Packing Rectangles into Bins

1
Packing Rectangles into Bins

• Nikhil Bansal (CMU)
• Joint with Maxim Sviridenko (IBM)

2
Problem

• Given Collection of rectangles (by height,
  width)
• Goal Pack in min of unit square bins
• Rectangles parallel to bin edges
• Cannot be rotated

Rectangles
4
5
3
6
1
2
6
6
1
1
2
Bins
2
3
3
4
4
5
5
3
Problem

• Given Collection of rectangles (by height,
  width)
• Goal Pack in min of unit square bins
• Rectangles parallel to bin edges
• Cannot be rotated

Rectangles
4
5
3
6
1
2
3
3
1
1
6
2
2
6
2-d Bin Packing
4
4
5
5
4
Motivation

• Many Applications
• Real World packing of (real world) bins
• Placing ads in newspapers
• Cloth cutting Minimizing sheets to buy
• Truck Loading
• Memory allocation in paging systems

5
1-d Bin Packing

• Special case of 2-d packing
• 1-dim bin packing
• Collection of intervals with length in 0,1,
• Pack into min of unit bins

4
3
!
2
2
4
1
1
3
6
1-d Bin Packing

• Special case of 2-d packing
• 1-dim bin packing
• Collection of intervals with length in 0,1,
• Pack into min of unit bins (NP-Hard)

4
3
!
2
2
4
1
1
3
7
Approximation algorithms

• An algorithm is a ? approximation (? gt 1) if
• runs in polynomial time
• Value of solution ? Opt
• Opt value of optimum solution
• Polynomial time approximation scheme (PTAS) is a
  family of 1? approximation algorithms for every
  ? gt 0.
• E.g., Running Time

8
1-d bin packing Inapproximability

• Cannot get a poly time approximation lt 3/2
• Partition problem (NP-Hard) Given s1,,sn , S
  s1s2,,sn
• 9 a subset of elements that sums to exactly
  S/2 ?

s1
s6
s2
s5
s3
s4
S/2
S/2
But maybe can get Opt 1 in poly time ?
9
Asymptotic PTAS

• Alg (1?) Opt f(?)
• f(?) depends only on ? e.g. (1/?)(1/?)
• For 1-d bin packing
• Alg (1?) Opt O(1/?2) de la Vega,
  Lueker 81

10
Outline

• Introduction
• APTAS for 1-d packing de la Vega, Lueker 81
• Results for 2-d packing
• Conclusions

11
APTAS for 1-d bin packing

• Theorem de la Vega, Lueker 81
• Alg(I) Opt(I)/(1-?) 1/?2

¼ Opt(I) (1?) f(?)
4
3
!
2
2
4
1
1
3
12
Main idea

• Simplify Original instance I -gt I
• I easy to solve
• Solutions of I and I close (within 1?)

13
Ideas applied to 1-d packing

• ? Small ? Big
• 1) I ! I with 1/?2 different big sizes solns.
  within 1?
• 2) I easy If k O(1) different big sizes, can
  get Opt k

14
1-d Rounding to a simpler instance
I
0
1
Various object sizes
15
1-d Rounding to a simpler instance

• Ib I restricted to bigs.
  Let b of bigs (i.e. ?)

Ib
0
1
?
1
Various object sizes
16
1-d Rounding to a simpler instance
Ib I restricted to bigs.
Let b of bigs (i.e. ?)
Partition big into 1/?2 groups, each group has
b ?2 objects
Ib
(here b?2 3)
0
1
?
17
1-d Rounding to a simpler instance
Ib I restricted to bigs.
Let b of bigs (i.e. ?)
Partition big into 1/?2 groups, each group has
b ?2 objects
Ib
0
1
?
Instance Ib Ignore largest b ?2 objects.
Round up sizes to smallest
size in next higher group
Ib
0
1
?
18
1-d Rounding to a simpler instance
Ib I restricted to bigs.
Let b of bigs (i.e. ?)
Partition big into 1/?2 groups, each group has
b ?2 objects
Ib
0
1
?
Instance Ib Ignore largest b ?2 objects.
Round up sizes to smallest
size in next higher group
Ib
0
1
?
Alg(Ib) Alg(Ib)
19
1-d Rounding to a simpler instance
Ib I restricted to bigs.
Let b of bigs (i.e. ?)
Partition big into 1/?2 groups, each group has
b ?2 objects
Ib
0
1
?
Instance Ib Ignore largest b ?2 objects.
Round up sizes to smallest
size in next higher group
Ib
0
1
?
Alg(Ib) Alg(Ib) b ?2
20
1-d Rounding to a simpler instance
Ib I restricted to bigs.
Let b of bigs (i.e. ?)
Partition big into 1/?2 groups, each group has
b ?2 objects
Ib
0
1
?
Instance Ib Ignore largest b ?2 objects.
Round up sizes to smallest
size in next higher group
Ib
0
1
?
Alg(Ib) Alg(Ib) Alg(Ib) b ?2
21
1-d Rounding to a simpler instance
Ib I restricted to bigs.
Let b of bigs (i.e. ?)
Partition big into 1/?2 groups, each group has
b ?2 objects
Ib
0
1
?
Instance Ib Ignore largest b ?2 objects.
Round up sizes to smallest
size in next higher group
Ib
0
1
?
Alg(Ib) Alg(Ib) Alg(Ib) b ?2
Opt(Ib) b ? ) b ?2 ? Opt(Ib)
22
1-d Solving the few and big case

• Ib 1/?2 different sizes gt ?. Call these
  s1,,sk .
• Configuration A way to pack a bin (Eg C 3
  s1, 17 s3, 5 s18 )
• Configurations (1/?2)1/? O(1)
• xi of bins with configuration i
• nj of objects of size sj in instance
• cij of objects of size sj in configuration i.
• Minimize ?i xi
• ?i cij xi nj 8 j 2
  1,..,1/?2
• xi 0 8 i, xi 2 Z

IP for Ib
23
1-d Few and Big using LP

• Minimize ?i xi
• ?i cij xi nj 8 j 2
  1,..,1/?2
• xi 0 (Relaxed to
  be fractional)
• Clearly, LP (Ib) OPT(Ib)
• xi could be fractional.
• Round up to next integer ( Eg 17.34 -gt 18)
• Adds configurations (1/?2)1/? O(1)
• In fact, adds 1/?2 (non-zero xis in basic
  soln)

24
1-d Filling in the smalls

• So, Alg(Ib) Opt(I)/(1-?) 1/?2
• Packing smalls
• In each bin, fill as many smalls as possible.
• If bins not enough, open new bins to fill smalls.
• Proof
• If no new bins opened, done.
• If new bins opened, all bins (except maybe last)
  filled 1-?
• So, Alg(I) Area(I)/(1-?) 1
• Opt(I)/(1-?)
  1

25
1-d Overview

• 0) Partition into small and big
• 1) Pack small objects later
• 2) Round large objects to O(1) sizes.
• Solve the few and big case almost
  optimally.

26
Outline

• Introduction
• APTAS for 1-d packing de la Vega, Lueker 81
• Results for 2-d packing
• Conclusions

27
Outline

• Introduction
• APTAS for 1-d packing de la Vega, Lueker 81
• Results for 2-d packing
• Square packing problem
• General rectangle packing problem
• Conclusions

28
Square Packing Problem

• Given a collection of squares (hypercubes for
  dgt2)
• Pack into min of unit bins.

1
1
1
4
2
2
!
2
5
3
3
5
4
5
4
6
6
7
7
6
3
7
29
Square Packing Previous Work

• 2-dim 2.125 Chung, Garey,
  Johnson 82
• 1.454 Epstein, van
  Stee 03
• d-dim (3/2)d Coppersmith,
  Raghavan 89
• 2-(2/3)d Kohayakawa et al
  02
• We give an APTAS (i.e., Alg(I) 1?
  Opt(I) f(?) )

30
Outline

• Introduction
• APTAS for 1-d packing de la Vega, Lueker 81
• Results for 2-d packing
• Square packing problem
• O(1) Approximation
• APTAS
• General rectangle packing problem
• Conclusions

31
O(1) Approximation Square Packing

• Opt Total area of squares in I.
• If we can use constant fraction of area in each
  bin
• ) O(1) approximation.
• Eg If use 1/100th area in each bin ) 100
  approx

32
O(1) approximation Square Packing

• If big (length 1/3), put in its own bin.
• Use 1/9th area.

Big uses gt 1/9th area
Need a way to pack small objects efficiently.
33
Shelf Packing

• Given a rectangular region of size a b
• Goal Pack squares of length s

a
b
34
Shelf Packing

• Given a rectangular region of size a b
• Goal Pack squares of length s
• Algorithm Decreasing size shelf packing.

• Take squares in decreasing size
• Place sequentially

a
1
3
2
b
35
Shelf Packing

• Given a rectangular region of size a b
• Goal Pack squares of length s
• Algorithm Decreasing size shelf packing.

• Take squares in decreasing size
• Place sequentially
• If next does not fit,
• open a new shelf

a
1
3
2
b
36
Shelf Packing

• Given a rectangular region of size a b
• Goal Pack squares of length s
• Algorithm Decreasing size shelf packing.

• Take squares in decreasing size
• Place sequentially
• If next does not fit,
• open a new shelf

4
a
8
1
3
b
37
Shelf Packing

• Given a rectangular region of size a b
• Goal Pack squares of length s
• Algorithm Decreasing size shelf packing.

• Take squares in decreasing size
• Place sequentially
• If next does not fit,
• open a new shelf

a
b
38
Shelf Packing

• Given a rectangular region of size a b
• Goal Pack squares of length s
• Algorithm Decreasing size shelf packing.

Wasted Space s(ab)
a
b
39
Shelf Packing

• Given a rectangular region of size a b
• Goal Pack squares of length s
• Algorithm Decreasing size shelf packing.

Wasted Space s(ab) Right side At most s a
a
b
40
Shelf Packing

• Given a rectangular region of size a b
• Goal Pack squares of length s
• Algorithm Decreasing size shelf packing.

Wasted Space s(ab) Right side At most s
a Top s16 b
a
b
41
Shelf Packing

• Given a rectangular region of size a b
• Goal Pack squares of length s
• Algorithm Decreasing size shelf packing.

Wasted Space s(ab) Right side At most s
a Top s16 b Shelf 1 (s1 s3) b
a
b
42
Shelf Packing

• Given a rectangular region of size a b
• Goal Pack squares of length s
• Algorithm Decreasing size shelf packing.

Wasted Space s(ab) Right side At most s
a Top s16 b Shelf 1 (s1 s3) b Shelf 2 (s4
s8) b
a
b
43
Shelf Packing

• Given a rectangular region of size a b
• Goal Pack squares of length s
• Algorithm Decreasing size shelf packing.

Wasted Space s(ab) Right side At most s
a Top s16 b Shelf 1 (s1 s3) b Shelf 2 (s4
s8) b . Adding all, at most (s1-s16) b
a
b
44
Square Packing Packing Smalls

• If all squares s, waste in a b region
  s(ab)
• If ab1, s 1/3 ) Waste 2/3
• O(1) approximation follows.

45
Outline

• Introduction
• APTAS for 1-d packing de la Vega, Lueker 81
• Results for 2-d packing
• Square packing problem
• O(1) Approximation
• APTAS
• General rectangle packing problem
• Conclusions

46
Square Packing
0
1
Squares with various lengths
47
Square Packing
G0
G1
Gm
G1/? -1
?m1
?01
?1
?m
?1/?
?2


0
48
Square Packing
G0
G1
Gm
G1/? -1
?m1
?01
?1
?m
?1/?
?2


0
Small
Medium
Big
Gm ?m1,?m) be such that area ? (total
area)
49
Square Packing
G0
G1
Gm
G1/? -1
?m1
?01
?1
?m
?1/?
?2


0
Small
Medium
Big
Gm ?m1,?m) be such that area ? (total
area)
Pack medium in separate bins using O(1) approx
Adds at most O(?) to the number of bins. Ignore
medium jobs from now on.
50
Square Packing
G0
G1
G1/? -1
?m1
?01
?1
?m
?1/?
?2


0
Small
Big
Gm ?m1,?m) be such that area ? (total
area)
Pack medium in separate bins using O(1) approx
Adds at most O(?) to the number of bins. Ignore
medium jobs from now on.
51
Square Packing
G0
G1
G1/? -1
?m1
?01
?1
?m
?1/?
?2


0
Small
Big
Gm ?m1,?m) be such that area ? (total
area)
Pack medium in separate bins using O(1) approx
Adds at most O(?) to the number of bins. Ignore
medium jobs from now on.

• ?0 1 , ?i ? ?i-12 (1,?,?3,?7,)

52
Square Packing

• Packing bigs Rounding technique similar to 1-d
  case,
• ! O(1) different lengths
• Opt(Ib) Opt(Ib) Opt(Ib)/(1-?)
• LP to find close to optimum packing of Ib

G0
G1
G1/? -1
?m1
?01
?1
?m
?1/?
?2


0
Small
Big
53
Square Packing Filling in the smalls
Bigs use ¼ Opt bins What about smalls?
54
Square Packing Filling in the smalls
Divide the free space into rectangular
gaps. Fill smalls in gaps using decreasing size
shelf algorithm
Rectangular Gaps
55
Square Packing Filling in the smalls
Small ?m1 , big ?m
bigs 1/?m2, gaps 4/?m2 Recall Packing
s in a b wastes at most s(ab) ab 2,
s ?m1 ? ?m2 Total waste 2 s gaps
2 ? ?m2 4/?m2
8 ?
Rectangular Gaps
56
Square Packing

• If no new bins opened for smalls, done.
• If we do, waste in each bin (except maybe last)
  is O(?), hence done by area lower bound.
• Thm ALG(I) (1?) Opt(I)
• Generalizes to dgt2 dimensions with few
  modifications.

57
Square Packing Technique Summary

• Partition into big, medium, small
• Ignore mediums
• Gap between big and small
• Ideas from 1-d packing smalls with low waste

58
Outline

• Introduction
• APTAS for 1-d packing de la Vega, Lueker 81
• Results for 2-d packing
• Square packing problem
• General rectangle packing problem
• Conclusions

59
Rectangle packing

• Problem Pack rectangles into min of unit bins.
• (no rotation)
• 2.125 approximation Chung, Garey, Johnson
  82
• 2? Kenyon,
  Remila 96
• Best known 1.691 ? Caprara 02
• Prior to our work APTAS open

60
Results for Rectangle Packing

• Main Thm No APTAS exists for 2-d rectangle
  packing.
• Hence for dgt2.
• Proof Lets skip it.

61
Conclusion and Open Problems

• Impossibility of APTAS for rectangle packing
• APTAS for square packing
• 1-d Case (1?) Opt O(1/?2) de la Vega,
  Lueker 81
• Opt O(log2(Opt))
  Karmarkar, Karp 82
• Opt O(1) ?
• d 2 1.691 ? Caprara 02
• d gt 2 1.691d Csirik, Van Vliet
  93
• ?(cd) lower bounds for d-dim bin packing ?

62
Results for Rectangle Packing

• Main Thm No APTAS exists for 2-d rectangle
  packing.
• Hence for dgt2.
• Thm Kenyon, Correa 04
• APTAS if allow Alg to pack in (1?) (1?)
  size bins
• Alg(I) (1?) Opt(I)
  f(?)
• Thm APTAS if allow Alg to pack in 1 (1?)
  size bins.

63
Hardness

• 2-d Vector Packing Problem
• Given 2-d vectors (equivalently rectangles). In
  each bin,
• ? widths 1 , and ? heights 1.

Valid
Invalid
64
Hardness of Vector Packing

• Xx1,,xn, Yy1,,yn, Zz1,,zn, T
  t1,,tn
• Each gtb/5 and lt b/3
• Goal Max. number of disjoint 4-tuples that sum
  to exactly b
• SNP-Hard 9 ? Cannot distinguish if n such
  tuples or (1-?) n
• Woegingers construction 97
• Xi (.2 xi/5b, .3 xi/5b) Yi
  (.2 yi/5b, .3 yi/5b)
• Zi (.2 zi/5b, .3 zi/5b) Ti
  (.2 ti/5b, .3 ti/5b)
• Observation All these fit in a bin iff
  xiyjzktl b

65
Rectangle Packing

• 2 rectangles for each object in X,Y,Z,T
• (Thin, Tall) and (Fat, Short) for each object
  in X,Y,Z
• (Fat, Tall) and (Thin, Short) for each object
  in T

Details to show that other patterns do not work