Approximate Solutions to Bin Packing Problems

y z

Edward G Coffman Jr Janos Csirik Gerhard J Woeginger

March

Intro duction

The following abstract packing problem arises in a wide variety of contexts in the real world

A list L ha a a i of items must b e packed into ie partitioned among a minimum

1 2 n

cardinality set of bins B B sub ject to the constraint that the the set of items in any bin

1 2

ts within that bins capacity To give just a few of the applications mo deled by bin packing

we mention storage allo cation for computer networks assigning advertisements to newspap er

columns packing fruit slices into tins assigning commercials to station breaks on television

copying a collection of les to several oppy disks packing trucks with a given weight limit

and the cuttingsto ck problems of various industries like those pro ducing lumb er and cable

Let A denote an arbitrary packing algorithm and let OPT denote an algorithm that pro

duces optimal packings Let AL and OPTL denote the numb ers of bins used by algorithms

A and OPT To measure worstcase b ehavior the asymptotic worstcase ratio R of algorithm

A

A is dened by

m

R lim sup R

A A

m

where

m

R supfALOPTL j OPTL mg

A

L

This ratio which we shall just call the asymptotic ratio is the p erformance metric of choice

for bin packing algorithms the standard ob jectives of bin packing problems are algorithms

with small asymptotic ratios Although we fo cus on these ratios comments on absolute ratios

R sup fALOPTLg will sometimes b e appropriate Certain classes of algorithms

A

L

have the undesirable prop erty that a small asymptotic ratio is obtained at the exp ense of a

large absolute ratio

egcbelllabscom Bell Labs Lucent Technologies Mountain Ave Murray Hill NJ

USA

y

csirikinfuszegedhu Department of Computer Science University of Szeged Aradi vertanuk tere

H Szeged Hungary Supp orted by the Pro ject u of the AustroHungarian Action Fund

z

gwoegioptmathtugrazacat Institut fur Mathematik B TU Graz Steyrergasse A Graz

Austria Supp orted by the START program YMAT of the Austrian Ministry of Science and by the Pro ject

u of the AustroHungarian Action Fund

In some applications an a priori upp er b ound on the size of all items

is known To gauge the p erformance of approximation algorithms on lists of such items the

parametric asymptotic ratio R is intro duced as

A

m

R lim sup R

A A

m

where

m

R sup fALOPTL j OPTL m and all a L are g

i

A

L

The problem dened so far is quite general Implicitly we have allowed the bins B to have

i

varying capacities and b oth bins and items to b e multidimensional The bulk of this pap er

not to mention the bin packing literature is conned to the one dimensional problem with

equal bin capacities which we normalize to for convenience in this case any set of items

whose total size is at most ts into a bin Corresp ondingly items are always assumed to b e

numb ers drawn from an interval for some We keep with these classical assumptions

throughout the next section the third and last section will cover more general versions of bin

packing We note once and for all that the versions of bin packing of interest to us here are

NPhard except where stated otherwise

Many extensions and variants have b een instrumental in maintaining the interest in the

general area of bin packing We mention a few basic illustrations here others can b e found

in the more comprehensive treatment by Coman Galamb os Martello and Vigo Three

principal parameters of bin packing problems are the numb er of items the bin capacity and

the numb er of bins Cho osing the numb er of bins as the ob jective function gives the classical

problem but cho osing one of the two others as the ob jective function also gives interesting

problems Multipro cessor scheduling actually predates bin packing and is the problem with a

given numb er m of bins and the ob jective of nding the minimum common capacity such that

all items of the given list can b e packed into m bins The problem falls more prop erly within

scheduling theory and is surveyed in that context in the survey article by Lawler Lenstra

Shmoys and Rinno oy Kan The second problem is that of nding a maximum cardinality

subset of the given list of items which can b e packed into a given numb er of bins with a given

capacity see eg Coman Leung and Ting and Coman and Leung The dual of

packing is covering where the item sizes in a bin must sum to at least As a nal variant we

mention dual bin packing maximize the numb er of unit capacity bins covered by a given list

of items Assmann Johnson Kleitman and Leung

The classical problem

A bin packing algorithm is called online if it packs every item a solely on the basis of the sizes

i

of the items a j i ie without any information on subsequent items The decisions

j

of an online algorithm are irrevo cable packed items cannot b e repacked at later times A

bin packing algorithm that can use full knowledge of all items in packing L is called oline

Below we cover online algorithms rst and then discuss a numb er of oline algorithms

Online algorithms

Simple algorithms

The simplest approximation algorithm for bin packing is probably the NEXT FIT NF algo

rithm NF makes a single scan through the list L and shortsightedly packs the items one after

the other into a unique active bin In case an item do es not t into the active bin the bin

is closed never to b e used again and an empty bin is op ened and b ecomes the new active

bin The running time of NF is linear in the numb er of items packed It is easy to see that

1 1 1

NF has an asymptotic ratio R The list L h i that consists of n pairs

NF

2 2 2

1 1

where shows that the asymptotic ratio cannot b e b etter than Moreover

2 n

the total contents assigned to two consecutive active bins is always at least and this yields

an upp er b ound of on the asymptotic ratio Johnson gave a complete analysis of

the parametric asymptotic ratio of NF he showed that for R and for

NF

R

NF

An obvious drawback of NF is that it never uses the empty space in closed bins A simple

mo dication gives the FIRST FIT FF algorithm FF also makes a single scan through the

list L but it never closes an active bin When packing a new item FF puts it into the lowest

indexed bin into which it will t A new bin is started only if the item do es not t into

any nonempty bin The running time of FF is O n log n and thus greater than the O n

running time of NF However FF has a much b etter asymptotic ratio Johnson et al

proved the following Let m b e a p ositive integer with m m Then for m

R holds and for m R m m holds

FF FF

Other classical online algorithms are BEST FIT BF WORST FIT WF and ALMOST

WORST FIT AWF BF b ehaves like FF except that it puts the next item into the bin into

which it will t with the smallest gap left over ties are broken arbitrarily WF puts the next

item into a nonempty bin with the largest gap starting a new bin only if this largest gap is

not big enough AWF tries rst to put the next item into a nonempty bin with the second

largest gap if the item do es not t there then AWF b ehaves like WF All three variants b elong

to the class of socalled ANY FIT AF algorithms An AF algorithm scans once through the

list L while packing the items It never puts an item a into an empty bin unless the item

i

do es not t into any partially lled bin Similarly an ALMOST ANY FIT AAF algorithm

is an AF algorithm that never puts an item into a partially lled bin with the lowest level

unless there is more than one bin having this level or unless the bin of lowest level is the only

one that has enough ro om Johnson proved a b eautiful and surprising result An AF

algorithm can never have an asymptotic ratio b etter than FF and all AAF algorithms have

the same asymptotic ratio as FF these statements hold even for the parametric ratios As an

easy consequence we get that the AAF algorithms BF and AWF have asymptotic ratios

The asymptotic ratio of the AF algorithm WF however is the same as NF

Bounded space online algorithms

An online bin packing algorithm is said to use k boundedspace if for each new item the choice

of bins into which it may b e packed is restricted to a set of k or fewer active bins A bin

b ecomes active when it receives its rst item once a bin is closed it can never b ecome active

k AFF ABF ABB H SH Champion

k k k k k

ABB

ABB

ABB

ABB H SH

SH

SH

SH

SH

H SH

Table Asymptotic ratios for b oundedspace bin packing algorithms rounded to ve decimal

places

again NEXT FIT uses b oundedspace whereas FF WF and AWF all use unb ounded space

There are four very natural b oundedspace bin packing algorithms that are dened via simple

packing rules for items and simple closing rules for bins A new item can always b e packed into

the lowest indexed bin as in FF or into the bin with the smallest remaining gap as in BF If

the new item do es not t into any active bin some active bin has to b e closed in this case one

can always cho ose the lowest indexed bin the FIRST bin or the fullest bin the BEST bin

The corresp onding four algorithms are called AFF AFB ABB and ABF Here A stands

k k k k

for Algorithm the second letter denotes the packing rule Best t or First t the third letter

denotes the closing rule Best bin or First bin and k is the upp er b ound on the numb er of

active bins

Algorithm AFF This algorithm was dened under the name NEXTk FIT by Johnson

k

in Provably tight b ounds were not in hand until almost years later Csirik

17

and Imreh constructed a sequence of worstcase examples that show R

AF F

10

k

3

and Mao proved a matching upp er b ound on R

AF F

10(k 1)

k

17 3

Algorithm ABF Mao proved that R

k

AB F

10 10k

k

Algorithm AFB Zhang adapted the analysis of Mao to this algorithm and

k

17 3

proved that R

AF B

10 10(k 1)

k

Algorithm ABB This algorithm was investigated by Csirik and Johnson In com

k

parison to the other three algorithms ABB uses the b etter packing and closing rules

k

and it has the b etter asymptotic ratio R holds for any k

AB B

k

So except for small k the asymptotic ratios of all four algorithms are around since they

are of the ANY FIT typ e they of course cannot outp erform FF

Lee and Lee intro duced a new class of b ounded space algorithms using bin reservation

techniques Their algorithm HARMONIC is based on a partition of the interval into k

subintervals where the partitioning p oints are k to each of these subintervals

there corresp onds a single active bin and only items b elonging to this subinterval are packed

into this bin If a new item arrives that do es not t into its corresp onding active bin the

tends to the numb er bin is closed and a new bin is activated In it is proved that R

H

k

P

1

h This numb er is dened by h where t and t t t

1 i+1 i i

i=1

t 1

i

for i it is a prominent numb er in bin packing Wo eginger intro duced the

SIMPLIFIED HARMONIC SH algorithm a mo dication of HARMONIC with a dierent

k

interval structure and with a slightly b etter asymptotic ratio for small values of k

A summary of the asymptotic ratios of the b oundedspace algorithms for some small values

of k is given in Table The asymptotic ratios of all ve algorithms always remain ab ove h

In fact Lee and Lee showed that a b oundedspace algorithm cannot have an asymptotic

ratio b etter than h

Better algorithms

The rst online algorithm for bin packing with R h was Yaos REVISED FF RFF

A

Both the denition and the analysis of RFF are fairly involved it is something of a cross

b etween FF and HARMONIC with the asymptotic ratio R All other known online

RF F

algorithms that b eat this h b ound are variants of HARMONIC with a sp ecial treatment of

the large items Lee and Lee describ ed the REFINED HARMONIC RH algorithm

which was based on H and proved the asymptotic ratio R Ramanan

20

RH

Brown Lee and Lee intro duced the MODIFIED HARMONIC MH with R

MH

Finally in Richey intro duced the HARMONIC algorithm our current champion

The denition of HARMONIC uses a partition of into more than intervals Its

design and analysis were p erformed with the help of linear programming Its asymptotic ratio

is less than

Finally we discuss lower b ounds on the p erformance of any online bin packing algorithm

Roughly sp eaking a lower b ound argument can pro ceed as follows Supp ose an online algo

rithm A is confronted with a huge set of tiny items If A packs these tiny items very tightly

it will not b e able to nd an ecient packing for the larger items that might arrive later if

such items actually do arrive A is going to lose On the other hand if A leaves lots of ro om

for large items while packing the tiny items the large items might not arrive in that case A

is again going to lose Yao formulated this idea mathematically he used as the

size of the tiny items and and as sizes of large items He proved that with

certain lists of such items the asymptotic ratio of every online bin packing algorithm A must

satisfy R Brown and Liang indep endently generalized this lower b ound to

A

Ten years later van Vliet found an elegant linear programming formulation

for the BrownLiang construction Van Vliet gave an exact analysis and increased the lower

b ound to R This is currently the b est lower b ound known for online bin packing

A

Oline algorithms

Simple algorithms

The FF and BF algorithms do not work well on lists ordered by increasing item size Intuitively

the large items at the end of the list should b e combined with the small items at the b eginning

of the list whereas FF and BF pro duce very dense packings for the small items and very p o or

packings for the large items Hence it is natural to design oline algorithms that rst sort

the list by decreasing size and afterwards b ehave like one of the simple online algorithms

This yields FIRST FIT DECREASING FFD BEST FIT DECREASING BFD and NEXT

FIT DECREASING NFD Not surprisingly the p erformance of NFD is not very strong

Baker and Coman proved that R h holds On the other hand the improvement

NFD

of FFD and BFD over the online packing algorithms is dramatic Johnson showed that

R R The pro of of this result is notorious more than pages of

FFD BFD

tedious case analysis In fact Johnson proved the inequality FFD L OPTL

The additive constant of in this inequality was later reduced to by Baker and then to

by Yue the pro ofs in are simpler than Johnsons but are based on similar ideas

but slightly simpler

Better algorithms

After Johnsons theorem in the early s quite a few years passed b efore a p olynomial

time algorithm b eating the b ound was published Yao showed that the b ound was

b eaten by his REVISED FFD RFFD algorithm that b ehaves as FFD but has a sp ecial

10

treatment for the items of size The running time deteriorates to O n log n and the

7

asymptotic ratio is improved only very slightly to but the result shows that there

is nothing magic ab out the b ound

Fernandez de la Vega and Lueker constructed a p olynomial time approximation scheme

PTAS for short for bin packing In other words they proved that for there exists an

approximation algorithm A that has a running time p olynomial in the size of the input list L

but exp onential in and that computes a packing with A L OPTL const

where const as This result was considerably strengthened by Karmarkar and

2

Karp who designed another PTAS that guarantees A L OPTL log OPTL The

algorithm in uses a lot of tricks elimination of small items rounding techniques linear

programming formulations the ellipsoid metho d and so on The running time of the algorithm

9

is roughly O n In this area the development stops with Karmarkar and Karp although many

questions remain op en Is it p ossible to guarantee that A L OPTL log OPTL in

2

p olynomial time ie replace the log in by a simple log We feel that the answer to this

question should b e YES Is it p ossible to guarantee that A L OPTL in p olynomial

time The answer to this question might indeed b e YES and it seems to b e completely out

of reach of our current techniques

Generalizations

We now relax certain of the assumptions of the classical problem

Variable sized bin packing

In variable sized bin packing the items are packed into bins of several dierent typ es B B

1 r

with sizes sB sB sB There is an innite supply of bins of each typ e

1 2 r

The goal is to pack the items a into a set of bins with smal lest total size observe that

i

the sp ecial case of a single bin typ e B of size sB is the classical one dimensional bin

1 1

packing problem For a list L of items and an approximation algorithm A denote by sA L

the total size of bins used by algorithm A Denote by sOPT L the total size of bins used in

an optimal packing The quality of algorithm A is measured by

v ar

R lim supfsA LsOPT L j OPTL k g

A

k

In the online version of variable sized bin packing every time a new bin is op ened the online

algorithm decides which bin typ e to use next

Friesen and Langston give three simple approximation algorithms for variable sized bin

packing with the asymptotic ratios and Only the rst of these three algorithms

is online This algorithm always cho oses the largest bin size when a new bin is op ened and

otherwise b ehaves just like NF Not surprisingly it has the same asymptotic ratio as NF It can

b e shown that any online algorithm that always cho oses the largest p ossible bin size has an

asymptotic ratio of at least and also any online algorithm that always cho oses the smallest

available bin size has an asymptotic ratio of at least Kinnersley and Langston design

a hybrid strategy called FFf FFf uses the packing strategy of FF The bin typ e selection is

based on a socalled l ling factor f with f Supp ose that FFf must start a new

bin as item a arrives If a then FFf starts a new bin of size If a then

i i i

FFf cho oses the smallest bin size in the interval a a f in case no such bin size exists FFf

i i

cho oses size The asymptotic ratio of FFf is at most f and there exist values of f

eg f for which this b ound is tight Zhang shows that the asymptotic ratio of

1

FF equals thus matching the guarantee of FF

2

Csirik designed an online algorithm for variable sized bin packing that is based on the

HARMONIC algorithm For every item size a the algorithm computes a corresp onding bin

i

typ e B All items that are assigned to the same bin typ e B are then packed by a HARMONIC

j j

algorithm into bins of size sB For any collection of bin typ es the asymptotic ratio of this

j

algorithm can b e made arbitrarily close to h The algorithm p erforms even b etter for

sp ecial collections of bin typ es eg for two bin typ es with sizes sB and sB the

1 2

asymptotic ratio is These constitute the strongest results currently known for the online

version of variable sized bin packing

For the oline version of variable sized bin packing Murgolo constructs a p olynomial

time approximation scheme This scheme is based on and extends the techniques of Karmarkar

and Karp

Packings in Higher Dimensions

There are several generalizations of bin packing to higher dimensions we will rst discuss

packings in which items are vectors and then discuss packings of rectangles in two dimensional

arenas

Vector Packing

In vector packing instead of an item b eing a single numb er it is a ddimensional vector

a v a v a where v a holds for j d The goal is to pack all

i 1 i d i j i

items into the minimum numb er of bins in such a way that in every bin the sum of all vectors

is at most one simultaneously in every co ordinate The vector packing problem arises as a

crucial subproblem in scheduling with resource constraints cf Garey Graham Johnson and

Yao

Kou and Markowsky call an approximation algorithm A for ddimensional vector pack

ing reasonable if A never yields packings in which the contents of two nonempty bins could

b e combined into a single bin in one dimension this obviously corresp onds to ANY FIT algo

rithms Kou and Markowsky show that any reasonable vector packing algorithm A ob eys the

b ound R d Since the obvious generalization of FF to ddimensional vector packing

A

is reasonable ddimensional FF has an asymptotic ratio of at most d Garey Graham

Johnson and Yao p erformed an exact analysis of ddimensional FF and showed that it has

a slightly b etter asymptotic ratio of d This currently provides the b est asymptotic

ratio known for ddimensional online vector packing Galamb os Kellerer and Wo eginger

and Blitz van Vliet and Wo eginger discuss lower b ounds for online vector packing Their

lower b ounds are rather weak and tend to as d go es to innity For d dimensions the

b est known lower b ound is

For oline algorithms the metho d of Fernandez de la Vega and Lueker yields asymp

totic ratios of d in d dimensions where can b e made arbitarily close to Yao

proved that in a reasonable but somewhat restricted mo del of computation any oline ap

d proximation algorithm with on log n running time must have an asymptotic ratio R

A

Chekuri and Khanna give p olynomial time approximation algorithms for ddimensional vec

tor packing whose asymptotic ratios grow with O ln d Their approach is based on a linear

programming relaxation of this problem Clearly their algorithm do es not fall into Yaos

restricted mo del of computation It is an op en problem whether R const is p ossible for

A

some p olynomial time approximation algorithm A in any dimension d where the value of

const is indep endent of d In fact it would even b e interesting to obtain asymptotic ratios

growing like O ln ln d Wo eginger see also Chekuri and Khanna observed that unless

P NP there cannot exist a p olynomial time approximation scheme for dimensional vector

packing

Packing in Strips and Two Dimensional Bins

Strip packing originated in the work of Baker Coman and Rivest In this extension to

two dimensions the items are rectangles and the goal is to pack them into a unitwidth semi

innite strip so as to minimize the total length of the strip spanned by the packing Packed

rectangles can not overlap each other or the b oundaries of the strip We consider the strip to

b e vertical and require that rectangles b e packed orthogonally in their given orientations ie

their edges must b e parallel to the b ottom or sides of the strip and no rotations are allowed If

we place b oundaries at the integers and require that no rectangle overlap any such b oundary

then we have the twodimensional packing setup of Chung Garey and Johnson where

the ob jective is to minimize the numb er of bins unit squares needed ie the nearest integer

no smaller than the height of the corresp onding strip packing

The bottomleft algorithm of Baker Coman and Rivest and the leveloriented or shelf

algorithm by Coman Garey Johnson and Tarjan were fundamental to packing in two

dimensions In a b ottomleft packing each successive item is placed as near the b ottom of the

strip as p ossible and then as far left at that height as p ossible If the rectangles are packed in

decreasingwidth order an asymptotic ratio of is achieved As the name suggests in level

oriented packings each rectangle is placed on one of a sequence of levels The b ottommost

level is the b ottom of the strip a higher level is a horizontal b oundary running through the

tallest rectangle in the next lower level Thus levels corresp ond to horizontal bins in the

onedimensional problem The algorithms NF FF and BF can b e adapted in the obvious

way to level packings Prior to applying any of these algorithms the list may b e ordered by

decreasing height or width To cite a typical result see Coman Garey Johnson and Tarjan

we mention that FF with rectangles in decreasing height order has the asymptotic

ratio of FF in the one dimensional case Baker Brown and Katse extended the basic

ideas and designed an algorithm with an asymptotic ratio of As a nal note we mention

that a p olynomialtime approximation scheme for a somewhat restricted sp ecial case of strip

packing was devised by Fernandez de la Vega and Zissimop oulos Finally Kenyon and

Remila succeeded in constructing an approximation scheme for the general strip packing

problem

Shelves are preset levels and were intro duced as the basis of online leveloriented algorithms

by Baker and Schwarz As an example assume that is an a priori b ound on rectangle

k

heights and supp ose that the only shelf heights p ossible are r k for some given r

k +1 k k

Rectangles with heights in r r must b e placed on shelves of height r starting a new

such shelf on top of the current packing whenever necessary according to the given packing

algorithm A principal result of Baker and Schwarz is that the FF shelf algorithm has an

asymptotic ratio of r but an absolute ratio that grows like r In an analogous

way Csirik and Wo eginger transform the HARMONIC algorithm into the corresp onding

HARMONIC shelf algorithm The asymptotic ratio of the HARMONIC shelf algorithm can

b e made arbitrarily close to Achieving an asymptotic ratio b etter than for online

strip packing is probably dicult in it is argued that no shelf algorithm can do this in

which case a completely dierent approach will b e necessary

The algorithmic techniques of one dimensional and strip packing are easily applied to two

dimensional bin packing Combining FFD bin packing with FFDH strip packing yields an

asymptotic ratio known to b e in the interval cf Chung Garey and Johnson

Shelf algorithms were adapted for online two dimensional bin packing by Copp ersmith and

Raghavan Variants and extensions of their algorithms were intro duced by Csirik Frenk

and Labbe and by Li and Cheng in the latter pap er an algorithm was presented

39

with the provably tight asymptotic ratio of The HARMONIC algorithm see Section

40

was extended to the two dimensional problem by Li and Cheng their algorithm has

2

an asymptotic ratio of the square of the corresp onding result in one dimension

Csirik and van Vliet subsequently describ ed an improvement that substantially reduces

the absolute ratio of the LiCheng algorithm while leaving the asymptotic ratio unchanged

Lower b ounds for online two dimensional bin packing were taken up by Galamb os

He proved a lower b ound of A later collab oration with van Vliet yielded The

PhD thesis of van Vliet contains The b est b ound currently known is R

A

Blitz van Vliet and Wo eginger

Final Remarks

While worstcase analysis of bin packing remains quite active it must b e noted that in the last

years or so averagecase analysis has received at least as much attention The monograph

by Coman and Lueker on the sub ject describ es basic results and analytic techniques A

more recent survey of averagecase results for onedimensional bin packing can b e found in

To illustrate typical averagecase estimates supp ose the n items of L are indep endent uniform

random draws from and consider the wasted space W in the classical NF FF and BF

packings ie the dierence in the numb er of bins used and the total size of the items Then we

NF FF 23

have EW n n Coman So Hofri and Yao EW n Shor

n n

p

34

BF

Coman Johnson Shor and Web er and EW n log n Leighton and

n

OPT

Shor Shor Rhee and Talagrand These results can b e compared to EW

n

p

n Lueker

Our discussion of generalizations can b e continued to problems of three or more dimensions

In the large the algorithmic extensions from one to two dimensions can b e essentially rep eated

as extensions from d to d dimensions for d as one might exp ect asymptotic ratios tend

to worsen as d increases We refer the reader to Copp ersmith and Raghavan Galamb os

and van Vliet and Li and Cheng

References

S B Assmann D S Johnson D J Kleitman and J YT Leung On a dual version of

the onedimensional bin packing problem J Algorithms

B S Baker A new pro of for the FirstFitDecreasing binpacking algorithm J Algorithms

B S Baker and E G Coman A tight asymptotic b ound for NextFitDecreasing bin

packing SIAM J Alg Disc Meth

B S Baker D J Brown and H P Katse A algorithm for twodimensional packing

J Algorithms

B S Baker E G Coman Jr and R L Rivest Orthogonal packings in two dimensions

SIAM J Comput

D Blitz A Van Vliet and G J Wo eginger Lower b ounds on the asymptotic worstcase

ratio of online bin packing algorithms Unpublished manuscript

D J Brown A lower b ound for online onedimensional bin packing algorithms Technical

Rep ort R Co ordinated Sci Lab Urbana Illinois

B S Baker and J S Schwarz Shelf algorithms for twodimensional packing problems

SIAM J Comput

C Chekuri and S Khanna On multidimensional packing problems in Proceedings of the

tenth Annual ACMSIAM Symposium on Discrete Algorithms

F R K Chung M R Garey and D S Johnson On packing twodimensional bins SIAM

J Alg Disc Meth

E G Coman Jr M R Garey D S Johnson and R E Tarjan Performance b ounds

for leveloriented twodimensional packing algorithms SIAM J Comput

E G Coman Jr G Galamb os S Martello and D Vigo Bin packing approximation

algorithms Combinatorial Analysis in Handbook of Combinatorial Optimization DZ

Du and P M Pardalos Eds Kluwer Academic Publishers

E G Coman Jr M R Garey and D S Johnson Bin Packing Approximation Al

gorithms A Survey In Approximation Algorithms for NPHard Problems D Ho chbaum

ed PWS Publishing Co Boston MA

E G Coman Jr D S Johnson P W Shor and R R Web er FirstFit Bin Packing

A Tight Bound on Exp ected Wasted Space Probab Th and Related Fields

E G Coman Jr and J Y Leung Combinatorial analysis of an ecient algorithm for

pro cessor and storage allo cation SIAM J Comput

E G Coman Jr J Y Leung and D W Ting Bin packing maximizing the numb er

of pieces packed Acta Informatica

E G Coman Jr and G S Lueker Probabilistic Analysis of Packing and Partitioning

Algorithms Wiley New York

E G Coman Jr K So M Hofri and A C Yao A sto chastic mo del of binpacking

Inf and Cont February

D Copp ersmith and P Raghavan Multidimensional online bin packing Algorithms and

worstcase analysis Oper Res Lett

J Csirik An online algorithm for variablesized bin packing Acta Informatica

J Csirik and B Imreh On the worstcase p erformance of the NkF bin packing heuristic

Acta Cybernetica

J Csirik and D S Johnson Bounded space online bin packing b est is b etter than rst

In Proc nd ACMSIAM Symp Discrete Algorithms

J Csirik J B G Frenk and M Labbe Two dimensional rectangle packing On line

metho ds and results ARIDAM V Rutgers University

J Csirik and A van Vliet An online algorithm for multidimensional bin packing Oper

Res Lett

J Csirik and G J Wo eginger Shelf algorithms for online strip packing Inform Process

Lett

G Galamb os A lowerb ound for the twodimensional online rectangle binpacking

Acta Cybernetica

G Galamb os H Kellerer and G J Wo eginger A lower b ound for online vector packing

algorithms Acta Cybernetica

G Galamb os and A van Vliet Lower b ounds for and dimensional online bin

packing algorithms Computing

M R Garey R L Graham D S Johnson and A C C Yao Resource constrained

scheduling as generalized bin packing J Comb Th Ser A

W Fernandez de la Vega and G S Lueker Bin packing can b e solved within in

linear time Combinatorica

W Fernandez de la Vega and V Zissimop oulos An approximation scheme for strippacking

of rectangles with b ounded dimensions Technical Rep ort LRI University of Paris XI

France

D K Friesen and M A Langston Variable sized bin packing SIAM J Comput

D S Johnson Nearoptimal bin packing algorithms PhD thesis MIT Cambridge MA

D S Johnson Fast algorithms for bin packing J Comput System Sci

D S Johnson A Demers J D Ullman M R Garey and R L Graham Worstcase

p erformance b ounds for simple onedimensional packing algorithms SIAM J Comput

N Karmarkar and R M Karp An ecient approximation scheme for the onedimensional

bin packing problem In Proc rd Ann Symp on Foundations of Computer Science

C Kenyon and E Remila Approximate strippacking to app ear in Math Oper Res

N G Kinnersley and M A Langston Online variablesized bin packing Discr Appl

Math

L T Kou and G Markowsky Multidimensional bin packing algorithms IBM J Research

and Development

E L Lawler J K Lenstra A H G Rinno oy Kan and D B Shmoys Sequencing and

scheduling Algorithms and complexity in Handbook of Operations Research and Man

agement Science Vol Logistics of Production and Inventory S C Graves A H G

Rinno oy Kan and P H Zipkin editors North Holland

C C Lee and D T Lee A simple online bin packing algorithm J Assoc Comput

Mach

K Li and KH Cheng Generalized FirstFit algorithms in two and three dimensions Int

J Foundations Comp Sci

K Li and K H Cheng On three dimensional packing SIAM J Comput

K Li and KH Cheng A generalized HARMONIC algorithm for online multidimensional

bin packing Technical Rep ort TR UHCs University of Houston January

K Li and K H Cheng Heuristic algorithms for online packing in three dimensions J

Algorithms

T Leighton and P Shor Tight b ounds for minimax grid matching with applications to

the average case analysis of algorithms Combinatorica

F M Liang A lower b ound for online bin packing Inform Process Lett

G S Lueker An averagecase analysis of bin packing with uniformly distributed item

sizes Technical Rep ort University of California at Irvine Department of Information

and Computer Science

W Mao BestkFit bin packing Computing

W Mao Tight worstcase p erformance b ounds for NextkFit bin packing SIAM J Com

put

F D Murgolo An ecient approximation scheme for variablesized bin packing SIAM J

Comput

P Ramanan D J Brown C C Lee and D T Lee Online bin packing in linear time

J Algorithms

W T Rhee and M Talagrand Exact b ounds for the sto chastic upward matching problem

Trans Amer Math Soc May

M B Richey Improved b ounds for harmonicbased bin packing algorithms Discr Appl

Math

P W Shor The averagecase analysis of some online algorithms for bin packing Combi

natorica

A Van Vliet An improved lower b ound for online bin packing algorithms Inform Pro

cess Lett

A Van Vliet Lower and upper bounds for online bin packing and scheduling heuristics

PhD thesis Erasmus University Rotterdam The Netherlands

G J Wo eginger Improved space for b oundedspace online bin packing SIAM J Discr

Math

G J Wo eginger There is no asymptotic PTAS for twodimensional vector packing Inform

Process Lett

A C C Yao New algorithms for bin packing J Assoc Comput Mach

11

M Yue A simple pro of of the inequality FFD L OPTL L for the FFD

9

binpacking algorithm Acta Math App Sinica

G Zhang A tight worstcase p erformance b ound for AFBk Technical Rep ort Insti

tute of Applied Mathematics Academia Sinica Beijing China

G Zhang Worstcase analysis of the FFH algorithm for online variablesized bin packing

Computing