A tale of 2-dimensional (Guillotine) bin packing Nikhil Bansal (IBM) Andrea Lodi (Univ. of Bologna, Italy) Maxim Sviridenko (IBM)

2-d bin Packing 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 Bins

Many Applications  Cloth cutting, steel cutting, wood cutting  Truck Loading  Placing ads in newspapers  Memory allocation in paging systems  …

Guillotine Bin Packing Guillotine Cut: Edge to Edge cut across a bin

Guillotine Bin Packing Guillotine Cut: Edge to Edge cut across a bin k-stage Guillotine Packing [Gilmore, Gomory] k recursive levels of guillotine cuts to recover all items stage 4-stage

Non-guillotine Packing

Motivation for Guillotine Packings Practical constraint (simplicity of cutting tool) Many heuristics are “small” stage packing algorithms Many column generation techniques Nice combinatorial properties (2-stage: Shelf Packing) Shelf Packing

Asymptotic approximation ratio 1-d Packing: Cannot distinguish in poly time if need 2 or 3 bins (Partition Problem) Asymptotic Approximation ratio (  ): If Alg (I) ·  ¢ Opt(I) + O(1) Asymptotic PTAS: If Alg(I) · (1+  ) Opt(I) + f(  )

Previous Work on APTASes 1-d Packing: (1+  ) Opt + 1 [de La Vega, Lueker 81] Strip Packing: APTAS [Kenyon, Remila 96] Square Packing: APTAS [B, Sviridenko 04] [ Correa, Kenyon 04] General 2-d Packing: No APTAS [B, Sviridenko 04] 2-stage (or Shelf) Packing: APTAS [Caprara, Lodi, Monaci 05] Shelf Packing

Our Result APTAS for arbitrary 2-d guillotine packing 1) Approximate with O(1) stage packing 2) APTAS for O(1) stage packing. Bit surprising as no APTAS for general 2-d packing. A large stage guillotine packing

Talk outline APTAS for 1-d packing APTAS for square packing APTAS for strip packing Why no APTAS for general 2-d packing ? Our algorithm and analysis Grouping and rounding Fractional packing Size classification

1-d: Rounding to simple instance Partition big into O(1) groups, with equal objects 0 1  0 1  I’ I... I’ ¸ I

1-d: Rounding to a simpler instance Partition big into O(1) groups, with equal objects 0 1  0 1  I’ I... I’ ¸ I I’ – { } · I I’ ¼ I I’ can be solved optimally

Filling in smalls Take solution on bigs. Fill in smalls (i.e. <  ) greedily. 1) If no more bins need, already optimum. 2) If yes, every bin (except maybe 1) filled to 1-  Alg(I) · Size(I)/(1-  ) +1 · Opt/(1-  ) +1

Strip Packing (Cutting Stock) Given rectangles (height · 1, width · 1) Strip: Width 1, infinite height Place rectangles (no rotation) to minimize height

Strip Packing Kenyon Remila 96: (1+  ) Opt + 1/  2 Key insight: Can split a rectangle arbitrarily along the height Individual height does not matter width cumulative height

Square Packing: (Small, big) not enough 0 1 I Various square sizes Identical to 1-d case: Round bigs, easy to solve exactly Problem: Not clear how to pack small in gaps without waste? Small, medium and big

General 2-d Packing Packing with arbitrary rectangles 1.698… [Caprara 02] No APTAS (from 3D Matching) [B., Sviridenko 04] 01 1no order width height Each item has unique identity Bin packs well if and only if the “right” items chosen

Guillotine Packing Arbitrary -> O(1) stage packings [Transformation Procedure] APTAS for O(1) stage packings. [ Rounding relies on Opt’s structure]

Size Classification H V B S Big: Width > , Height >  Horizontal: Width > , Height <<  Vertical: Height > , Width <<  Rest medium: O(  ) area, ignored Ignore smalls Allow V and H to be fractionally packed (individual identities not so important) V H  <<  medium

Transformation procedure (Idea) If only bigs (i.e. (h,w) >  ) -> depth < 1/ , degree < 1/  If high depth or degree, lots of regions corresponding to V or H Since fractional (structure does not matter), can prune the tree Guillotine Tree: Leaf = object, Interior node = region

APTAS for O(1) stage packing Lemma: Almost Opt solution with O(1) different types of regions. Do not know Opt, but can guess regions from a polynomially large set. k=O(1) possible types of guillotine trees (O(1) stage, O(1) degree, O(1) different regions) Guess n 1, n 2, …, n k : number of each tree type used. Matching rectangles to leaves in the trees Make V, H integral. Packing S (smalls) suitably.

Constant number of region types Apply 1-d rounding recursively on regions Round heights of shelves (width irrelevant) There is an almost Opt solution with O(1) different region types

Concluding Remarks Extensions: 90 degree rotations are allowed (using ideas of Van Stee, Jansen 05) Running time: n raised to a huge tower of 1/  ’s Can the algorithm be made practical? Can use to improve guarantee for general 2-d packing? 1.698… [Caprara 02] Best known gap between guillotine and general is 4/3 ?

Questions ?