Round and Approx: A technique
for packing problems
Nikhil Bansal (IBM Watson)
Maxim Sviridenko (IBM Watson)
Alberto Caprara (U. Bologna, Italy)
Problems
Bin Packing: Given n items, sizes s1,…,sn, s.t.
0 < si · 1. Pack all items in least number of unit size bins.
D-dim Bin Packing (with & without rotations)
1
4
3
2
5
3
6
2
4
1
6
5
Problems
d-dim Vector Packing: Each item d-dim vector.
Packing valid if each co-ordinate wise sum ·1
Bin: machine with d resources
Item: job with resource requiremts.
Valid
Invalid
Set Cover: Items i1, … , in
Sets C1,…,Cm.
Choose fewest sets s.t. each item covered.
All three bin packing problems, can be viewed as set cover.
Sets implicit: Any subset of items that fit feasibly in a bin.
Short history of bin-packing
Bin Packing: NP-Hard if need 2 or 3 bins?
(Partition Prob.)
Does not rule out Opt + 1
Asymptotic  approximation:  OPT + O(1)
Several constant factors in 60-70’s
APTAS: For every  >0, (1+) Opt + O(1) [de la Vega, Leuker 81]
Opt + O(log2 OPT)
[Karmarkar Karp 82]
Outstanding open question: Can we get Opt + 1
No worse integrality gap for a natural LP known
Short history of bin-packing
2-d Bin Packing: APTAS ) P=NP [B, Sviridenko 04]
Best Result: Without rotations: 1.691… [Caprara 02]
With rotations: 2
[Jansen, van Stee 05]
d-dim Vector Packing: No APTAS for d=2
Best Result: O(log d) for constant d
If d part of input, d1/2 - ) P=NP
Best for d=2 is 2 approx.
[Woeginger 97]
[Chekuri Khanna 99]
Our Results
1) 2-d Bin Packing : ln 1.691 + 1 = 1.52
Both with and without rotations
(previously 1.691 & 2)
2) d-Dim Vector Packing: 1 + ln d
(for constant d)
For d=2: get 1+ ln 2 = 1.693
(previously 2)
General Theorem
Given a packing problem, items i1,…,in
1) If can solve set covering LP
min C xC
s.t.
C: i 2 C xC ¸ 1
8 items i
2)  approximation : Subset Oblivious
Then (ln  + 1) approximation
d subset oblivious approximation for vector packing
1.691 algorithm of Caprara for 2d bin packing is subset ob.
Give 1.691 subset ob. approx for rotation case (new)
Subset Oblivious Algorithms
Given an instance I, with n items
 (I) = all 1’s vector
(S) incidence vector for subset of items S.
There exist k weight (n - dim) vectors w1, w2,…,wk
For every subset of items S µ I, and  > 0
1) OPT (I) ¸ maxi ( wi ¢ (I) )
2) Alg (S) ·  maxi (wi ¢ (S)) +  OPT(I) + O(1)
An (easy) example
Any-Fit Bin Packing algorithm:
Consider items one by one. If current item does not fit in any
existing bin, put it in a brand new bin.
No two bins filled · 1/2
(implies ALG · 2 OPT + 1 )
Also a subset oblivious 2 approx
K=1: w(i) = si (size of item i)
1) OPT(I) ¸ i 2 I si = w ¢ (I)
[Volume Bound]
2) Alg(S) · 2 w ¢  (S) + 1
[ # bins · 2 ( total volume of S) + 1 ]
Non-Trivial Example
Asymptotic approx scheme of de la Vega, Leuker
For any  > 0,
Alg · (1+) OPT + O(1/2)
We will show it is subset oblivious
1-d: Algorithm
I
0
1
1-d: Algorithm
bigs
I
0

1
1-d: Algorithm
Partition bigs into 1/2 = O(1) groups, with equal objects
I
0

1
.
I’
0

.
.
1
I’ ¸ I
1-d: Algorithm
Partition bigs into 1/2 = O(1) groups, with equal objects
I
0

1
.
I’
0
.
.

1
I’ ¸ I
I’ ¼ I
I’ – { } · I
I’ has only O(1/2) distinct sizes
LP for the big items
1/2 items types. Let ni denote # of items of type i in instance.
LP:
min C xC
s.t.
C ai,C xC ¸ ni
C indexes valid sets (at most (1/2)(1/) )
ai,C number of type i items in set C
At most 1/2 variables non-zero.
Rounding: x ! d x e
Solution (big) · Opt (big) + 1/2
8 size types i
Filling in smalls
Take solution on bigs. Fill in smalls (i.e. <) greedily.
1)
2)
If no more bins need, already optimum.
If needed, every bin (except maybe one) filled to 1-
Alg(I) · Volume(I)/(1-) +1
· Opt/(1-) +1
We will now show this is a subset oblivious algorithm !
Subset Obliviousness
LP:
min xC
C ai,C xC ¸ ni
Dual: max ni wi
i ai,C wi · 1
8 item types i
for each set C
If consider dual for subset of items S
Dual: max |type i items in S| wi
i ai,C wi · 1
for each set C
Dual polytope independent of S: Only affects objective function.
Subset Obliviousness
LP:
min xC
C ai,C xC ¸ ni
Dual: max ni wi
i ai,C wi · 1
8 item types i
for each set C.
Define vector Wv for each vertex of polytope (O(1) vertices)
LP*(S) = maxv Wv ¢  (S)
(LP Duality)
Alg(S) · LP*(S) + 1/2 = maxv Wv ¢  (S) + 1/2
Opt(I) ¸ LP(I) = maxv Wv ¢  (I)
Handling smalls: Another vector w, where w(i) = si
General Algorithm
Theorem: Can get ln  + 1 approximation, if
1) Can solve set covering LP
2)  approximate subset oblivious alg.
Algorithm:
Solve set covering LP, get soln x* .
Randomized Rounding with parameter  > 0, i.e. choose set C
independently with prob  xC*
Residual instance: Apply subset oblivious  approx.
Proof of General Theorem
After randomized rounding,
Prob. element i left uncovered · e-
Pf: Prob = C: i 2 C (1-  xC) · e-
( as C: i 2 C xC ¸ 1 )
E ( wi ¢ (S)) · e- wi ¢ (I)
wi ¢ (S) sharply concentrated
(variance small: proof omitted)
maxi (wi ¢  (S)) ¼ e- maxi (wi ¢ (I) )
· e- OPT(I)
But subset oblivious  algorithm implies
Alg(S) ·  maxi (wi ¢ (S)) ·  e- OPT(I)
Proof of General Algorithm
Expected cost = Randomized Rounding + Residual instance cost
¼  LP cost +  e- Opt
Gives  +  e- approximation
Optimizing , gives 1 + ln  approx.
Wrapping up
d-dim vector packing: Partition Instance I into d parts I1,…,Id
Ij consists of items for which jth dim is largest
Solving Ij is just a bin packing problem
1+ for bin packing gives d+ subset oblivious algorithm
2-d bin Packing: Harder
Framework for incorporating structural info. into set cover.
Other Problems?
Questions?