
a2AP(C)
⇣
2da 1
⌘
™
Æ
¨
, (6)
where eC is the number of edges in the connected component C join-
ing two articulation points. Observe that 2|CC(G)| is the number of
solutions implicitly explored by the original PX.
P. Equation (5) is a consequence of Lemma 3.1 where the
maximum over all the articulation points in a connected component
is taken. Equation (4) is a sum of the contribution to the objective
value of each connected component plus the sum of the evaluation
Number of solutions
considered by PX Edges joining two
articulation points
Degree of an articulation point
in the recominbation graph
Connected
component
Figure 6: Two articulation points not joined by an edge. The
analyses of a1 and a2 explore only two common combina-
tions: the ones found in the parent solutions.
In summary, the number of explored combinations in one con-
nected component C is:
’
a2AP(C)
(2da +1
2) 2eC +2 = 2
©
≠
´
1 eC +
’
a2AP(C)
(2da 1)
™
Æ
¨
. (7)
The product of Equation (7) extended to all the connected com-
ponents of G is Equation (6). ⇤
A more practical (easier to remember) lower bound for the num-
ber of explored solutions is provided in the next corollary.
C 3.3. Given two solutions x and , a lower bound of
the number of solutions implicitly evaluated in APX is:
E(x, ) 2|CC(G)|
÷
C2CC(G)
|AP(C)|0
2(1 + |AP(C)|) (8)

Deterministic Recombination and
Iterated Local Search (DRILS)
4: child PX(current, next);
5: if child = current or child = next then
6: current next;
7: else
8: current HBHC(child);
9: end if
10: end while
PX PX
Figure 4: Graphical illustration of DRILS. Curly arrows rep-
resent HBHC while normal arrows represent a perturbation
ipping N random bits.
graphical illustration of the algorithm is presented in Figure 4 and
Hill Climber
Perturbation
(! N bits flipped)
Random Solution Local Optimum

Experimental Results
An NK Landscape is a pseudo-Boolean optimization problem with objective function:
where each subfunction f(l) depends on variable xl and K other variables
MAX-SAT consists in finding an assignment of variables to Boolean (true and false)
values such that the maximum number of clauses is satisfied
A clause is an OR of literals: x1 ∨ ¬x2 ∨ x3
S2(x) = f (x 2) f (x) + f (x 2) f (x) + f (x 2) f (x)
S1,2(x) = f(1)
(x 1, 2) f(1)
(x)+f(2)
(x 1, 2) f(2)
(x)+f(3)
(x 1, 2) f(3)
(x)
S1,2(x) 6= S1(x) + S2(x)
f(x) =
NX
l=1
f(l)
(x)
1
x7
x8 x9
x10
(a) Sample VIG
x1
x2
x3x4
x5
x6
x7
x8 x9
x10
(b) Selected and adjacent variables
x1
x2
x3x4
x5
x6
x7
x8 x9
x10
(c) Sample random VIG
Random model

Example for NKQ Landscapes with N=100 000 and K=2 (DRILS+APX)
There are 4339 nodes grouped in 858 components with 1825 articulation points (in red)
that the logarithm is around two times the number of components.
This means that APX implicitly explores a set of solutions with a
size that is around the square of the number of solutions explored
by PX. According to the data in Table 1, this number is between
2254 = 1076 and 215 987 = 104 813.
Figure 7 shows the recombination graph for two local optima
during a run of DRILS+APX. For visualization purposes, we chose
one of the smallest recombination graphs we observed in the ex-
periments, for an instance with N = 105 variables and K = 2.
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Figure 7: Example of recombination graph during a
DIRLS+APX run for an NK instance with N = 105 and K = 2,
showing the connected components and articulation points
(red nodes). The graph contains 858 components and 4 339
nodes, of which 1 825 are articulation points (42%).
105
2 10
3 10
4 2
5 1
106
2 2
3 5
4 9
5 1
and K = 3 (average o
DRILS+APX outperform
Columns sixth and
runtime of one applica
the same order of magn
sometimes is slower th
computation does not h

APX runtime is in the same order of magnitude than that of PX
number of explored solutions, we compute the binary logarithm
and provide the average. This makes it possible to easily compare
the number of solutions explored by APX and PX, since the number
of components (third column) is the binary logarithm of the number
of solutions explored by PX.
Table 1: APX Statistics.
N K #Comp. #APs da log2 E(x, )
105
2 662 687 2.25 1 311
3 503 1 151 2.37 1 105
4 138 196 2.33 286
5 119 218 2.36 254
106
2 7 774 10 836 2.28 15 987
3 4 515 21 793 2.35 9 454
4 1 748 6 281 2.38 3 907
5 1 105 7 207 2.34 2 341
We can observe in Table 1 that the number of articulation points
can be similar to the number of components, but it can also be
several times larger, indicating that each connected component can
on Points Analysis GECCO ’18, July 15–19, 2018, Kyoto, Japan
he number of articu-
mponents. While the
mber of articulation
erage degree of the
minimum value). It is
m value we observed
re probable when K
utions, we observe
mber of components.
of solutions with a
solutions explored
number is between
or two local optima
purposes, we chose
observed in the ex-
bles and K = 2.
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Table 2: Number of NKQ instances where any of the algo-
rithms statistically outperforms the other or the two are
similar. The average runtime of one execution of APX and
PX is also shown.
DRILS performance Runtime (ms)
N K APX PX Sim. APX PX
105
2 10 0 0 55 46
3 10 0 0 67 73
4 2 0 8 55 52
5 1 1 8 63 52
106
2 2 3 5 1 383 970
3 5 0 5 1 785 2 485
4 9 0 1 1 360 1 439
5 1 0 9 1 633 1 559
and K = 3 (average over 100 samples). We can clearly see how
DRILS+APX outperformed DRILS after a few seconds.
24515≈101359 solutions: 101349 ≈ (1080)16 solutions per nanosecond

y = 2,06x + 65,302
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
log2E(x,y)
# Components
Number of Explored Solutions (APX vs. PX)
on. The computation eort required to do this analysis
eases the time in a constant factor compared to PX. There
nge in the asymptotic behaviour of the operator run time,
O(N) for k-bounded pseudo-Boolean functions and O(N2)
neral case.
PERIMENTS
to experimentally analyze the performance of APX, we
it in the Deterministic Recombination and Iterated Local
DRILS) algorithm. DRILS [1] uses a rst improving move
er to reach a local optimum. Then, it perturbs the solution
mly ipping N bits, where is the so-called perturbation
then applies local search to the new solution to reach
ocal optimum and applies Partition Crossover to the last
optima, generating a new solution that is improved further
hill climber. This process is repeated until a time limit is
The pseudocode is shown in Algorithm 1.
tion to the original DRILS algorithm, we implement a vari-
e the Partition Crossover operator in Line 4 of Algorithm 1
d by APX. This version is called DRILS+APX in the rest
per. In all the runs we set a time limit of 60s (1 minute).
algorithms are stochastic, we performed 10 independent
ach instance and algorithm. We used NP-hard problems to
he performance of APX: Random NKQ Landscapes with
d MAX-SAT.
mputer used for the experiments is a multicore machine
Intel Xeon CPU (E5-2670 v3) at 2.3 GHz, a total of 48
in the case K = 2, 3 and = 0.01 in the case K = 4, 5. These v
were taken from the recommendations in [1].
Table 1 shows averages over all the recombinations appe
in all the runs for each combination of N and K. In the case
number of explored solutions, we compute the binary loga
and provide the average. This makes it possible to easily com
the number of solutions explored by APX and PX, since the nu
of components (third column) is the binary logarithm of the nu
of solutions explored by PX.
Table 1: APX Statistics.
N K #Comp. #APs da log2 E(x, )
105
2 662 687 2.25 1 311
3 503 1 151 2.37 1 105
4 138 196 2.33 286
5 119 218 2.36 254
106
2 7 774 10 836 2.28 15 987
3 4 515 21 793 2.35 9 454
4 1 748 6 281 2.38 3 907
5 1 105 7 207 2.34 2 341
We can observe in Table 1 that the number of articulation
can be similar to the number of components, but it can a
several times larger, indicating that each connected compone

y = 2,06x + 65,302
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
log2E(x,y)
# Components
|PX|

APX runtime is in the same order of magnitude than that of PX and the number of
solutions explored is squared!
|APX| ≈ |PX|2
y = 2,06x + 65,302
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
log2E(x,y)
# Components
|PX|
|APX|

DRILS and DRILS+APX solving NKQ Landscapes
106
4 9 0 1 1 360 1 439
5 1 0 9 1 633 1 559
Columns sixth and seventh of Table 1 also report the average
runtime of one application of APX and PX. The numbers are in
the same order of magnitude although sometimes PX is faster and
sometimes is slower than APX. Thus, we can claim that the extra
computation does not have a big impact in the runtime.
DRILS
DRILS+APX
0 10 20 30 40 50 60
4.55× 107
4.60× 107
4.65× 107
4.70× 107
Time (s)
Averagefitness
Figure 8: Average tness over time obtained by DRILS and
DRILS+APX in all the instances and all the runs of NKQ
Landscapes for N = 106 and K = 3.
Enhancing Partition Crossover with Articulation Points Analysis GECCO ’18, July 15–19, 2018, Kyoto, Japan
have several articulation points. The trend in the number of articu-
lation points is not as clear as the number of components. While the
number of components decrease with K, the number of articulation
points can increase or decrease with K. The average degree of the
articulation points is slightly larger than 2 (its minimum value). It is
not common to see high degrees; the maximum value we observed
in the experiments was 13. High values are more probable when K
is high. Regarding the number of explored solutions, we observe
that the logarithm is around two times the number of components.
This means that APX implicitly explores a set of solutions with a
size that is around the square of the number of solutions explored
by PX. According to the data in Table 1, this number is between
2254 = 1076 and 215 987 = 104 813.
Figure 7 shows the recombination graph for two local optima
during a run of DRILS+APX. For visualization purposes, we chose
one of the smallest recombination graphs we observed in the ex-
periments, for an instance with N = 105 variables and K = 2.
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Table 2: Number of NKQ instances where any of the algo-
rithms statistically outperforms the other or the two are
similar. The average runtime of one execution of APX and
PX is also shown.
DRILS performance Runtime (ms)
N K APX PX Sim. APX PX
105
2 10 0 0 55 46
3 10 0 0 67 73
4 2 0 8 55 52
5 1 1 8 63 52
106
2 2 3 5 1 383 970
3 5 0 5 1 785 2 485
4 9 0 1 1 360 1 439
5 1 0 9 1 633 1 559
Columns sixth and seventh of Table 1 also report the average
runtime of one application of APX and PX. The numbers are in
the same order of magnitude although sometimes PX is faster and
sometimes is slower than APX. Thus, we can claim that the extra
N=1 Million K=3

DRILS and DRILS+APX solving MAX-SAT (instances from MAX-SAT Evaluation 2017)
Mann-Whitney with signicance level 0.05) between the algorithms.
Three dierent values for the perturbation factor ( ) were used:
0.10, 0.20 and 0.30.
Table 3: Number of MAX-SAT instances where any of the
algorithms statistically outperforms the other or the two are
similar. The last two columns report the runtime of APX and
PX.
DRILS performance Runtime (µs)
Instances APX PX Sim. APX PX
Unweighted
0.10 78 1 81 463 454
0.20 82 2 75 684 729
0.30 85 2 73 849 1 060
Weighted
0.10 26 19 87 1 425 882
0.20 49 14 69 1 859 1 416
0.30 77 5 50 2 365 1 713
DRILS+APX seems to be better in the unweighted instances than
in the weighted ones, compared to DRILS. Unweighted instances are
Gray-B
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Source Code in GitHub
https://github.com/jfrchicanog/EfficientHillClimbers

Conclusions
• The Variable Interaction Graph provides useful information to improve the search
• Articulation Points Partition Crossover squares the number of solutions considered by
PX in around the same time
• APX is specially good in Unweighted MAX-SAT instances (many plateaus)
• Take home message: use Gray-Box Optimization if you can
• Plateaus exploration in MAX-SAT guided by APX
• New ways of perturbing the solution to maximize the components in (A)PX
• Look at the Variable Interaction Graph of industrial problems
Future Work

AcknowledgementsNTIDAD
Universidad de Málaga
VERSIÓN VERTICAL EN POSITIVO
VERSIÓN VERTICAL EN NEGATIVO
Thanks for your attention!!!

Enhancing Partition Crossover with Articulation Points Analysis

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