round and approx: a technique for packing problems
DESCRIPTION
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 s 1 ,…,s n , s.t. 0 < s i · 1. Pack all items in least number of unit size bins. - PowerPoint PPT PresentationTRANSCRIPT
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Round and Approx: A technique for packing problems
Nikhil Bansal (IBM Watson)Maxim Sviridenko (IBM Watson)
Alberto Caprara (U. Bologna, Italy)
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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
6
5
32
12 3
4 5 6
12
3
4 5
6
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Problemsd-dim Vector Packing: Each item d-dim vector. Packing valid if each co-ordinate wise sum ·1
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.
Valid Invalid
Bin: machine with d resourcesItem: job with resource requiremts.
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Short history of bin-packing
Bin Packing: NP-Hard if need 2 or 3 bins? (Partition Prob.)
Does not rule out Opt + 1Asymptotic 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 + 1No worse integrality gap for a natural LP known
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Short history of bin-packing2-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 [Woeginger 97]
Best Result: O(log d) for constant d [Chekuri Khanna 99] If d part of input, d1/2 - ) P=NP
Best for d=2 is 2 approx.
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Our Results1) 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)
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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 packing1.691 algorithm of Caprara for 2d bin packing is subset ob.Give 1.691 subset ob. approx for rotation case (new)
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Subset Oblivious AlgorithmsGiven an instance I, with n items(I) = all 1’s vectorS) 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)
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An (easy) exampleAny-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 approxK=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 ]
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Non-Trivial Example
Asymptotic approx scheme of de la Vega, LeukerFor any > 0, Alg · (1+) OPT + O(1/2)
We will show it is subset oblivious
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1-d: Algorithm
0 1I
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1-d: Algorithm
0 1I
bigs
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1-d: AlgorithmPartition bigs into 1/2 = O(1) groups, with equal objects
0 1
0 1I’
I
. . .
I’ ¸ I
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1-d: AlgorithmPartition bigs into 1/2 = O(1) groups, with equal objects
0 1
0 1I’
I
. . .
I’ ¸ I I’ – { } · I
I’ ¼ I I’ has only O(1/2) distinct sizes
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LP for the big items1/2 items types. Let ni denote # of items of type i in instance.
LP: min C xC s.t. C ai,C xC ¸ ni 8 size types i
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
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Filling in smallsTake solution on bigs. Fill in smalls (i.e. <) greedily.
1) If no more bins need, already optimum.2) 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 !
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Subset ObliviousnessLP: min xC
C ai,C xC ¸ ni 8 item types i
Dual: max ni wi
i ai,C wi · 1 for each set C
If consider dual for subset of items SDual: 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.
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Subset ObliviousnessLP: min xC
C ai,C xC ¸ ni 8 item types i
Dual: max ni wi i ai,C wi · 1 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
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General AlgorithmTheorem: 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.
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Proof of General TheoremAfter 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 impliesAlg(S) · maxi (wi ¢ (S)) · e- OPT(I)
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Proof of General AlgorithmExpected cost = Randomized Rounding + Residual instance cost¼ LP cost + e- Opt
Gives + e- approximationOptimizing , gives 1 + ln approx.
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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 problem1+ for bin packing gives d+ subset oblivious algorithm
2-d bin Packing: Harder
Framework for incorporating structural info. into set cover.Other Problems?
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Questions?