Abstract
The objective scaling ensemble approach is a novel two-phase heuristic for integer linear programming problems shown to be effective on a wide variety of integer linear programming problems. The technique identifies and aggregates multiple partial solutions to modify the problem formulation and significantly reduce the search space. An empirical analysis on publicly available benchmark problems demonstrate the efficacy of our approach by outperforming standard solution strategies implemented in modern optimization software.
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Appendices
Appendices
1.1 A. Experimental results of Easy instances
Name | \(\gamma \) (%) | \(G_{\text {OSEA}}\) (%) | \(T_{\text {OSEA}}\) | \(G_{\text {Standard}}\) (%) | \(T_{\text {Standard}}\) |
|---|---|---|---|---|---|
30_70_45_095_100 | 11.82 | 0.00 | 2.21 | 0.00 | 2.79 |
30n20b8 | 99.09 | 0.00 | 0.38 | 0.00 | 2.58 |
50v-10 | 88.78 | 0.12 | 0.92 | 0.32 | 60.00 |
aflow40b | 87.20 | 0.00 | 0.94 | 0.00 | 60.00 |
air04 | 94.12 | 0.00 | 11.34 | 0.00 | 9.22 |
app1-2 | 90.18 | 0.00 | 1.34 | 0.00 | 47.06 |
beasleyC3 | 64.36 | 0.00 | 0.24 | 0.00 | 10.53 |
berlin_5_8_0 | 64.23 | 0.00 | 0.03 | 0.00 | 60.00 |
biella1 | 93.09 | 1.41 | 11.60 | 5.87 | 60.00 |
binkar10_1 | 42.65 | 0.00 | 0.40 | 0.00 | 8.32 |
co-100 | 96.65 | 0.01 | 25.07 | 45.55 | 60.02 |
core2536-691 | 90.62 | 0.00 | 16.56 | 0.00 | 31.48 |
cov1075 | 0.00 | 0.00 | 5.97 | 0.00 | 3.32 |
dfn-gwin-UUM | 58.95 | 0.00 | 2.00 | 0.00 | 60.00 |
eil33-2 | 97.87 | 5.43 | 5.56 | 0.00 | 36.46 |
eilB101 | 90.74 | 3.13 | 2.27 | 3.02 | 60.00 |
enlight13 | 35.80 | 0.00 | 0.01 | 0.00 | 3.44 |
enlight15 | 53.78 | 0.00 | 0.01 | 0.00 | 14.42 |
gmu-35-40 | 92.92 | 0.02 | 0.02 | 0.03 | 60.00 |
gmu-35-50 | 95.98 | 0.01 | 0.03 | 0.03 | 60.00 |
go19 | 16.99 | 0.00 | 60.00 | 1.18 | 60.00 |
harp2 | 94.85 | 0.00 | 0.11 | 0.00 | 25.31 |
ic97_potential | 2.63 | 0.18 | 60.00 | 0.08 | 60.00 |
iis-100-0-cov | 0.00 | 0.00 | 60.00 | 0.00 | 60.01 |
iis-bupa-cov | 14.97 | 0.00 | 60.00 | 0.00 | 60.00 |
iis-pima-cov | 55.14 | 2.94 | 60.00 | 2.94 | 60.00 |
k16x240 | 87.95 | 0.00 | 0.02 | 0.00 | 60.00 |
lectsched-4-obj | 43.26 | 0.00 | 0.21 | 0.00 | 0.65 |
m100n500k4r1 | 95.20 | 4.17 | 0.01 | 4.17 | 60.00 |
macrophage | 53.41 | 0.27 | 0.06 | 0.00 | 60.00 |
mc11 | 48.53 | 0.00 | 1.33 | 0.00 | 9.79 |
mine-166-5 | 41.08 | 0.00 | 0.30 | 0.00 | 19.57 |
mine-90-10 | 40.11 | 0.07 | 0.14 | 0.07 | 60.00 |
mzzv11 | 96.57 | 3.91 | 0.24 | 0.00 | 14.93 |
n3div36 | 99.69 | 0.00 | 24.96 | 0.15 | 60.01 |
n3seq24 | 98.91 | 2.25 | 33.61 | 2.61 | 60.04 |
n4-3 | 24.14 | 0.85 | 60.00 | 0.19 | 60.00 |
n9-3 | 43.65 | 3.17 | 60.00 | 9.84 | 60.00 |
neos-1109824 | 91.78 | 0.00 | 2.08 | 0.00 | 26.80 |
neos-1112782 | 88.94 | 0.00 | 0.15 | 0.00 | 8.67 |
neos-1112787 | 89.29 | 0.00 | 0.13 | 0.00 | 1.83 |
neos-1224597 | 66.51 | 0.00 | 0.25 | 0.00 | 0.25 |
neos-1225589 | 81.48 | 0.00 | 0.08 | 0.00 | 0.14 |
neos-1337307 | 59.33 | 0.00 | 24.75 | 0.00 | 25.67 |
neos-1396125 | 10.49 | 0.00 | 25.63 | 0.00 | 10.39 |
Name | \(\gamma \) | \(G_{\text {OSEA}}\) | \(T_{\text {OSEA}}\) | \(G_{\text {Standard}}\) | \(T_{\text {Standard}}\) |
|---|---|---|---|---|---|
neos-1440225 | 67.21 | 0.00 | 0.76 | 0.00 | 23.18 |
neos-1616732 | 0.00 | 0.62 | 60.00 | 0.62 | 60.00 |
neos-1620770 | 0.00 | 0.00 | 60.00 | 0.00 | 60.01 |
neos-506428 | 94.65 | 77.78 | 7.77 | 77.78 | 60.01 |
neos-520729 | 99.44 | 0.00 | 14.53 | 0.00 | 60.02 |
neos-555424 | 49.78 | 0.00 | 0.31 | 0.00 | 5.06 |
neos-631710 | 92.67 | 46.30 | 32.32 | 63.49 | 60.03 |
neos-686190 | 90.61 | 0.59 | 2.46 | 10.98 | 60.00 |
neos-693347 | 49.96 | 1.68 | 60.00 | 1.27 | 60.05 |
neos-777800 | 94.89 | 0.00 | 7.04 | 0.00 | 1.17 |
neos-824661 | 92.85 | 0.00 | 3.73 | 0.00 | 9.13 |
neos-824695 | 91.61 | 0.00 | 1.09 | 0.00 | 2.70 |
neos-826650 | 85.79 | 0.00 | 0.25 | 0.00 | 60.00 |
neos-826694 | 86.73 | 0.00 | 1.02 | 0.00 | 1.97 |
neos-826812 | 86.81 | 0.00 | 0.64 | 0.00 | 0.87 |
neos-826841 | 84.78 | 0.00 | 0.26 | 0.00 | 40.41 |
neos-885524 | 99.78 | 0.00 | 1.55 | 0.00 | 0.58 |
neos-932816 | 93.66 | 0.00 | 0.88 | 0.00 | 0.91 |
neos-933638 | 83.64 | 0.36 | 60.00 | 0.00 | 33.14 |
neos-933966 | 89.00 | 0.00 | 31.99 | 0.00 | 6.68 |
neos-934278 | 77.74 | 8.45 | 60.00 | 0.00 | 42.49 |
neos-935627 | 47.56 | 10.51 | 60.00 | 0.19 | 60.00 |
neos-935769 | 41.23 | 1.95 | 60.00 | 0.00 | 25.78 |
neos-937511 | 58.84 | 0.00 | 60.00 | 0.00 | 10.30 |
neos-941262 | 46.16 | 1.10 | 60.02 | 1.10 | 60.00 |
neos-941313 | 79.80 | 0.00 | 32.07 | 0.00 | 39.15 |
neos-957389 | 91.98 | 0.00 | 0.39 | 0.00 | 1.16 |
neos15 | 0.00 | 0.90 | 60.00 | 0.90 | 60.00 |
neos16 | 32.10 | 0.22 | 60.00 | 0.22 | 60.00 |
neos18 | 4.83 | 0.00 | 0.44 | 0.00 | 41.12 |
net12 | 58.07 | 0.00 | 0.75 | 16.08 | 60.00 |
nobel-eu-DBE | 93.91 | 2.05 | 45.11 | 0.82 | 60.00 |
noswot | 78.00 | 0.00 | 0.01 | 0.00 | 42.38 |
nu60-pr9 | 90.84 | 1.58 | 0.78 | 3.18 | 60.35 |
p80x400b | 76.01 | 0.17 | 0.07 | 0.00 | 60.00 |
pg | 0.00 | 0.25 | 60.00 | 0.25 | 60.00 |
pg5_34 | 0.00 | 0.03 | 60.00 | 0.03 | 60.00 |
pigeon-10 | 82.00 | 0.00 | 0.04 | 0.00 | 60.00 |
pigeon-11 | 82.03 | 0.00 | 0.02 | 0.00 | 60.00 |
pw-myciel4 | 33.24 | 0.00 | 60.00 | 0.00 | 60.00 |
pw-myciel4 | 33.24 | 0.00 | 60.00 | 0.00 | 60.00 |
rail507 | 98.24 | 0.00 | 48.07 | 0.00 | 60.01 |
ran14x18 | 74.42 | 0.64 | 0.16 | 0.64 | 60.00 |
ran16x16 | 72.83 | 0.10 | 0.06 | 0.00 | 48.51 |
reblock166 | 45.54 | 0.03 | 0.83 | 0.06 | 60.00 |
reblock67 | 38.36 | 0.21 | 0.06 | 0.59 | 60.00 |
rmatr100-p10 | 0.00 | 0.00 | 28.41 | 0.00 | 47.48 |
rmatr100-p5 | 0.00 | 6.69 | 60.00 | 6.96 | 60.00 |
rmine6 | 31.66 | 0.01 | 0.24 | 0.00 | 60.00 |
rococoB10-011000 | 87.43 | 2.54 | 0.15 | 7.00 | 60.00 |
rococoC10-001000 | 85.95 | 0.04 | 0.08 | 0.50 | 60.00 |
satellites1-25 | 85.80 | 0.00 | 20.80 | 117.24 | 60.00 |
Name | \(\gamma \) | \(G_{\text {OSEA}}\) | \(T_{\text {OSEA}}\) | \(G_{\text {Standard}}\) | \(T_{\text {Standard}}\) |
|---|---|---|---|---|---|
satellites2-60-fs | 82.49 | 144.19 | 37.60 | 139.58 | 60.50 |
sp97ar | 97.61 | 0.26 | 22.35 | 0.88 | 60.13 |
sp98ic | 98.70 | 0.19 | 25.07 | 0.65 | 60.00 |
sp98ir | 92.81 | 0.13 | 1.40 | 0.00 | 30.46 |
tanglegram1 | 84.22 | 0.00 | 0.87 | 0.00 | 4.48 |
tanglegram2 | 87.36 | 0.00 | 0.11 | 0.00 | 0.55 |
toll-like | 55.22 | 0.33 | 0.08 | 1.61 | 60.00 |
uct-subprob | 5.85 | 1.57 | 60.00 | 0.95 | 60.37 |
umts | 85.21 | 0.12 | 1.77 | 0.13 | 60.00 |
wachplan | 86.88 | 0.00 | 6.99 | 0.00 | 60.00 |
zib54-UUE | 0.00 | 2.23 | 60.00 | 2.23 | 60.00 |
1.2 B. Experimental results of Hard instances
Name | \(\gamma \) (%) | \(G_{\text {OSEA}}\) (%) | \(T_{\text {OSEA}}\) | \(G_{\text {Standard}}\) (%) | \(T_{\text {Standard}}\) |
|---|---|---|---|---|---|
a1c1s1 | 9.90 | 0.26 | 60.00 | 0.02 | 60.00 |
b2c1s1 | 3.91 | 5.84 | 60.00 | 6.75 | 60.03 |
bg512142 | 5.67 | 5.44 | 60.00 | 8.75 | 60.00 |
dg012142 | 7.68 | 20.70 | 60.00 | 25.81 | 60.00 |
dolom1 | 81.32 | 92.38 | 60.00 | 97.80 | 60.02 |
germany50-DBM | 1.33 | 2.13 | 60.00 | 1.46 | 60.00 |
d10200 | 79.27 | 0.10 | 23.79 | 0.10 | 60.00 |
dc1c | 95.24 | 5.95 | 19.50 | 91.22 | 60.00 |
janos-us-DDM | 25.04 | 0.05 | 60.00 | 0.08 | 60.00 |
lotsize | 44.77 | 0.47 | 60.00 | 1.22 | 60.00 |
eilA101-2 | 99.54 | 7.68 | 23.50 | 7.68 | 60.14 |
n3-3 | 46.72 | 3.71 | 60.00 | 10.18 | 60.00 |
neos-948126 | 33.25 | 4.22 | 60.00 | 4.22 | 60.01 |
neos-984165 | 39.81 | 23.97 | 60.00 | 23.97 | 60.01 |
mkc | 96.67 | 0.02 | 0.11 | 0.00 | 60.00 |
rmatr200-p10 | 24.30 | 2.04 | 60.00 | 10.36 | 60.01 |
nu120-pr3 | 92.10 | 3.99 | 1.12 | 4.38 | 60.43 |
p100x588b | 75.65 | 0.65 | 0.29 | 1.71 | 60.00 |
p6b | 86.36 | 0.00 | 0.02 | 0.00 | 60.00 |
pigeon-12 | 82.07 | 0.00 | 0.03 | 0.00 | 60.00 |
pigeon-13 | 82.10 | 0.00 | 0.03 | 0.00 | 60.00 |
protfold | 83.98 | 34.78 | 0.07 | 40.91 | 60.00 |
queens-30 | 95.89 | 8.11 | 0.03 | 8.11 | 60.07 |
r80x800 | 84.42 | 0.06 | 0.30 | 0.06 | 60.00 |
reblock354 | 39.29 | 0.01 | 0.28 | 0.04 | 60.02 |
reblock420 | 45.93 | 15.28 | 10.03 | 15.68 | 60.06 |
rmatr200-p20 | 20.87 | 0.12 | 60.01 | 4.67 | 60.00 |
rmatr200-p5 | 18.83 | 6.84 | 60.01 | 9.24 | 60.01 |
seymour | 29.68 | 0.24 | 60.00 | 0.24 | 60.10 |
rococoC11-011100 | 90.29 | 3.14 | 0.18 | 6.97 | 60.15 |
tw-myciel4 | 38.03 | 0.00 | 0.14 | 0.00 | 60.00 |
wnq-n100-mw99-14 | 90.58 | 14.24 | 22.73 | 9.44 | 60.03 |
1.3 C. Experimental results of Open instances
Name | \(\gamma \) (%) | \(G_{\text {OSEA}}\) (%) | \(T_{\text {OSEA}}\) | \(G_{\text {Standard}}\) (%) | \(T_{\text {Standard}}\) |
|---|---|---|---|---|---|
core4872-1529 | 67.32 | 3.96 | 60.00 | 3.59 | 60.01 |
bab1 | 98.70 | 31.16 | 22.35 | 31.16 | 60.01 |
cdma | 83.99 | 32138.99 | 20.17 | 32138.99 | 60.01 |
dc1l | 95.68 | 8.00 | 60.06 | 10.30 | 61.98 |
d20200 | 85.20 | 0.30 | 5.37 | 0.31 | 60.00 |
ex1010-pi | 86.93 | 9.93 | 60.00 | 11.02 | 60.00 |
ger50_17_trans | 97.54 | 10.82 | 60.00 | 15.56 | 60.00 |
momentum3 | 68.69 | 76.50 | 60.02 | 76.50 | 60.03 |
methanosarcina | 56.53 | 100.00 | 0.28 | 100.00 | 60.00 |
n3700 | 92.85 | 23.33 | 60.00 | 24.64 | 60.00 |
n3705 | 92.32 | 24.13 | 60.00 | 24.22 | 60.00 |
n370a | 91.65 | 23.71 | 60.00 | 25.09 | 60.00 |
neos-937815 | 48.89 | 0.56 | 60.00 | 0.56 | 60.00 |
ns4-pr3 | 0.99 | 0.20 | 60.00 | 0.19 | 60.00 |
ns4-pr9 | 0.00 | 0.16 | 60.00 | 0.16 | 60.00 |
pigeon-19 | 82.33 | 5.56 | 0.07 | 5.56 | 60.00 |
ramos3 | 18.69 | 44.77 | 60.13 | 48.30 | 60.11 |
rmine10 | 34.92 | 2.45 | 14.60 | 2.46 | 60.01 |
rmine14 | 96.73 | 2426.06 | 1.54 | 2426.06 | 60.08 |
rococoC12-111000 | 93.73 | 25.93 | 0.28 | 25.54 | 60.00 |
rvb-sub | 99.39 | 94.70 | 22.05 | 93.47 | 60.01 |
siena1 | 79.86 | 92.48 | 60.00 | 97.73 | 60.12 |
sing2 | 47.56 | 2.30 | 60.00 | 9.22 | 60.01 |
stockholm | 55.71 | 99.50 | 60.01 | 99.49 | 60.05 |
sts405 | 0.00 | 60.98 | 60.04 | 60.98 | 60.01 |
sts729 | 0.00 | 62.44 | 60.01 | 62.44 | 60.12 |
t1717 | 97.80 | 43.80 | 60.02 | 43.80 | 60.01 |
t1722 | 97.21 | 35.94 | 60.00 | 32.35 | 60.00 |
usAbbrv.8.25_70 | 69.84 | 21.49 | 0.08 | 22.13 | 60.00 |
van | 0.00 | 95.17 | 60.01 | 95.17 | 60.01 |
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Zhang, W., Nicholson, C.D. Objective scaling ensemble approach for integer linear programming. J Heuristics 26, 1–19 (2020). https://doi.org/10.1007/s10732-019-09418-9
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DOI: https://doi.org/10.1007/s10732-019-09418-9