steven s. selden and rob van stee- new bounds for multi-dimensional packing

8/3/2019 Steven S. Selden and Rob van Stee- New Bounds for Multi-dimensional Packing

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New Bounds for Mult i -d imensional PackingS t e v e n S . S e l d en * R o b v a n S t e er

A b s t r a c tN e w upper and lower bo un ds are presented for a multi-dimensional generalization of bin packing called bo xpacking.

Several variants of this problem , including bo un de dspace bo x packin g, square packing, variable sized bo xpacking an d resource au gm en te d box packing are alsostudied. T h e ma i n results, stated for d ---- 2, areas follows: A ne w up pe r bo un d of 2.66013 for onlinebox packing, a ne w 14/9 + 6 polynomial time oi~ineappro ximat ion algorithm for square packing, a ne wupper bo un d of 2.43828 for online square packing, ane w lower bo un d of 1.62176 for online square packing,a ne w lower bo un d of 2.28229 for bo un de d space onlines q ua r e p a c k in g a n d a n e w u p p e r b o u n d o f 2 . 32 5 71 f oronline two-sized bo x packing.1 I n t r o d u c t i o nBin packing is one of the oldest and most well-studiedproblems in comput er science [9, 4]. The s tu dy of thisproblem dates back to th e early 1970's, when computerscience was still in its formative phase--ideas whichoriginated in the stu dy of the bin packing problem havehelped shape computer science as we know it today. Theinfluence and importan ce of this problem axe witnessedby the fact that it has spawned of[ whole areas ofresearch, including the fields of online algorithms andapproximation algorithms. In this paper, we stu dya natural generalization of bin packing, called boxpacking.Pr ob le m Def ini tio n: Let d :> 1 be an integer. Inthe d - d i m e n s i o n a l b o x p a c k i n g problem, we receive asequence a of p/eces Pl ,P 2, -. ., PN- We use the wordspiece and i t e m synonymously. Each piece p has a fixed~a r tm en t of Computer Science, 298 Coates Hall,Louisiana State University, Bato n Rouge, LA 70803, U.S.A.s s e i d e n 0 a ~ . o r g . T h i s r e s e a r c h w a s partially upporte d by t h eR e s e a rc h C o m p e t it i v e n e ss u b p r o g r a m o f t h e L o u i s ia n a o a r d o fRegents.~lnstitut fiir lnformatik, Albert-Ludwigs-U niversit~t,Georges-K~hler-Allee, 79110 Freibu rg, Germany. [email protected], de. This work done while the authorwas at the CWI, The Netherlands. Research supported by t h eNetherlands Organ ization for Scientific Research (NWO), projectnumber StON 612-30-0{}2.

s i z e , which is sl(p ) "s d( 2) . I.e. si(P) is th e size ofp in the i th dimension. We have an infinite number ofb i n s each of which is a d-dimensional unit hyper-cube.Each piece must be assigned to a bin and a position( x l ( p ) , . . . , x d ( p ) ) , where 0 _< xi(P) an d x i ( p ) + s i (p ) ~ 1for 1 < i < d. Further, the positions must be assignedin such a way tha t no two items in the same bin overlap.A bin is e m p t y if no piece is assigned to it, otherwiseit is u s e d . The goal is to minimize the number of binsused. Note th at for d = 1, the box packing problemreduces to exactly the classic bin packing problem.

In this pap er, we focus mai nly on th e case of d = 2.We say that item p has w i d t h w ( p ) = s l O a) a n d h e i g h th(p) = s2(P). We note tha t ma ny of our result s are moregeneral, however, we focus on the two-dimensional casein order to avoid the cumbersome notation required bya more general treatment.There are a number of variants of this problemwhich axe of interest: 1 In the o n l i n e version of thisproblem, each piece must be assigned in turn, withoutknowledge of the n ext pieces. 2) In the s q u a r e p a c k i n gproblem we have the rest riction th at h(p) ~- w(p / for allitems p. 3 i In t he t w o - s i z e d b o x p a c k i n g problem, binshave one of two sizes, eithe r 1 1 or 1 x z. Th e algo rith mchooses the size of a bin when it is allocated and the costof a bin is equal to its area. 4) In the r e s o u r c e a u g m e n t e db o x p a c k i n g problem, the algorithm is allowed to havelarger bins than the adversary. The cost of each bin isone. 51 In the b o u n d e d s p a c e variant, an algorithm hasonly a const ant numbe r of bins available to accept itemsat any point during processing.The offline versions of these problems are NPohard, while even with unlimited computational abilityit is impossible in general to produce the best possiblesolution online. We therefore consider both online andoifline approximation algorithms.The standard measure of algorithm quality for boxpacking is the a s y m p t o t i c w o r s t c a se p e r f o r m a n c e r a t io/~A, which we now define. For a given inpu t sequence a,let co sta (a) be the number of bins used by algorithm 4on a and cost(a) be the minimum possible number ofbins used to pack pieces in a. Then

I cost~t(~) [ }R ~ =limsupsupn_~o ~. ~ co st (a l = n .


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In the case of a rand omized a lgor i thm we replacecostA (a) with F~[cost .a(a)] in the preceding defini t ion.P r e v i o u s R e s u l t s : T h e cl as si c o n li n e b i n p a c k i ngproblem was f i rs t inves t igated by Johnson [16], whoshowed tha t the NEXT FIT a lgor i thm has pe r formancera t io 2 . Subsequent ly , i t was shown by Johnson,D e m e r s, U l l m an , G a r e y a n d G r a h a m t h a t t h e FIRSTFIT a lgor i thm has pe r formance ra t io 17/10 ]17] . Yaoshowed that REVISED FIRST FIT has performance rat io5/3 , and fur the r showed tha t no onl ine a lgor i thm hasper forma nce ra t io le s s than 3 /2 ]27]. B rown and Liangindepen dent ly improved th i s lower bound to 1 .53635 [2 ,19] . The lower boun d cur rent ly s t ands a t 1 .54014,du e to va n Vliet [25]. De fin e 7r~+1 = 7ri(~i - 1) + 1,~1 2, an d Hoo ~ - w ,= ~-~,=1 1/(Tr, 1) 1.69103. Le e an dLee showed tha t the HARMONIC algori thm , which u sesboun ded space , achieves a pe r formance ra t io a rb i t ra r i lyclose to YI [18]. A sequ ence o f furt her resul ts hasbrought the upper bound down to 1 .58889 [18 , 21 , 22 ,24].

Whi le box packing i s a na tura l next s t ep f rom binpacking , the problem seems to be mo re d i ff i cu l t, and th enum ber of resu l t s is smal le r . Th e o i~ ine problem wasin t roduc ed by Chung, Gare y and Joh nson [3], whi le theonl ine problem was f i r s t inves t iga ted by Coppersmi thand Raghav an [5] . Th ey g ive an a lgor i thm based ofNEXT FIT wi th pe r formance ra t io 13/4 - - 3 .25 for d - -2. Csir ik, Frenk and Lab be [7] give an alg ori thm basedon FIRST FIT wi th pe r formance ra t io 49/16 ~- 3 .0625for d =- 2. Th e bes t resul t to d ate is tha t o f Csir ikand van Vl ie t [8] . Th ey present an a lgor i thm based onHARMONIC with perf orm ance r at io (II ~) d for a l l d > 2(2.85958 for d = 2) . For bou nde d space algori thm s, alower bound of (IIoo)d is implied by [8] . Several generallower bounds have been shown [14, 15, 26, 1] . The bes tof these for d -- 2 is 1.907 [1] , while the b es t lower bo undfor large d is less than 3.

For onl ine square packing, even less is known. Thefol lowing resul ts are known for d -- 2: Co ppe rsm ith andRaghavan f5] show an upp er bo und of 43/16 = 2 .6875and a lower bound o f 4 /3 . The u pper bo und i s improvedto 100/39 < 2 .56411 by Fuj i t a and Hada [13]. For theof l l ine problem, Fer re i ra , Miyazawa and Wakabayashigive a 1.988-approximation algori thm [12].

As far as we know, the only resul ts for onl ine two-s ized box packing and onl ine resource augmented boxpacking a re for the o ne-d imens iona l case .O u r R e s u l t s : I n t h i s p a p e r , w e p r e s en t a n u m b e r o fresul ts for onl ine and otf l ine box and square packing:

We show tha t i f we have a one d imens iona l onl inebin packing a lgor i thm ,4 chosen f rom a ce r t a inclass of a lgori thm s (which we define precisely la ter)

and the pe r form ance ra t io of ,4 is r , t hen we cancons t ruc t an onl ine box a lgor i thm for d = 2 wi thper forma nce ra tio a rb i t ra r i ly c lose to r I Ic . Thi simproves the up per bou nd for onl ine box packingto 2.73220.We go on to examine a s imple and na tura l random-ized var i an t of th i s a lgor i thm. Th e per formancera t io of the random ized var i an t i s shown to be a tmost 2.66013. I t i s poss ible to derandomize this a l -gor i thm, the re fore 2 .66013 is a l so an upp er bo undon the de te rmini s t i c pe r formance ra t io .We show tha t no onl ine box packing a lgor i thmwhich uses a t mos t d - 1 open b ins has fin it eper formance ra t io .For the offi ine squa re pack ing prob lem in twodimens ions , we g ive an 14/9 + z polynomia l t imeapproximat ion a lgor i thm.

For the onl ine square packing problem in twodimens ions , we show an u ppe r b ound of 2.43828. We show an improved lower bound of 1 .62176 for

onl ine square packing. We show the f i r st lower bou nd for boun ded space

onl ine square packing , name ly 2 .28229.For onl ine two-s ized packing with s izes 1 1 and1 z we show tha t i f the a lgor i thm may chose z ,then a performance rat io of 2.32571 is poss ible .

We show the f i r st uppe r bou nd for onl ine resourceaugmented box packing .

Our resu l ts a re de r ived us ing a nu mb er of genera lt echniques which should prove use fu l in making fu r the rprogres s on mul t i -d imens iona l packing problems .2 O n l i n e A l g o r i t h m sOur onl ine algori thms, which we describe f i rs t , are al lbased on the fol lowing ideas : We say that an i temis s m a l l if i t~ wid th is at mo st e, whe re c :> 0 is ause r de f ined paramete r . O therwise , the i t em i s large.We pack i tems which are large and small different ly.Specif ical ly, we pack small i tems us ing an algor i thm w ecal l GEOMETRIC NEXT FIT, described in Sect ion 2.2.To pack large i tems, we use a different a lgori thm, whichwe cal l A B, described in Sect io n 2.3. Bot h of theseuse as sub-rout ines onl ine one-dimensional bin packingalgori thms. We theref ore begin in Sect ion 2.1 bydescribing a class of one-d imensio nal onl ine bin p ackinga lgor i thms .


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2 . 1 T h e I M PR O VE D H A R M O N IC C l a s s o f A l g o -r i t h m s W e s h a ll u s e a s s u b r o u t i n e s o n e - d im e n s i o n a lb i n p a c k i n g a l g o r i t h m s f r o m a f a m i l y o f a l g o r i t h m sw h i c h w e c a l l IM P R O VE D H A R M O N I C , w h i c h w e d e s c r i b ei n t h i s s e c t io n . T h i s s e t o f a l g o r i t h m s i s a s u b s e t o f t h ec l a s s S U P E R H A R M O N IC d e f i n e d b y S e i d e n [ 24 ]. T h er e s t r i c t i o n t o t h i s s u b s e t y i e ld s a l g o r i t h m s w h i c h a r ee a s i e r t o d e s c r i b e a n d a n a l y z e . T h e I MP R OV E D H A R -M O N I C c l a s s i n c l u d e s H A R M O N I C , R E F I N E D H A R M O N I Ca n d M O D I F IE D H A R M O N IC a m o n g o t h e r s .

T h e f u n d a m e n t a l i d e a o f a l l IM P R O VE D H A RM O N I C( a n d S U P E R H A R M O N I C ) a l g o r i t h m s i s t o f i r s t c l a s s i f yi t e m s b y s i z e , a n d t h e n p a c k i t e m s a c c o r d i n g t o t h e i rc l a s s ( a s o p p o s e d t o d i r e c t l y l e t t i n g s i z e i n f l u e n c e p a c k -i n g d e c i s io n s ) . A n i n s t a n c e , 4 o f t h e I MP R OV E D H A R -M O N IC a l g o r i t h m i s d e s c r i b e d b y a n u m b e r o f p a r a m -e t e r s : a n i n t e g e r n A _> 4 , a r ea l nu m b er A A G [ ~ , ]a n d r e a l n u m b e r s ~ t - - - ~ t E [0 , 1 ]. T o f a c i l i t a t e t h ec l a s s i f i c a t i o n o f i t e m s , w e d e f i n e t h e i n t e r v a l t r~ t o b e(K "l-1 , ~'~] fo r .~ = I , . . . , I%.4 -1- I , w h er e

t ~t = 1 t ~ = l - - A

1 f o r 5 < i < n + lt h = i - 2t ~ "}'2 ~--- 0

W e f u r t h e r d e f i n e e ~ = t ~ + l . N o t e t h a t t h e s e i n te r v al sa r e d i s j o i n t a n d t h a t t h e y c o v e r ( 0, 1 ]. A n I M PR O V EDH A R M O N I C a l g o r i t h m . 4 a s s i g n s e a c h p i e c e a t ype de-p e n d i n g o n i t s s iz e . A n i t e m o f s i ze s h a s t y p e v A (s ) = ji f a n d o n l y i f s E I~ 4. W h e n i t i s c l e a r t h a t w e a r e d i s -c u s s i n g a l g o r i t h m , 4 , w e s h a l l d r o p t h e . 4 s u b s c r i p t .

N o t e t h a t i f w e p l a c e a n i t e m o f t y p e 2 i n a b in ,t h e a m o u n t o f s p a c e l e ft o v e r is A . I f p o s s i b l e , w ew o u l d l i k e t o u s e t h i s s p a c e t o p a c k c e r t a i n i t e m s . T oa c c o m p l i s h t h i s , w e a s si g n e a c h i t e m a c o l o r , r ed o r blue.T h e a l g o r it h m a t t e m p t s t o p a c k r e d i t em s w i t h t y p e 2i t e m s .

T o a s s i g n c o l or s t o i t e m s , t h e a l g o r i t h m u s e s t w os e t s o f c o u n t e r s , e t , . . . , e ~ a n d s t , . . . , s ~ , a l l o f w h i c ha r e i n i t ia l l y ze r o . s l k e e p s t r a c k o f t h e t o t a l n u m b e ro f t y p e i it e m s . T h e n u m b e r o f t y p e i i t e m s w h i c h g e tc o l o r e d r e d is c o u n t e d b y e l . F o r 1 < i < n , t h e i n v a r i a n te i = [ a l s l ] i s m a i n t a i n e d , i .e . t h e p e r c e n t a g e o f t y p ei i t e m s c o l o re d r e d i s a p p r o x i m a t e l y ~ i . S i n c e i t e m s o ft y p e s 1 , 2 a n d 3 m a y n o t f i t i n a b i n w i t h a t y p e 2 i t e m ,w e r e q u i r e t h a t c~ = (x2 ------ 3 ---- 0. I t e m s o f t y p e i ~ 4a r e g u a r a n t e e d t o fit i n a b i n w i t h a t y p e 2 i t em . D e f i n e~/i = [A /t ~] f o r i > 4 . T h i s i s t h e n u m b e r o f r e d i t e m s o ft y p e i t h a t t h e a l g o r i t h m p l a c es t o g e t h e r i n a b i n w i t ha t y p e 2 i t e m .

T h e n u m b e r o f t y p e i i t e m s w h i c h f it in a b in i s

/~i = [ 1 / t i J - T y p e i i t e m s w h i c h g e t c o l o r e d b l u e a r ep l a c e d f l i i n a b i n , a s i n t h e H A R M O N I C a l g o r i t h m .

W e e x p l a i n m o r e p r e c i se l y t h e m e t h o d b y w h i ch r e di t e m s a r e p a c k e d w i t h t y p e 2 i t e m s . W h e n a b i n i so p e n e d , i t i s a s s i g n e d t o a group. T h e b i n g ro u p s a r en a m e d :

1 , 3 , 4 , . . . , n + 1 ;C2,73;(? , i ) , for a i ~ 0, 1 < i < n;(2 , i) , for a i ~ t 0, 1 < i < n.

T h e g r o u p i c o n t a i n s b i n s w h i c h h o l d o n l y b l u ei t e m s o f t y p e i . T h e r e i s o n e o p e n b i n i n e a c h o f t h e s eg r o u p s ; t h i s b i n h a s f e w e r t h a n fl i i t e m s . T h e c l o s e db i n s a l l c o n t a i n / 3 i i t e m s . T h e g r o u p ( 2 , j ) c o n t a i n s b i n sw h i c h c o n t a i n a b l u e i t e m o f t y p e 2 a lo n g w i t h r e d i t e m so f t y p e j . A c l o se d b i n i n t h i s g r o u p c o n t a i n s o n e t y p e 2i t e m a n d " yj t y p e j i t e m s . T h e r e i s a t m o 6 t o n e o p e n b i ni n a n y o f t h e s e g r o u p s . T h e g r o u p ( 2 , ? ) c o n t a i n s b i n sw h i c h h o l d o n l y a b l u e i t e m o f t y p e 2 . T h e s e b i n s a r e a l lo p e n , a s w e h o p e t o a d d r e d i t e m s t o t h e m l a t er . T h eg r o u p ( ? , j ) c o n t a i n s b i n s w h i c h h o l d o n l y r e d i t e m s o ft y p e j . A g a i n , t h e s e b i n s a r e a ll o p e n , b u t o n l y o n e h a sf e w e r t h a n 7 j i t e m s . W e w i l l t r y t o a d d a t y p e 2 i t e mt o t h e s e b i n s i f p o s s ib l e .

W e c a l l b i n s i n t h e l a s t t w o g r o u p c l a s se s i n d e t e r -m i n a t e . E s s e n t i a l l y , t h e a l g o r i t h m t r i e s t o m i n i m i z e t h en u m b e r o f i n d e t e r m i n a t e b i n s , w h i l e m a i n t a i n i n g a ll t h ea f o r e m e n t i o n e d i n v a r i a n t s . I .e . w e t r y t o p l a c e r e d a n dt y p e 2 i t e m s t o g e t h e r w h e n e v e r p o ~ i b l e ; w h e n t h i s i sn o t p o s s i bl e w e p l ac e t h e m i n i n d e t e r m i n a t e b i n s in h o p et h a t t h e y c a n l a t e r b e c o m b i n e d . A f o r m a l d e s c r i p t i o no f I M P R O V ED H A R M O N I C i s d i s p l a y e d i n F i g u r e 1 .

A n I M PR O V ED H A R M O N IC a l g o r i t h m c a n b e a n a -l y z ed u s i n g t h e m e t h o d o f w e i g h t i n g s y s t e m s d e v e l o p e di n [ 24 ]. T h e f u l l g e n e r a l i t y o f w e i g h t i n g s y s t e m s i s n o tr e q u i r e d h e r e , s o w e a d o p t a s l i g h t l y d i f fe r e n t n o t a t i o nt h a n t h a t u s e d i n [ 24 ], a n d r e s t r i c t o u r s e l v e s t o a s u b -c l a s s o f w e i g h t i n g s y s t e m s . H o w e v e r , t h e d e f i n i t io n sh e r e a r e c o n s i s t e n t w i t h t h o s e i n [ 2 4 ] .

A w e i g h t i n g s y s t e m f o r a IM P R OV E D H A R M O N ICa l g o r i t h m , 4 is a p a i r (W.4 , VA) . WA and V .4 a r ew e i g h t i n g J u n c t i o n s w h i c h a s s i g n e a c h i t e m p a r e a ln u m b e r b a s e d o n i t s si ze . T h e w e i g h t i n g f u n ct i o n s f o ra n I M P R O VE D H A R M O N I C a l g o r i t h m . 4 a r e d e f i n e d a sf o l lows .

I if x E I2 ,1 - ~W A ( : r ) = /~ i i f x E I 1 , I 3 , I 4 , . . . , I , ~ ,

x i f x E I n + l .


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Ini t ia l ize e l ~- 0 and s i ~- 0 for 1 < i < n.F or each p i ece p :

i ~- ~(p).I f i = n + 1 place p us ing NEXT FIT.Else:

si *-- si + 1.I f e i < [ o t i s / ] :

ei ~-- ei + 1.Col or p r ed .I f t he r e i s an op en b i n i n g roup (2 , i )o r ( ? , i ) w i t h f e w e r t h a n 7 i t y p e ii t ems , t hen p l ace p i n t h i s b i n .E l se i f t he r e i s some b i n i n g ro up(2 , ? ) t hen p l ace p i n i t an d chan get he g rou p of th i s b i n t o (2 , i ) .O t he rwi se , ope n a new gro up ( ? , i ) b i nand place p in i t .

E lse:Col or p b l ue .If i = 2:

I f t h e r e i s s o m e b i n i n g r o u p ( ? , j )t h e n p l a c e p i n it a n d c h a n g e t h eg r o u p o f t h is b i n t o ( 2, j ) .Otherwise, o p e n a n e w gro up (2 , ? )bin and place p there.

Else:If here is an o pe n bin in gro up i withfewer than/~i items , then place p inthis bin.I f n o t , o p e n a n e w g r o u p i b i n a n dplace p there .

F i gu re 1 : T he IM PROVED HARM ONIC Al go r i t hm.

0 i f x E Iu ,1 - a i a iV A ( x ) = - -- fi i- + - - i f x E I l , I a , 1 4 , . . . , l n ,7i

x i f x E 1.+1 .- ;T h e f o l lo w i n g l e m m a f o ll ow s d i r e c t l y f r o m L e m m a 4of [24]:LEMMA 2.1. If . ,4 is an IMPROVED HARMONIC al-g o r i t h m t h e n f o r a ll a , w e ha v e c o s t A ( a ) VH(X) for all x E [0, 1]. T h e r esu lts o f [18i m p l y t h a t t h e v a l u e o f P ( W H ) is

i(2.t) n . = ~ _1 n - 1.= r~ 1 + (~i+~ - 1) (n - 2)'wh ere i i s the in teg er sat i s fy ing ~ri < n < 7r i+l. No tet h a t t h i s i s c o n s i s t e n t w i t h o u r e a r l ie r d e f i n i t io n o f H asince Hoo = l im,~_,~o H,.

E x a m p l e 2 MOD IF IED H A R M O N I C ( M H ) i s de-f i ned by n = 39 , A = 265 / 684 and c~1 = a 2 = a s = 0;a a = 1/9 ; ots = 1/1 2; a 6 = ot7 = 0; ot~ = ( 3 9 -i ) / ( 3 7 ( i - - 1)) , for 8 < i < 38; a a9 = 0. Th e res ul t so f [ 21] i mp l y t h a t t he v a l ue o f max{7~ (W M H) , P ( V M n ) }i s 538 / 333 < 1 .61562 .2 .2 T h e GIi ~OME TRIC NE XT F IT Al go r i t h m GP .-OM E TRIC NE XT F IT i s a genera l i za t i on o f NE XT F IT .T h e a l g or i t h m has a r ea l pa r a met e r 6 E (0 , 1 ). W e sha l la s s u m e t h a t a l l i t e m s p r o c e s s e d b y G E O M E T R I C N E X TF IT a r e sma l l . W e sa y t ha t a p i ece p has c l a s s i i f andon ly i f e(1 - 6) I -1 < w(p ) < e(1 - 6) i . S inc e eve ry i te mproces sed b y GE OM E T RIC NE XT F IT has w(p ) < ~ , ev-e ry i t em has a c l a s s i > 0 . E ve ry i t em i s packed accord -


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ing to its class. We say that class i is a c t i v e if we haveprocessed at least one small item with class i.

An i sl ice is a box of height 1 and wi dth e(1 - 5)i.T he s i ze of an i slice is defined to be e(1 _5 )i . We divideeach bin used by the alg orit hm into several slices. Inparticular, we always have one bin which we call f u l l yo p e n . When we need to allocate a slice, we first try toallocate i t from this bin. Let S be the sum of the sizesof the sl ices already al located in the fully open bin. Ifthe size of the new slice is at most 1 -- S, we allocatethe slice in the fully open bin. Otherwis e, we allocatea new bin and allocate the new slice wit hin it. Thenew bin becomes fully open, while the old fully openbin becomes ei ther o p e n or c l osed a s shall be explainedlater. Intuitively , we are using NEXT FIT to allocateslices in bins.

We run an independent copy of .A for each activeclass. We den ote t he c opy of .A assoc iated wi th class ias ~4n+1,~, for reasons t hat shall be com e clear in Sec-tion 2.3. Pieces o f class i axe packed into i slices usinga l g o r i t h m A n + L ~ - W e c a n i m a g i n e t h a t A n + 1 , i d o e s n o tk n o w t h a t i t e m s h a v e a w i d t h , i t t r e a t s h e i g h t a s si ze ,a n d t r e a t s s l ic e s s b i n s.

A s m e n t i o n e d b e fo r e, a b i n o t h e r t h a n t h e o n e f ul lyo p e n b i n i s e i t he r o p e n o r c l os e d . S u c h a b i n i s d e f in e dt o b e o p e n if it c o n ta i n s a n o p e n s l ic e w i t h r e s pe c t t os o m e a l g o r i t h m ~ 4 ~ + i, i . I .e . If t h e r e i s t h e p o s si b i li t yt h a t s o m e a l g o r i t h m J t ~+ l ,~ w i l l p l a ce a n i t e m i n a n islice con tai ned in it. Otherw ise, it is closed.

I n e s se n ce , G E O ME T R I C N E X T F I T p a c k s i t e m s i n tos l ic e s u s i n g , 4, t h e n p a c k s s l ic e s i n t o b i n s u s i n g N EXTFIT. In general , i f we have some algori thms 4 and B forthe one-dimensional bin packing problem, then we canuse them to construct an algori thm for two dimensionsas follows: We someh ow classify items. We use Jt topack i tems of the same class into sl ices, an d th en use Bto pac k slices into bins. In the next s ection, we use thisa lgori thm design paradigm to const ruct an a lgori thmfor packing large items.2.3 Th e .4 x B Al go ri th m Let ~4 and B beins tan ces o f IMPROVED HARMONIC. We co ns tr uc t analgori thm .4 x B for the two-dimensional box packingpro blem from ~4 and B.

A x B oper ates as follows: We r un n B+ 1 inde-pendent copies of A , which we denote ~41,. . . , -Ans+l-Small items are given to GEOMETRIC NEXT FIT, whichalso uses A as its sub-r outin e algorithm . A large itemp is given to alg or ith m Yl,~(~(p)). I.e. item s are pro-cessed according to the type of their width. Algor i thmJti packs th e ite ms it receives in slices of height 1 andwidth t~. When A is used to pack an item into a slice,i t considers only the i tem's height . I.e. i t considers the

size of the item to be h(p). The size of a slice is definedto be its width. Wh en a new slice is alloca ted by someAi, we allocate it from a bin using algo rit hm B. Asfar as B is concerned, the slices are items, and B knowsnothing of the internals of a slice.

From now on, we consider the case where ,4 is HAR-MONIC (H) , w hile B is so me IMPROVED HARMONIC algo -ri thm. It ma y he possible to get improved performvalcewithout this restrict ion, but the analysis would seem tobe significantly more complicated. We shall also assumetha t nA ---- nB, and hencef orth we refer to this co mm onvalue as n.

We are ready to start analyzing H x B. Duringcourse of processing, H1,. . . , H~+l make al location re-quests for slices to B. Define ~ to be the sequence ofslices received by B. Define 0i to be the sub- sequen ceof a of items p with T~(w(p)) = i. Fro m the definitionof H x/ 3 we have

costHxB(a) = costB(~) + costoNF( en+l)We first consider the cost incurred by B: If s E is

a slice, we use w ( s ) to denote its width (size). Define ito be the sub- sequence of of slices s with rB (w( s)) = i.To start , t hrou gh a long series of manipu lat ions we findthat

~t

tt


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( )~ ( a - ~ y ~ w.(h(p))+o0)= ~ e ( 1 - 5 ) ~ E WH(h(p))

j - - 1 p E ~ o n + l , /

+ O ( 1 ) y ~ e ( 1 - 5) ~j = l

= Z E e (1- - t~)JWtt(h(p)) + O(1)

< 1 -1 _S k E w (p )W I- l(h (p) )+O(1 )j = l PEOn+I,$1-- 1 - J Z w(p)WH(h(P)) +0 (1) "PEQ,~+IThere fore , the nu mbe r of b ins used by GEOMETmO

NEXT FIT iscos tcN F(0 ,+I ) < (1 -- 5)(1 -- e) + O(1)

1- - I - - ( ~ E W l t x B ( P ) + O ( 1 ) -pEQ,~+IPut t ing these two resul t s toge ther , we f ind tha t

cos t lq8(a) is a t most

i= 1 pEot i= 1 PE@i )+ 1 1 - W . x (p ) + o ( x)

-< 1---a m ~ w~~(p), ~v a ~s (p ) +o(1) .LpE~ peer(2.2)

We are us ing the fact that WttxB(p) = VHB(p) for a l lp E ~ + 1- As is the case with a weight ing sys tem for theone d imens iona l problem, we can conc lude tha t the cos tof H x B is upper b ounde d b y the sum o f the we ight s ofi tems in a .

We now re la te the a lgor i thm' s cos t to th a t of theopt imal of l tine a lgor i thm. To fac ih ta te th i s , we need thefo llowing def in i tions: Le t m = cos t (a ) b e the num ber ofbins in the opt im al off i ine solut ion. For 1 < i < m , le tXi be the mu l t i - se t of i t ems conta ined in the i th b in ofthe op t ima l off l ine solut ion. We now rewri te (2.2) as

1~ _ ~ ma x Z WttB(p),i=l pEXI

- }E E V r ] x s ( p) + O ( 1 )i = l p ~ X i

1 { p ~-- - ma x WHB(p),1 g 1 _ < i _ < ,~ iE VHB (P)/es t (er) + O(1).pEX~ )

We conc lude tha t the p e r form ance ra t io of H x B i s upperb o u n d e d b y t h e v a l ue o f t h e m a t h e m a t i c a l p r o g r a m :Maximize1 { }_ ~ m ax E WHB(P), ~ VHB(p)

pEX pEXover al l f ini te mult isets of i tems X which f i t in as ingle ( two-dimensional) bin. We cal l this opt im izat io nproblem Q. The rea der should note the s imi la r i ty tothe progra m P defined earl ier .

The fo l lowing l emma i s the key to proving mos t ofour onl ine upper bounds :LEMMA 2.2. Let f and g be funct ions mapping from(0,11 to R +. Let F and G be the values o f th emathematical programs P( f) and P(g), respect ively.T h en t he ma x imu m o f ~ p e x f (h (p ) ) g (w ~ 9 ) ) o ver a llfinite m ultisets o f item s X which fit in a single bin is atmos t F G.

P r o o f T h e p r o o f is s i m il a r to t h e p r o o f o f L e m m a 2in [8] . We cons ider modi f i ed mul t i s e ts of i tems X ' andX" . The i t ems in each se t a re the same, however , thewidth s and h eights of i tems are modified as follows:For each i tem p E X the re is an i tem p' E X ~ withw(p ' ) = w(p) and h(p ' ) = f (h~)) . For each i t em p e Xt h e r e i s a n i t e m p " E X " w i t h w(p") = g(w(p)) andh(p") = f(h(p)). N o t e t h a t t h e a r e a o f i t e m p " i sf (h(p))g(w(p)) . We show tha t the i t ems in X" f i t i nbox of he ight F and width G , which proves the des i redresul t .

Towards th i s goa l , we fi r st show tha t the i t em s in X 'f i t in a box of height F and wid th 1. Since the i tems in Xfit in a bin, th ere is som e pai r of functio ns ~b : X H I0, 1]an d ~o : X ~ [0, 1] such th at (~(p ), ~o(p)) is the po sitio nof p . The pos i t ions mus t obey q~(p) + w(p) < 1 and~o(p) + h(p) _< 1 for al l p e X. Fur ther mo re, thecoordina tes mus t be such tha t th e a rea of the reg iondescr ibed by{ ( x , y)l x E [~(P)' (p) + w(p )]n[(q), (q) + w(q)], }y e [~v(p),~(p) + h (y)l n [~(q), ~(q) + h(q)] ,is zero for a ll p, q E X. Intui t ively, we describe a binus ing a coordina te sys tem wi th or ig in (0 ,0) in the b in ' s


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l o w e r l e f t c o r n e r . F o r e a c h i t e m p , ( p ) i s t h e h o r i z o n t a lc o o r d i n a t e o f t h e l o w e r l e f t c o r n e r o f p , w h e r e a s ~ a(p ) i st h e v e r t i c a l c o o r d i n a t e . T h e f i rs t r e s t r i c ti o n s i m p l y s a y st h a t e a c h i t e m m u s t b e w h o l l y c o n t a i n ed i n th e b i n. T h es e c o n d c o n d i t i o n p r e v e n t s t w o i t e m s f r o m o v e r l a p p i n g .

W e m a y a s s u m e w i t h o u t l o ss of g e n e r al i ty t h a t f o ra l l i t em s p E X , e i th e r ~b(p) = 0 or the r e ex is t s ani t e m q s u c h t h a t ( p ) - - (q ) + w(q). I . e . e a c h i t e mi s a s fa r t o t h e l e f t a s p o s s i b l e . I f t h i s w e r e n o t t h ec a se , w e c a n s i m p l y m o v e p l e ft u n t i l i t is t r u e , w i t h o u te f f e ct i n g t h e v a l i d i t y o f t h e p a c k i n g . S i m i la r l y, w e a l s oa s s u m e w i t h o u t l o s s o f g e n e r a l i t y f o r a ll i t e m s p E X ,e i t h e r ~ a(p ) = 0 o r t h e r e e x i s t s a n i t e m q s u c h t h a t~o(p) = ~(q) + h(q). W e say that q ~ - p i f a n d o n l y i f ( p ) = (q)+w(q). W e s a y t h a t p i s leftmost if ~a(p) ----0a n d rightmost i f t h e r e i s n o q s u c h t h a t p * - q . Ahorizontal chain i s a s e q u e n c e o f i t e m s P l ~ - P2 ~ - " ' " ptw h e r e p ~ i s l e f t m o s t a n d P t i s r i g h t m o s t . A n a l o g o u s l y ,we sa y th a t q & p i f an d on ly i f ~a( p) = ~o( q) + h(q) andd e f i n e bottommost, topmost and vertical chain.G i v e n a n d ~ , w e c o n s t r u c t ~ a n d ~ J , w h i c h p a c kt h e i t e m s i n X ' i n t o a b i n o f h e i g h t F a n d w i d t h 1 . T h i si s done as f o l lows : ~b~(p ) i s a s s igne d ( p) f or a l l 14 q X ' .F o r a ll b o t t o m m o s t i t e m s q~ w e s e t w'(q') = 0 . O n c e w eh a v e c o m p u t e d ~ f ( q f ) f o r a ll q~ s u c h t h a t q p w e a s s i g n~ ( p t ) t h e v a l u e

max qa'(q') + h(q').q' , qlPS i n ce t h e r e l a t i o n J. i n d u c e s a d i r e c t e d a c y c li c g r a p h o nX , t h i s p r o c e s s t e r m i n a t e s . F u r t h e r , i t i s e a s i ly s e e nt h a t n o i te m s o v e r l a p i n t h e r e s u l t i n g p a ck i n g . S u p p o s et h a t t h e p a c k i n g d o e s n o t f i t i n t h e r e q u i r e d b o x . O n l yt h e v e r t i c a l c o o r d i n a t e s o f i t e m s c h a n g e , s o t h i s m e a n st h e r e i s s o m e i t e m r ' w i t h ~ o '( r~) -l-h(r') > F. W i t h l o s so f g e n e r a li t y , r t i s t o p m o s t . T h e n b y t h e c o n s t r u c t i o n o f~ d, t h e r e i s a v e r t i c a l c h a i n f l ~ " *" ~ f t = r s u c h t h a t

tZ h(p;)= l ( h O , , ) ) > f .i = l i - - ItB u t s i n c e ~ -~ i= l h ( p i ) < 1 , t h i s c o n t r a d i c t s t h e f a c t t h a t

~ ( f ) h a s v a l u e F .No w, g iv en b ' an d ~a , we con s t r u c t ~b" an d ~" ,w h i c h p a c k t h e i t e m s i n X" i n t o a b i n o f h e i g h t F a n dw i d t h G . T h e c o n s t r u c t i o n i s a n a l o g o us t o t h e o n e j u s tg i v e n : ~ a '( 1 4 ') i s a s s i g n e d ~ ' ( p ' ) f o r all t 4' ~ X ' . F o ra l l l e f t m o s t i t e m s q" w e s e t ~'(q') = O. O n c e w e h a v ec o m p u t e d dp(q') f or a l l q" s u c h t h a t q ~ - p w e a s s i g nqV' (14' ) the value

max ~bt t (q ' ) + w(q") .q " , q * - - p

O n c e t h e c o n s t r u c t i o n i s c o m p l e t e , i f t h e r e i s s o m er i g h t m e s t i t e m r " w i t h ~ b "( r ~r) + w ( r " ) > G t h e n t h e r ei s a h o r i z o n t a l c h a i n p~' *- - - . ~ -- p~l _ r " such th a t

l t= > c

i=1 i=1tSince ~-~i=lw(pi) ~_ I, thi s contradicts t he fa ct that

/)(9) has value G .2 . 4 O n l i n e U p p e r B o u n d s G i v e n t h e r e su l ts o f t h ep r e v i o u s s e c t i o n , w e a xe a b le t o p r o v e a n u m b e r o f u p p e rb o u n d s .T H E O R E M 2 . 1 . For online box packing, for all ~ > O,the asymptot ic performance ratio of H MH is at most2 . 7 3 2 2 0 / ( 1 - 6 ) and at least 2 . 7 2 9 9 0 .P r o o f W e s h o w o n l y t h e u p p e r b o u n d a n d re f e r t o th ef u l l p a p e r f o r t h e l o w e r b o u n d .

W e h a v e n = 3 9 a n d t h e r e f o r e I I ~ = 4 3 8 / 2 5 9 b y(2.1 ). F ro m [21], max{~:~(WMH),~1:~(VMH)} _< 53 8/ 33 3.N o t e t h a t Q c a n b e d e c o m p o s e d i n to t w o m a t h e m a t i c a lp r o g r a m s w h i c h h a v e t h e f o r m r e q u i r e d b y L e m m a 2 .2 .I n t h e f i rs t w e h a v e f = W H a n d g = W M H -H e n c e m a x p e x W H x M H ( P ) 7, the asymptotic performance ratioof H H is at most 2.43828 and at least L~8~9.


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W e have two mor e r esu l t s tha t f o l low d i r ec t ly f r omL e m m a 2 . 2.

The VARIABLE HARMONIC (VH)algor i thm [6, 23] ,i s a n o p t i m a l b o u n d e d s p a c e a l g o r i t h m f o r th e o n e -d imen s iona l two- s ized on l ine b in packing pr oblem . I t spe r f or mance r a t io i s a f unc t ion R ~ H ( z ) . W e s h o w t h efirst res ult for d >__ 2. For d ---- 2, an alg or ith m is allow edto ch oose whe the r i t opens a b in of size 1 x 1 or a b inof s ize 1 x z when i t opens a new b in .THEOREM 2.3 . F o r t h e t w o - s i z e d t w o - d i m e n s i o n a l o n -l i n e b o x p a c k i n g p r o b l em , t h e p e r f o r m a n c e r a t io o f H xV H R ff (z )The r esu l t s o f [23] imply tha t f o r z = 5 /7 th i s i s a t m os t2.32571.

I n [ 10], fo r the one- d imens iona l r e sour ce au gme ntedb i n p a c k i n g p r o b l e m w h e r e t h e a l g o r i t h m i s a l l o w e dbins of s ize z , i t i s shown th a t the pe r f or m ance r a t ioof HARMONIC is a function R ~ ( z ) . A g a i n w e s h o w t h efirst r esu lt for d _> 2:THEOREM 2.4 . F o r t w o - d i m e n s i o n a l r e s o u rc e a u g -m e n t e d o n l i n e b o x p a c k in g , t h e a s y m p t o t i c p e r f o r m a n c er a t io o f H x H i s ( R ~ ( z ) ) 2 w h e n th e a lg o r i th m u s e s z x zb ins .3 A n O f fl i ne A p p r o x i m a t i o n A l g o r i t h m f o rSquare P a c k i n gW e d e v e l o p a 1 4 / 9 + e a p p r o x i m a t i o n a l g o r i t h m c a l l e dSQUARE SCHEME for t he t w o - d i m e n s i o n al s q u a r e p a c k -ing pr oblem f or any e > 0 . The a lgor i thm uses s ever a li d e a s u s e d i n t h e d e s ig n o f a p p r o x i m a t i o n s c h e m e s f o rone- d imens iona l b in packing [ 111 .

F r o m n o w o n , t h e s i z e of an i t em i s de f ined to bei t s he igh t ( which i s the s ame as it s wid th) . Th e bas icidea is as follows: we def ine A - - e2/( 4(4 + ~)2) < 1/4.A n i t e m i s large i f i t s s ize is g r ea te r th an A, and s m a l lother wise .

For t he la r ge i t ems , we c r ea te a ( 1 + e /2) ~app r ox ima t ion us ing br u te for ce . To do th i s , we usea s tand ar d technique : W e r ound the s izes of a l l squar esso tha t we have a cons tan t n um ber of l a r ge s izes . W ee n u m e r a t e a l l p o s s i b l e p a c k i n g p a t t e r n s f or i t ems wi ththese f ixed s izes and t r y a l l the poss ib i l it i e s . I n f ac t , thep a c k i n g p a t t e r n s u s e d f o r l a rg e i t e m s c a n b e r e s t r i c t e dt o w h a t w e c a l l d o m i n a n t p a t t e r n s , e s s e n ti a l ly p a t t e r n si n w h i c h t h e c a r d i n a l it y o f a n y l a rg e i t e m p r e s e n t c a n n o tbe inc r eased .

Smal l i t ems a r e then packed f i r s t in to b ins wi thce r ta in la r ge i t ems , and then , i f neces sa r y , in to b insby themse lves . Ca l l a pa t te r n which conta ins a s ing lei t e m o f s iz e g r e a te r t h a n ~ a s in g le to n . T o p a c k t h esmal l i t ems , we f i r s t use those b ins packed accor d ing to

a s ing le ton pa t te r n . To do th i s , supp ose we have a b inw h i c h c o n t a i n s o n l y a n i t e m o f s i ze y E ( 1 , 1 - v ~ ] .2W e p lace th i s i t em in the lower le f t cor ner and d iv idet h e r e m a i n i n g a r e a i n t o t w o r e c t a n g u l a r a r e a s , o n e o fw i d t h 1 a n d h e i g h t 1 - y a n d t h e o t h e r o f w i d t h 1 - y a n dhe ight y . W e sor t the sma l l i t ems and use the NEXT FI TDECREASI NG a lgor i th m to pack i t ems in thes e a r eas , a sdesc r ibed in [ 20]. I f we r un ou t o f these r ec tang ula ra r eas , then we cont inue to use NEXT FI T DECREASI NGb u t w i t h n e w l y a l l o c a t e d b i n s.

I n the case tha t a l l smal l i t ems can be packed ins ing le ton b ins , the so lu t ion f or l a r ge i t ems i s a gooda p p r o x i m a t i o n a n d w e a r e d o n e. T h e o t h e r c a s e, w h e r en e w b i n s m u s t b e a l l o c a t e d t o h o l d s o m e s m a l l i t e m s ,pr oves to be mor e com pl ica ted . Th e f o l lowing r esu l t ,d i r ec t ly adapted f r om [ 20] , p r oves to be ve r y use f u l :LEMMA 3.1. (M EIR ~ MOSER) Let ~ be a l is t o fs q u a r e s, o ] wh ic h th e la r g e s t i s o f s i z e z . ~ c a n b e p a c k e din a rec tangle o f he ig ht x >_ z and w id t h y > z us in gNEXT FIT DECREASING i f t h e to ta l a r e a o f i t e ms in

a t m o s t z 2 + z ) ( y -W e f i r s t bound the ava i lab le a r ea in non- s ing le ton b ins .LEMMA 3.2. T h e t o t a l u n u s e d a r e a o f a n y d o m i n a n tp a t t e r n w h i c h i s n o t a s i n g l e t o n i s a t m o s t s _9"P r o o f T h e r e a r e f iv e c a s es , d e p e n d i n g o n t h e n u m b e ro f i t e m s o f s i ze g r e a t e r t h a n 1 / 3 i n t h e p a t t e r n . T h e r eaxe a t mos t f our such i t ems , so the r e a r e f ive cases .

I n the f i r s t case , the r e a r e exac t ly f our i t ems ofs i ze g r e a t e r t h a n 1 / 3 , a n d t h e u n u s e d a r e a i s a t m o s t1 - 4 / 3 2 = s / 9 .

I n t h e o t h e r c a s e s , w e us e L e m m a 3 .1 t o s h o w t h a tthe to ta l s ize of the i t ems in the b in i s a t l eas t 4 /9 . Forde ta i l s , we r e f e r the r ead er to the f u l l pape r . W e n o w b o u n d t h e u s a b l e a r e a i n s i n g l e t o n b i n s .LEMMA 3.3. Fo r 0 < A


9/10

4 9 4

T h e r a t i o o f t h i s t o t h e a v a i l a b le a r e a i sA 2 + ( 1 - - A ) ( l - - y - - A )

1 - - yT h i s i s s t r i c t l y d e c r e a s i n g i n y f o r 0 < A < 1 / 4 a n d s oi t i s a t a t m o s t

1 -- ~ -- 2A + A3/2 + 2 A2.... > 1 - 2 v ~ ,l _ v / ~for A < 1 /4 .N o w c o n s i d e r t h e a r e a o f si z e y ( 1 - - y ) . B y s i m i l a ra r g u m e n t s , t h e r a t i o o f a re a u s e d t o t o t a l a r e a i s

A 2 + ( y _ A ) ( I _ y _ A ) = 1 A ( 1 - - 2 A )y ( 1 - - y ) y ( 1 - - y )

> 1 - - 2 v ~ + 2 A a /2 > 1 - 2 V ~ ,- - l _ v / ~

for A < 1 /4 .T h e t o t a l a r e a a v a i l a b l e in t h e b i n i s 1 - y 2 .D e f i n e Z b( y) t o b e t h e n u m b e r o f si n g l e t o n b i n s

c o n t a i n i n g a n i t e m o f s iz e y in t h e s o l u t i o n f o r t h e l a r g ei t e m s a n d - - ~ -~112


10/10

495

B o u n d e d s p .U n b o u n d e d

d = l d = 2 d = 3 d = 41.69103 2.28229 2.73429 3.067211.54014 1.62176 1.60185 1.55690

Table 1 : Lower boun ds f or on l ine squar e pac kingto [ 26] , i t i s pos s ib le to f or mula te l inea r p r ogr ams toca lcu la te lower bounds f r om th i s s equence , which wegive in Table 1 .5 C o n c l u s i o n sW e h a v e i n t r o d u c e d n e w t e c h n i q u e s f or t h e m u l t i d i m e n -s i o n a l b i n p a c k i n g p r o b l e m a n d u s e d t h e m t o d e r i v e an u m b e r o f r es u l ts . I t s h o u l d b e p o s s i b le t o a p p l y t h e s etechniques f ur the r .

O n e o p e n p r o b l e m i s t h a t t h e a n a l y s i s o f , 4 Bd o e s n o t g o t h r o u g h i f w e c h os e B t o b e H A R M O N I C + + ,w h i c h i s c u r r e n t l y t h e b e s t p e r f o r m i n g o n e - d i m e u s i o n a la l g o r i t h m . T o g e t b e t t e r r e s ul t s , w e n e ed s o m e s t r o n g e rw a y o f b o u n d i n g t h e w e i g h t i n a b in t h a n L e m m a 2 .2 .R e f e r e n c e s

[11 BL ITZ~ D.~ VAN VLIE T, A.~ AND WOEGING ER~ G . J.Lower bounds on the asymp totic wors t-case rat io of on-l ine bin packing algori thms. Unpublished manuscr ipt ,1996.[2] BROWN, D . J . A lower boun d for on-line one-dimensional bin packing algorithms. Tech. Rep. R-864,Coo rdina ted Sei. Lab., University of I llinois at U rban a-Champaign, 1979.[3] CHUNG, F. R. K., GAR EY, M. R., ANt) JOHNSON,D. S. On packing two-dimensional bins. SIAM Journalon Algebraic and Discrete Methods 3 (1982), 66-76.[4] CO~FMAN, E. G. , GA meY , M. R., AND JOHNSON,D .S . Appr oximat ion a lgori thms f or b in packing:A survey. In Approximation Algorithms ]or NP.hardProblems, D. Hochbaum, Ed. PWS Publishing Com-pany, 1997, eh. 2.[5] C O P P E R S M I T H , D ., A N D P tdk GH A V A N , P . M u l tidi m e n-sional online bin pack ing: A lg orithms and worst caseanalysis. Op e r a t i o n s R e s e a r c h L e t t e r s (19 89), 1 7 - 2 0.

[ 6 1 CsImK , J. A n on-line algorithm for variable-sized inpacking. A c t s I n f o r m at i c a 2 6 , 8 (19 89), 97- 7 09.

[ 7] C SIR I K , J .~ F R E N K , J ., A N D L A B B E , M . T w o -dimensional rectangle packing : on-line e tho ds and re-sults. D i s c r e t e A p p l i e d M a t h e m a t i c s ~(5, (Sep 1 993),1 9 7 - 2 0 4 .{8] CS IRIK, J., AND VAN VLIET, A. An on -line a lgo rith mfor multidimensional bin packing. Operations ResearchLetters 13, 3 (Apr 1993), 149-158.[9] CsIm K, J. , AND WOE~INGER, G. On-line packi ngand covering problems. In On-Line Algorithms--Th eState of the Art, A. Fiat an d G. Wo eginger, Eds. ,Lecture Notes in Computer Science. Springer-Verlag,1998, eh. 7.[10] CSIRIK, J., AND WOEGINGER, G. P~esource aug me n-tat ion for online bounded space bin packing.

In Proceedings of the 27th International Colloquiumon Automata, Languages and Programming (Jul 2000),pp. 296-304.[ II] D E L A V E G A , W . F ., A N D L U E K E R , G . S. B i n p ac k in g

can b e solved within 1 + e in linear time. C o m b i n a t o r i e a1, 4 (1981), 349-355.[121 FERR EmA , C. E., MIVAZAWA, F. K., AND WAK -ABAYASHI, Y. Packing squares into squares. PesquisaOperaeional 19, 2 (1999), 223--237.[13] FUJITA, S., AND HADA, T. Two -dimen sional on-linebin packing problem with rotatab le i tems. In 6thAnnual International Conference on Computing andCombinatories (2000), pp. 210-20.[14] GALAMBOS, G. A 1.6 low er-bou nd for th e two -dimensional online rectangle bin-packing. Acta Cyber-netics 10, 1-2 (1991), 21-24.[ 1 5 ] G A L A M B O S , G., A N D V A N V L I E T , A . L o w er b o u n ds for

1, 2 and 3 -dimensional online b in packing algorithms.Computing 5~ (1994), 281-297.[16] JOHNSON, D. S. Fast a lgorith ms for bin packing.Journal Computer Systems Science 8 (1974), 272-314.[17] JOHNSON, D. S., DEMEaS, A., ULLMAN, J. D.,G A R E Y , M . R . , A N D G R A H A M , R . L . Worst-case per-formance b ounds for simple one-dimensional packingalgorithms. SlAM Journal on Computing 3 (1974),256-278.[18] LEE, C., AND LEE, D. A sim ple on-line bin-packin galgorithm. Journal of the ACM 32, 3 (Jul 1985), 562-572.[19] LIANC, F. M. A lower boun d for online bin packing.Information Processing Letters 10 (1980), 76-79.[20] MEIR, A., AND MOSER, L. On pa cking of squaresand cubes . Journal of Combinatorial Theory 5 (1968),126-134.[21] RAMANAN, P., BROW N, D., LE E, C., AND LEE,D. On-line bin packing in linear time. Journal ofAlgorithms 10, 3 (Sop 1989), 305-326.[22] RIC~IEV, M. B . Impro ved bou nds for harmon ic-basedbin packing algor i thms. Discrete Applied Mathematics34 (1991), 203-227.[23] SELDEN, S. S. An op tim al online algorit hm forboun ded space variable-sized bin packing. In Pro-ceedings of the 27th International Colloquium onAutomata, Languages and Programming (Jul 2000),pp. 283-295.[24] SEIDEN, S. S. On t he online bin packin g problem.In Proceedings of the 28th International Colloquiumon Automata, Languages and Programming (Jul 2001),pp. 237-249.[25] VAN VLIET, A. An imp rov ed lower bound for online binpacking algor i thms. Information Processing Letters 43,5 (Oct 1992), 277-284.[26] VAN VL IET, A. Lower and upper bounds for online binpacking and scheduling heuristics. PhD thes is , Eras-mus Univers ity, Rott erdam , T he Nether lands , 1995.[27] YAO, A. C . C . New algorith ms for bin packing.2ournal o] the ACM 27 (1980), 207-227.

steven s. selden and rob van stee- new bounds for multi-dimensional packing

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