Here we provide the instances used in the experiments presented in our paper as well as the solutions found by our methods. We have considered instances introduced by Galiev and Lisafina (2013), Kallrath and Rebennack (2014), and some instances we have created. The descriptions of the solutions for the two- and three-dimensional problems are available in the following formats, respectively.

d1 d2
m
a1 b1 x1 y1 t1
a2 b2 x2 y2 t2
...
am bm xm ym tm

d1 d2 d3
m
a1 b1 c1 x1 y1 z1 t1 u1 v1
a2 b2 c2 x2 y2 z2 t2 u2 v2
...
am bm cm xm ym zm tm um vm

The first line has the dimensions of the container. In the two-dimensional case, it can be the length and width of a rectangle or the lengths of the semi-axes of an ellipse. For three-dimensional instances, it can be the length, width and height of a cuboid or the radius of a sphere (in which case this line has only one number). The container is centered at the origin. The second line contains the number m of ellipsoids packed. Then, each one of the next m lines contains the lengths of the semi-axes (a, b) or (a, b, c), the center (x, y) or (x, y, z) and the rotations angles t or (t, u, v) of an ellipsoid. In the three-dimensional case, we have adopted a left-handed coordinate system and the ellipsoids were rotated through an angle t about the x-axis, through an angle u about the y-axis, and through an angle v about the z-axis, in this order.

Three-dimensional instances

We have considered two problems in the three-dimensional space. The first one is to pack a given set of ellipsoids within a minimizing volume sphere and the second problem is to pack a given set of ellipsoids within a minimizing volume cuboid.

Minimizing the volume of the sphere

Number of ellipsoids: 10
Semi-axis lengths: (1, 0.75, 0.5)
Radius of the sphere: 1.78458
Volume of the sphere: 23.806734
Solution description

Number of ellipsoids: 20
Semi-axis lengths: (1, 0.75, 0.5)
Radius of the sphere: 2.26366
Volume of the sphere: 48.587432
Solution description

Number of ellipsoids: 30
Semi-axis lengths: (1, 0.75, 0.5)
Radius of the sphere: 2.62078
Volume of the sphere: 75.402176
Solution description

Number of ellipsoids: 40
Semi-axis lengths: (1, 0.75, 0.5)
Radius of the sphere: 2.89455
Volume of the sphere: 101.58621
Solution description

Number of ellipsoids: 50
Semi-axis lengths: (1, 0.75, 0.5)
Radius of the sphere: 3.08312
Volume of the sphere: 122.76152
Solution description

Number of ellipsoids: 60
Semi-axis lengths: (1, 0.75, 0.5)
Radius of the sphere: 3.26101
Volume of the sphere: 145.26059
Solution description

Number of ellipsoids: 70
Semi-axis lengths: (1, 0.75, 0.5)
Radius of the sphere: 3.44474
Volume of the sphere: 171.22143
Solution description

Number of ellipsoids: 80
Semi-axis lengths: (1, 0.75, 0.5)
Radius of the sphere: 3.58268
Volume of the sphere: 192.62625
Solution description

Number of ellipsoids: 90
Semi-axis lengths: (1, 0.75, 0.5)
Radius of the sphere: 3.71427
Volume of the sphere: 214.63923
Solution description

Number of ellipsoids: 100
Semi-axis lengths: (1, 0.75, 0.5)
Radius of the sphere: 3.86848
Volume of the sphere: 242.49896
Solution description

Minimizing the volume of the cuboid

Number of ellipsoids: 10
Semi-axis lengths: (1, 0.75, 0.5)
Cuboid side lengths: (2.00000, 3.75737, 3.80478)
Volume of the cuboid: 28.592021
Solution description

Number of ellipsoids: 20
Semi-axis lengths: (1, 0.75, 0.5)
Cuboid side lengths: (3.95990, 3.44560, 3.91672)
Volume of the cuboid: 53.440997
Solution description

Number of ellipsoids: 30
Semi-axis lengths: (1, 0.75, 0.5)
Cuboid side lengths: (3.71991, 4.23900, 4.93161)
Volume of the cuboid: 77.765441
Solution description

Number of ellipsoids: 40
Semi-axis lengths: (1, 0.75, 0.5)
Cuboid side lengths: (4.13758, 4.49120, 5.45913)
Volume of the cuboid: 101.44566
Solution description

Number of ellipsoids: 50
Semi-axis lengths: (1, 0.75, 0.5)
Cuboid side lengths: (3.95279, 4.99581, 6.44175)
Volume of the cuboid: 127.20831
Solution description

Number of ellipsoids: 60
Semi-axis lengths: (1, 0.75, 0.5)
Cuboid side lengths: (4.27141, 5.02910, 7.07829)
Volume of the cuboid: 152.05152
Solution description

Number of ellipsoids: 70
Semi-axis lengths: (1, 0.75, 0.5)
Cuboid side lengths: (3.89904, 5.79302, 7.75238)
Volume of the cuboid: 175.10514
Solution description

Number of ellipsoids: 80
Semi-axis lengths: (1, 0.75, 0.5)
Cuboid side lengths: (4.33539, 5.81581, 7.88685)
Volume of the cuboid: 198.85788
Solution description

Number of ellipsoids: 90
Semi-axis lengths: (1, 0.75, 0.5)
Cuboid side lengths: (4.25842, 6.13744, 8.53894)
Volume of the cuboid: 223.17260
Solution description

Number of ellipsoids: 100
Semi-axis lengths: (1, 0.75, 0.5)
Cuboid side lengths: (5.17787, 5.73271, 8.26307)
Volume of the cuboid: 245.27508
Solution description

Two-dimensional instances

In the two-dimensional space, we have considered the following problems:

Given a rectangle and real numbers a and b, pack the maximum number of identical ellipses with semi-axes lengths a and b inside the rectangle.
Pack a given a set of ellipses inside a
1. minimizing area ellipse;
2. minimizing area rectangle.

Minimizing the area of the ellipse

Number of ellipses: 100
Semi-axis lengths: (2, 1)
Solution description

Packing as many ellipses as possible

Sh. I. Galiev and M. S. Lisafina (2013) considered the problem of packing the maximum number of identical ellipses inside a rectangle. The rectangle has length 6 and width 3 and the ellipses have eccentricity equals 0.74536.

Instance: GL1
Number of ellipses: 15
Semi-axis lengths: (0.68892, 0.45928)
Solution description

Instance: GL2
Number of ellipses: 19
Semi-axis lengths: (0.61237, 0.40825)
Solution description

Instance: GL3
Number of ellipses: 34
Semi-axis lengths: (0.45928, 0.30619)
Solution description

Instance: GL4
Number of ellipses: 50
Semi-axis lengths: (0.38273, 0.25515)
Solution description

Instance: GL5
Number of ellipses: 65
Semi-axis lengths: (0.33681, 0.22454)
Solution description

Instance: GL6
Number of ellipses: 79
Semi-axis lengths: (0.30619, 0.20412)
Solution description

Minimizing the area of the rectangle

Instances from J. Kallrath and S. Rebennack (2014) are associated with the problem of packing a given set of ellipses into a rectangle with minimum area. The instances are divided into three sets. The first one contains non-identical ellipses (prefix TC), the second one contains identical ellipses with semi-axes lengths 2 and 1 (prefix TS), and the third set contains instances with three non-identical ellipses with increasing eccentricity (prefix TE).

TC Instances

Instance: TC02a
Number of ellipses: 2
Rectangle's area: 18.0000000000000071
Rectangle's length: 5.9999999999999991
Rectangle's width: 3.0000000000000018
Solution description

Instance: TC02b
Number of ellipses: 2
Rectangle's area: 22.2315874004330603
Rectangle's length: 7.3929905048845788
Rectangle's width: 3.0071169962607907
Solution description

Instance: TC03a
Number of ellipses: 3
Rectangle's area: 21.3857734832136082
Rectangle's length: 5.3909432066481990
Rectangle's width: 3.9669817810063965
Solution description

Instance: TC03b
Number of ellipses: 3
Rectangle's area: 25.2246718757473047
Rectangle's length: 7.5489343744124735
Rectangle's width: 3.3414877682931925
Solution description

Instance: TC04a
Number of ellipses: 4
Rectangle's area: 23.1870753840489527
Rectangle's length: 6.0335319097230915
Rectangle's width: 3.8430351792260797
Solution description

Instance: TC04b
Number of ellipses: 4
Rectangle's area: 28.5407405651661641
Rectangle's length: 8.0065147800421119
Rectangle's width: 3.5646896745022993
Solution description

Instance: TC05a
Number of ellipses: 5
Rectangle's area: 24.5536799549524396
Rectangle's length: 5.4799328591765866
Rectangle's width: 4.4806534287797586
Solution description

Instance: TC05b
Number of ellipses: 5
Rectangle's area: 30.6491932166075216
Rectangle's length: 5.8021353047159590
Rectangle's width: 5.2823989112587473
Solution description

Instance: TC06
Number of ellipses: 6
Rectangle's area: 25.0833070727826453
Rectangle's length: 5.1389948364084237
Rectangle's width: 4.8809753407561391
Solution description

Instance: TC11
Number of ellipses: 11
Rectangle's area: 55.9165709930976149
Rectangle's length: 9.5491461972410061
Rectangle's width: 5.8556618401395246
Solution description

Instance: TC14
Number of ellipses: 14
Rectangle's area: 24.1716755436121780
Rectangle's length: 5.3498957555441304
Rectangle's width: 4.5181582311324329
Solution description

Instance: TC20
Number of ellipses: 20
Rectangle's area: 65.7013356728506892
Rectangle's length: 9.8339428454261011
Rectangle's width: 6.6810776415493667
Solution description

Instance: TC30
Number of ellipses: 30
Rectangle's area: 95.6112453750374414
Rectangle's length: 11.3938158750765712
Rectangle's width: 8.3915034632236321
Solution description

Instance: TC50
Number of ellipses: 50
Rectangle's area: 152.6929604015488451
Rectangle's length: 12.7330346911524170
Rectangle's width: 11.9918750011454822
Solution description

Instance: TC100
Number of ellipses: 100
Rectangle's area: 297.7055773358980559
Rectangle's length: 18.2387779649324919
Rectangle's width: 16.3226712835856347
Solution description

TS Instances

Instance: TS2
Number of ellipses: 2
Semi-axis lengths: (2, 1)
Rectangle's area: 16.0000000000000000
Rectangle's length: 2.0000000000000000
Rectangle's width: 8.0000000000000000
Solution description

Instance: TS3
Number of ellipses: 3
Semi-axis lengths: (2, 1)
Rectangle's area: 23.5141646667283588
Rectangle's length: 4.0045687762286137
Rectangle's width: 5.8718343923345762
Solution description

Instance: TS4
Number of ellipses: 4
Semi-axis lengths: (2, 1)
Rectangle's area: 31.0683820508490030
Rectangle's length: 7.7670955127122507
Rectangle's width: 4.0000000000000000
Solution description

Instance: TS5
Number of ellipses: 5
Semi-axis lengths: (2, 1)
Rectangle's area: 39.0164567453769493
Rectangle's length: 9.7353276228863770
Rectangle's width: 4.0077189239789712
Solution description

Instance: TS6
Number of ellipses: 6
Semi-axis lengths: (2, 1)
Rectangle's area: 46.0601824460689784
Rectangle's length: 7.8259221697493242
Rectangle's width: 5.8855916845317102
Solution description

Instance: TS7
Number of ellipses: 7
Semi-axis lengths: (2, 1)
Rectangle's area: 54.1367641016975583
Rectangle's length: 3.9999999999999742
Rectangle's width: 13.5341910254244766
Solution description

Instance: TS8
Number of ellipses: 8
Semi-axis lengths: (2, 1)
Rectangle's area: 60.6243483798902218
Rectangle's length: 7.7697443772545975
Rectangle's width: 7.8026181346922963
Solution description

Instance: TS9
Number of ellipses: 9
Semi-axis lengths: (2, 1)
Rectangle's area: 68.3970408978241977
Rectangle's length: 11.6323420708214460
Rectangle's width: 5.8799028159076627
Solution description

Instance: TS10
Number of ellipses: 10
Semi-axis lengths: (2, 1)
Rectangle's area: 75.3789423675233081
Rectangle's length: 9.6956258434446454
Rectangle's width: 7.7745308641925490
Solution description

Instance: TS11
Number of ellipses: 11
Semi-axis lengths: (2, 1)
Rectangle's area: 83.2299778175225242
Rectangle's length: 8.0724147481237321
Rectangle's width: 10.3104189284708934
Solution description

Instance: TS12
Number of ellipses: 12
Semi-axis lengths: (2, 1)
Rectangle's area: 89.6969863476838327
Rectangle's length: 11.4613861725028290
Rectangle's width: 7.8260155445139015
Solution description

Instance: TS13
Number of ellipses: 13
Semi-axis lengths: (2, 1)
Rectangle's area: 97.8414836597346067
Rectangle's length: 9.7465189503141243
Rectangle's width: 10.0386080567340645
Solution description

Instance: TS14
Number of ellipses: 14
Semi-axis lengths: (2, 1)
Rectangle's area: 105.4409899223218332
Rectangle's length: 7.7061251186282682
Rectangle's width: 13.6827508376986877
Solution description

Instance: TS15
Number of ellipses: 15
Semi-axis lengths: (2, 1)
Rectangle's area: 111.2680389453274330
Rectangle's length: 11.4393224941512006
Rectangle's width: 9.7268032265212874
Solution description

TE Instances

Instance: TE1.00
Number of ellipses: 3
Semi-axis lengths: (1.700, 1.700), (1.200, 1.200), (0.800, 0.800)
Rectangle's area: 22.1717085680107715
Rectangle's length: 5.5994935135331909
Rectangle's width: 3.9595917942265419
Solution description

Instance: TE0.99
Number of ellipses: 3
Semi-axis lengths: (1.700, 1.683), (1.200, 1.188), (0.800, 0.792)
Rectangle's area: 21.8416899820718413
Rectangle's length: 5.7149852101937446
Rectangle's width: 3.8218279100903190
Solution description

Instance: TE0.98
Number of ellipses: 3
Semi-axis lengths: (1.700, 1.666), (1.200, 1.176), (0.800, 0.784)
Rectangle's area: 21.5083272948845945
Rectangle's length: 5.6736548462796765
Rectangle's width: 3.7909121858175800
Solution description

Instance: TE0.97
Number of ellipses: 3
Semi-axis lengths: (1.700, 1.649), (1.200, 1.164), (0.800, 0.776)
Rectangle's area: 21.1766868267242430
Rectangle's length: 5.6325811589547827
Rectangle's width: 3.7596771762546468
Solution description

Instance: TE0.96
Number of ellipses: 3
Semi-axis lengths: (1.700, 1.632), (1.200, 1.152), (0.800, 0.768)
Rectangle's area: 20.8467173093865910
Rectangle's length: 5.5917608449198415
Rectangle's width: 3.7281131807212389
Solution description

Instance: TE0.95
Number of ellipses: 3
Semi-axis lengths: (1.700, 1.615), (1.200, 1.140), (0.800, 0.760)
Rectangle's area: 20.5183705222052062
Rectangle's length: 5.5511871817933960
Rectangle's width: 3.6962130532187230
Solution description

Instance: TE0.90
Number of ellipses: 3
Semi-axis lengths: (1.700, 1.530), (1.200, 1.080), (0.800, 0.720)
Rectangle's area: 18.8996014452201742
Rectangle's length: 5.3515468397006494
Rectangle's width: 3.5316146922255780
Solution description

Instance: TE0.80
Number of ellipses: 3
Semi-axis lengths: (1.700, 1.360), (1.200, 0.960), (0.800, 0.640)
Rectangle's area: 16.0999183753445294
Rectangle's length: 4.7512473928509520
Rectangle's width: 3.3885666319058769
Solution description

Instance: TE0.70
Number of ellipses: 3
Semi-axis lengths: (1.700, 1.190), (1.200, 0.840), (0.800, 0.560)
Rectangle's area: 13.7990860047184150
Rectangle's length: 4.0750719615004147
Rectangle's width: 3.3862189760294887
Solution description

Instance: TE0.60
Number of ellipses: 3
Semi-axis lengths: (1.700, 1.020), (1.200, 0.720), (0.800, 0.480)
Rectangle's area: 11.6500516331516195
Rectangle's length: 3.4489736422290713
Rectangle's width: 3.3778314483210119
Solution description

Instance: TE0.50
Number of ellipses: 3
Semi-axis lengths: (1.700, 0.850), (1.200, 0.600), (0.800, 0.400)
Rectangle's area: 9.7438388804918308
Rectangle's length: 3.3983557452298898
Rectangle's width: 2.8672215656553299
Solution description

Instance: TE0.40
Number of ellipses: 3
Semi-axis lengths: (1.700, 0.680), (1.200, 0.480), (0.800, 0.320)
Rectangle's area: 7.9165416652819207
Rectangle's length: 3.3936822424408239
Rectangle's width: 2.3327292008305820
Solution description

Instance: TE0.30
Number of ellipses: 3
Semi-axis lengths: (1.700, 0.510), (1.200, 0.360), (0.800, 0.240)
Rectangle's area: 6.1656614779925114
Rectangle's length: 3.4638965495145455
Rectangle's width: 1.7799785270309558
Solution description

Instance: TE0.20
Number of ellipses: 3
Semi-axis lengths: (1.700, 0.340), (1.200, 0.240), (0.800, 0.160)
Rectangle's area: 4.1578856444668446
Rectangle's length: 5.0796062533766912
Rectangle's width: 0.8185448708160936
Solution description

Instance: TE0.10
Number of ellipses: 3
Semi-axis lengths: (1.700, 0.170), (1.200, 0.120), (0.800, 0.080)
Rectangle's area: 2.0819262402802794
Rectangle's length: 5.0952497868913813
Rectangle's width: 0.4086014086368208
Solution description