Lower Bound for the Online Bin Packing Problem with Restricted Repacking

SIAM J. COMPUT. c© 2007 Society for Industrial and Applied MathematicsVol. 0, No. 0, pp. 000–000

LOWER BOUND FOR THE ONLINE BIN PACKING PROBLEM

WITH RESTRICTED REPACKING∗

JANOS BALOGH† , JOZSEF BEKESI† , GABOR GALAMBOS† , AND GERHARD REINELT‡

Abstract. In 1996 Ivkovic and Lloyd [A fundamental restriction on fully dynamic maintenanceof bin packing, Inform. Process. Lett., 59 (1996), pp. 229–232] gave the lower bound 4

3on the

asymptotic worst-case ratio for so-called fully dynamic bin packing algorithms, where the numberof repackable items in each step is restricted by a constant. In this paper we improve this result toabout 1.3871. We present our proof for a semionline case of the classical bin packing, but it worksfor fully dynamic bin packing as well. We prove the lower bound by analyzing and solving a specificoptimization problem. The bound can be expressed exactly using the Lambert W function.

Key words. bin packing, semionline algorithm, worst-case behavior, lower bound

AMS subject classifications. 68Q25, 68W25, 68W40

DOI. 10.1137/050647049

1. Introduction. The classical one-dimensional bin packing problem is amongthe most frequently studied combinatorial optimization problems. In its traditionaldefinition a list L = {x1, x2, . . . , xn} of elements (also called items) with sizes in theinterval (0, 1] and an infinite list of unit capacity bins are given. Each element xi fromthe list L has to be assigned to a unique bin such that the sum of the sizes of theelements in a bin does not exceed the bin capacity. The size of an element will alsobe denoted by xi. The bin packing problem consists of packing the items to the binsin such a way that as few bins as possible are used.

It is well known that finding an optimal packing is NP-hard [8]. Consequently, alarge number of papers have been published which look for polynomial time algorithmsthat find feasible solutions with an acceptable approximation quality.

For measuring the efficiency of algorithms there are two general methods: theinvestigation of the worst-case behavior or—assuming some probability distributionof the elements—a probabilistic analysis. In this paper we will concentrate on theasymptotic worst-case ratio of an algorithm. For a given list L, denote by A(L) andOPT (L) the number of bins used by algorithm A and the number of bins used inan optimal packing, respectively. Then the asymptotic worst-case ratio (AWR) ofalgorithm A is

RA := lim supk→∞

{

maxL

{

A(L)

k

∣

∣

∣

∣

OPT (L) = k

}}

.

If an algorithm has an AWR of RA, we also say that it is RA-competitive.

∗Received by the editors December 8, 2005; accepted for publication (in revised form) August20, 2007; published electronically DATE. This research was supported by the Hungarian NationalResearch Fund (projects T 048377 and T 046822) and by the MOB-DAAD Hungarian-GermanResearcher Exchange Program (project 21). A preliminary version of the paper was presented at the29th Hungarian Conference of Mathematics, Physics and Computer Science Teachers and appearedin electronic form in the proceedings of the conference published by the organizers.

http://www.siam.org/journals/sicomp/x-x/64704.html†Department of Computer Science, Gyula Juhasz Faculty of Education, H-6701 Szeged, Hungary

([email protected], [email protected], [email protected]).‡Research Group Discrete Optimization, Institute for Computer Science, University of Heidelberg,

D-69120 Heidelberg, Germany ([email protected]).

1

2 J. BALOGH, J. BEKESI, G. GALAMBOS, AND G. REINELT

In recent decades many different types of algorithms have been proposed and alsoseveral variants of the classical problem have been introduced and studied. Since theterminology used for classifying algorithms or problems is not unique, we start thepaper with a proposal for a new classification based on a different view of the problem.To this end, we distinguish two contributors: the scheduler, who provides informationon the problem data, and the loader, who packs the items. The actual problem thendepends on how the scheduler gives the data and which strategies the loader uses forpacking.

We suppose that the scheduler reveals information about the items at subsequentdiscrete time steps. The information consists of messages that certain elements (withrespective sizes) arrive and/or that certain elements can be deleted. The total numberof such time steps is at most n because information about several items can be givenin the same step. If the element xi appears in step j to be added to the list, then wesay that its arrival time ai is j. In some step k, the scheduler may also inform aboutthe fact that the item xi can be deleted. We say that the deletion time di of xi is k.From this time the item may be deleted from the bin into which it was packed earlier.Of course, ai < di, for 1 ≤ i ≤ n. By setting di = ∞ we indicate that no deletiontime for item xi has been given.

At every time step the loader decides on its actions for packing the items. If someelements are packed and some others are deleted, we assume that first the deletionoperations are performed.

In short, the scheduler decides on how the problem data is given, and the loaderfinds a feasible packing of all items depending on arrival and deletion times. Specialstrategies now lead to different variants of bin packing.

If ai = 1 and di = ∞ for every i, 1 ≤ i ≤ n, then the loader immediately receivescomplete knowledge about the list. In this case the scheduler produces an offlineproblem and the loader applies an offline algorithm.

If ai = i and di = ∞ and the loader packs the items one by one in each stepwithout knowing anything about subsequent elements (neither the sizes nor the num-ber of the elements), and the packed items may not be repacked anymore during thealgorithm, then we speak about an online problem and an online algorithm.

In the case of semionline (SOL) algorithms the scheduler specifies the problemonline as above, but the loader is allowed to apply at least one of the followingoperations: a repacking of some items, a lookahead into the next several elements,or some kind of preordering. SOL algorithms allowing only a restricted number ofelements to be repacked in each step are called c-repacking SOL algorithms.

If the scheduler also allows the deletion of elements, then we speak about a dy-namic bin packing problem (DBP), and if the loader exploits this additional informa-tion for deleting items then his algorithm is a dynamic bin-packing algorithm. In thiscase A(L) is considered as the maximum number of used bins during the packing. Anoffline DBP basically would be a special type of scheduling problem. Therefore, inthe context of bin packing, only online and SOL dynamic problems are investigated.

If, in the SOL case, only repacking is allowed for the loader, and the schedulermay specify some elements to be deleted, then Ivkovic and Lloyd in [12] use the termsfully dynamic bin packing problem (FDBP) and fully dynamic bin packing algorithm.The c-repacking FDBP can be defined similarly to the definition of c-repacking SOLproblems.

For offline bin packing algorithms the best results were achieved in [5] and [13],where the authors proved that for any ε > 0, there are algorithms which can solve

LOWER BOUND FOR ONLINE BIN PACKING WITH REPACKING 3

the bin packing problem in linear time, while their AWR is 1 + ε. Recently, the bestonline algorithm was given by Seiden in [15] and its AWR is 1.58889. The best-knownlower bound 1.5401 is due to van Vliet [16].

SOL algorithms were investigated in [9], [10], and [6]. The first and the second ref-erences use different types of lookaheads, and the last one allows repacking. Gambosi,Postiglione, and Talamo [7] also analyzed certain SOL algorithms. In their algorithmsrepacking was allowed in a special way: those elements which were “large enough”were repacked one by one, while the “small” elements were moved together in a bun-dle between the bins. In this sense these algorithms can repack even O(n) elementsin one step. They analyzed two algorithms. The faster one has linear time and is32 -competitive, and the other one runs in O(n logn) time and is 4

3 -competitive. Re-cently, except for trivial results, no good lower bound has been found for the efficiencyof SOL algorithms.

Various dynamic approximation algorithms were investigated in [3].Ivkovic and Lloyd constructed an approximation algorithm to solve the FDBP

(see [12]). Similar to the technique used in [7], they also repacked the small elementsusing a “bundle technique.” The competitive ratio of their algorithm is 5

4 and requiresΘ(log n) time per step; i.e., its running time is Θ(n log n). Investigating the lowerbounds, they proved that there is no FDBP algorithm with deletions and where onlya constant number of elements may be repacked with a better competitive ratio than43

[11] (i.e., for the c-repacking FDBP for any c). Using a similar construction inthe proof, we can apply this bound to SOL algorithms as well. This fact was alsomentioned by Csirik and Woeginger in [4].

In this paper we improve the lower bound to 1.3871 for the c-repacking SOLalgorithms, but it follows from the construction that this lower bound is also validfor c-repacking fully dynamic bin packing algorithms. This last fact is coming fromthe following observation. While we try to construct lower bounds either for thec-repacking SOL problem or the c-repacking FDBP, we realize that the two con-structions differ slightly from each other: to allow some deletions will not spoil thec-repacking SOL lower-bound construction for the c-repacking FDBP case.

During our analysis we will use different instruments: LP techniques will becombined with results from linear algebra, and finally we will solve a nonlinear opti-mization problem.

2. Construction of the linear program. For k ≥ 1, n ≥ 1, let x1, x2, . . . ,

xk

(

12 ≤ x1 < x2 < · · · < xk < 1

)

be fixed real numbers and c ≥ 1 be an arbitraryinteger. Define yi = 1 − xi (i = 1, . . . , k) and yk+1 = 0. Furthermore, let ε <

minj=1,...,k {yj − yj+1} be an arbitrary positive number and εj := εd n

2yje.

Denote the first list by L0. Define L0 as a list of N items of size a, where

a < εk

d nyk

ecand N := b

n2−ε

ac. We can see that L0 contains very small elements with

equal, suitably chosen size. The cumulative size of the small items converges to n2

from below if n → ∞. Denote by Li (1 ≤ i ≤ k, k ≥ 2) the lists containing bigelements. Define Li as a list of d n

2yie items of size xi + εi. This means that the sizes

of the elements are equal in list Li. Suppose we need to pack the elements of the listsL0, L1, . . . , Lk with an SOL algorithm which can repack only a constant number ofelements. We call these kinds of algorithms c-repacking algorithms. We analyze thebehavior of an arbitrary c-repacking algorithm for the lists L0 and L0Li (1 ≤ i ≤ k).In our analysis size(B) denotes the total size of items in bin B. We present the lowerbound as a solution of a linear programming problem.


Lemma 1.

limsupn→∞

A (L0)

OPT (L0)≥ 1 +

k∑

j=1

2zj

(

1

yj

− 1

)

.

Proof. Consider an arbitrary c-repacking algorithm A that has to pack list L0

first. It is clear that the above-defined a is a very small number, and the total sizeof the repackable items in d n

2yje (j = 1, . . . , k) steps is less than εk, which is the

smallest among ε1, . . . , εk. We want to give a lower bound on A (L0). Denote by zin

the cumulative size of the items that have been packed in yi-type bins (i = 1, . . . , k).Bin B is called a yi-type bin if size(B) ∈ (yi+1, yi]. It is easy to see that the totalnumber of yi-type bins is at least zin

yi. The cumulative size of the items that have not

been packed in any yi-type bin is Na − n∑k

j=1 zj . So we get that

A (L0) ≥k∑

j=1

zjn

yj

+ Na − n

k∑

j=1

zj = Na + n

k∑

j=1

zj

(

1

yj

− 1

)

.

Because of the definition of N we have Na ≥ n2− ε − a. This means that

(1) A (L0) ≥ n

1

2+

k∑

j=1

zj

(

1

yj

− 1

)

− ε − a.

In the following we investigate the behavior of the optimal algorithm on L0. In theoptimal packing each bin B is loaded such that size(B) > 1 − a. So we get

(2) OPT (L0) ≤Na

1 − a+ 1 ≤

n2 − ε

1 − a+ 1 ≤

n2

1 − a+ 1.

From (1) and (2) it follows that

(3) lim supn→∞

A (L0)

OPT (L0)≥ 1 +

k∑

j=1

2zj

(

1

yj

− 1

)

.

Lemma 2. For i = 1, . . . , k,

lim supn→∞

A (L0Li)

OPT (L0Li)≥ 1 + yi + 2yi

i−1∑

j=1

zj

(

1

yj

− 1

)

−k∑

j=i

zj

.

Proof. Consider now the packing of the concatenated list L0Li (1 ≤ i ≤ k) forfixed i. Let B be a bin which contains an element from Li. Because the size of thebig item is xi + εi, the total size of the small elements in the bin is at most yi − εi.This can also happen in such a way that, after L0 has been packed, the bin containsmore items, but during the processing of Li, some of them have been repacked toother bins. Since the list Li has exactly d n

2yie elements, at most cd n

2yie elements could

have been repacked. The size of each small item is a, so from the definition of a andεi we get that A can repack small items with at most εi cumulative size. So we cansay that after packing L0 size(B) ≤ yi must hold for B. From this it follows that the

cumulative size of the small items packed together with an item from Li is∑k

j=i zjn.


Other small items from L0 can be packed in yj -type bins, where j = 1, . . . , i− 1. The

total size of the remaining elements of L0 is Na− n∑k

j=1 zj. So we can estimate thenumber of bins used by A in the following way:

(4) A (L0Li) ≥n

2yi

+ n

i−1∑

j=1

zj

yj

+ Na − n

k∑

j=1

zj (i = 1, . . . , k) .

For the optimal packing of L0Li (1 ≤ i ≤ k) we get that

(5) OPT (L0Li) ≤⌈

n

2yi

⌉

+ Si ≤n

2yi

+ Si + 1,

where Si is the total number of bins which contain only items from L0 in an optimalpacking. A bin containing an Li-element will be denoted by Bi. We obtain for thecumulative size small(Bi) of the small elements in Bi that

small(Bi) ≥⌈

n

2yi

⌉

(yi − εi − a) ≥ n

2−⌈

n

2yi

⌉

εi −⌈

n

2yi

⌉

a

≥ Na −⌈

n

2yi

⌉

a ≥ Na − ε,

because a < ε⌈

n2yi

⌉ . So

(6) Si ≤ ε ≤ 1.

From (5) and (6)

(7) OPT (L0Li) ≤n

2yi

+ 2.

Combining (4) and (7) we get that

lim supn→∞

A (L0Li)

OPT (L0Li)≥ 1 + yi + 2yi

i−1∑

j=1

zj

(

1

yj

− 1

)

−k∑

j=i

zj

(8)

(i = 1, . . . , k) .

Theorem 3. The solution of the following linear programming problem is a lowerbound for any SOL c-repacking algorithm:

min b

b ≥ 1 + yi + 2yi

i−1∑

j=1

zj

(

1

yj

− 1

)

−k∑

j=i

zj

(i = 1, . . . , k) ,

b ≥ 1 +

k∑

j=1

2zj

(

1

yj

− 1

)

,(9)

zi ≥ 0, (i = 1, . . . , k) ,

k∑

j=1

zj <1

2.


Proof. The definition of zj (j = 1, . . . , k) obviously implies that∑k

j=1 zj < 12 .

The statement of the theorem then follows from Lemmas 1 and 2.

3. Solution of the linear program. To simplify our model we introduce anew variable z, which is the sum of all zj ’s, i.e., z =

∑k

j=1 zj . Using this substitutionand some reordering, we can rewrite (9) in the following form:

min b

−2yiz + 2yi

i−1∑

j=1

zj

yj

− b ≤ − (1 + yi) (i = 1, . . . , k) ,

−2z + 2

k∑

j=1

zj

yj

− b ≤ −1,(10)

−z +

k∑

j=1

zj = 0

zi ≥ 0, (i = 1, . . . , k) ,

z <1

2.

Instead of (10) we first solve a linear system of equations related to our problem.The number of equations is k + 2. The variables of the system are z, z1, . . . , zk, b,while y1, . . . , yk are some fixed parameters, satisfying the conditions of Theorem 3.

−2yiz + 2yi

i−1∑

j=1

zj

yj

− b = − (1 + yi) (i = 1, . . . , k) ,

−2z + 2

k∑

j=1

zj

yj

− b = −1,(11)

−z +

k∑

j=1

zj = 0.

Lemma 4. The system (11) has a unique solution.Proof. Let M be the matrix of (11). Then

(12) M =

−2y1 0 . . . 0 0 −1

−2y2 2 y2

y1. . . 0 0 −1

......

. . ....

......

−2yk 2 yk

y1. . . 2 yk

yk−10 −1

−2 2y1

. . . 2yk−1

2yk

−1

−1 1 . . . 1 1 0

.


Denote by det(M) the determinant of M . We prove that det(M) < 0, and so fromCramer’s rule the statement of the lemma follows. Let us expand det(M) using thelast column of M . So we obtain

(13) det(M) =k+1∑

i=1

− 1 (−1)k+2+i det(Mi) =k+1∑

i=1

(−1)k+1+i det(Mi),

where Mi is a (k + 1)× (k + 1) matrix, which is obtained from M by deleting its lastcolumn and ith row. Expand det(Mi) by its last row. So

(14) det(Mi) = (−1)(−1)k+2 det(Mi1) +

k+1∑

j=2

(−1)k+1+j

det(Mij),

where Mij is a k × k matrix, which is obtained from Mi by deleting its last row andjth column. Now we investigate how Mij looks:

Mij =

−2y1 0 . . . 0 0 . . . 0 0

−2y2 2 y2

y1. . . 0 0 . . . 0 0

......

. . ....

.... . .

......

−2yi−1 2yi−1

y1. . . 2

yi−1

yj−22

yi−1

yj. . . 0 0

−2yi+1 2 yi+1

y1. . . 2 yi+1

yj−22 yi+1

yj

. . . 0 0

......

. . ....

.... . .

. . ....

−2yk 2 yk

y1. . . 2 yk

yj−22 yk

yj. . . 2 yk

yk−10

−2 2y1

. . . 2yj−2

2yj

. . . 2yk−1

2yk

.

Case A. Suppose that j 6= i and j 6= i + 1 and consider only the ith and (i + 1)thcolumns of Mij. The two columns are the following:

0 0

......

0 0

2yi+1

yi−12

yi+1

yi

......

2 yk

yi−12 yk

yi

2yi−1

2yi

.

It is easy to see that these columns are not independent, so we obtain that det(Mij) =0 if j 6= i and j 6= i + 1.


Case B. If j = i, then Mii is a lower triangular matrix of the form

Mii =

−2y1 0 . . . 0 0 . . . 0 0

−2y2 2 y2

y1. . . 0 0 . . . 0 0

......

. . ....

.... . .

......

−2yi−1 2yi−1

y1. . . 2

yi−1

yi−20 . . . 0 0

−2yi+1 2 yi+1

y1. . . 2 yi+1

yi−22 yi+1

yi

. . . 0 0

......

. . ....

.... . .

. . ....

−2yk 2 yk

y1. . . 2 yk

yi−22 yk

yi. . . 2 yk

yk−10

−2 2y1

. . . 2yi−2

2yi

. . . 2yk−1

2yk

.

From this we get that det(M11) = 2k 1y1

, det(M(k+1)(k+1)) = −2kyk, and that for

2 ≤ i ≤ k, det(Mii) = −2k yi−1

yi.

Case C. Similarly, if j = i + 1, then Mi(i+1) is also a lower triangular matrix:

Mi(i+1) =

−2y1 0 . . . 0 0 . . . 0 0

−2y2 2 y2

y1. . . 0 0 . . . 0 0

......

. . ....

.... . .

......

−2yi−1 2 yi−1

y1. . . 2 yi−1

yi−20 . . . 0 0

−2yi+1 2yi+1

y1. . . 2

yi+1

yi−22

yi+1

yi−1

. . . 0 0

......

. . ....

.... . .

. . ....

−2yk 2 yk

y1. . . 2 yk

yi−22 yk

yi−1. . . 2 yk

yk−10

−2 2y1

. . . 2yi−2

2yi−1

. . . 2yk−1

2yk

.

It is easy to see that det(Mi(i+1)) = −2k if i 6= 1 and det(M12) = 2k.Using these results, we get from (14) that

det(M1) = −2k

(

(−1)k+2 1

y1+ (−1)

k+3

)

,

det(Mi) = −2k

(

(−1)k+1+i yi−1

yi

+ (−1)k+1+i+1

)

, i = 2, . . . , k,

det(Mk+1) = −2kyk.


So from (13) we obtain

det(M) = −2k

(

(−1)2(k+1) 1

y1+ (−1)

2(k+1)+1

)

+ −2k

(

k∑

i=2

(

(−1)2(k+1+i) yi−1

yi

+ (−1)2(k+1+i)+1

)

+ (−1)2(k+1)yk

)

= −2k

(

1

y1+

k∑

i=2

yi−1

yi

+ yk − k

)

.

Since yi−1 > yi (i = 2, . . . , k) and 12

≥ y1 we get 1y1

+∑k

i=2yi−1

yi> k, and so

det(M) < 0.Lemma 5. The solution of system (11) satisfies the conditions of the linear

program (10) and

b = 1 +1 − yk

1y1

+k∑

i=2

yi−1

yi+ yk − k

.

Proof. By substituting the right-hand side of (11) into M , we obtain the matrixMb,

Mb =

−2y1 0 . . . 0 0 −1 − y1

−2y2 2 y2

y1. . . 0 0 −1 − y2

......

. . ....

......

−2yk 2 yk

y1. . . 2 yk

yk−10 −1 − yk

−2 2y1

. . . 2yk−1

2yk

−1

−1 1 . . . 1 1 0

,

and

det(Mb) = det(M) +

k∑

i=1

yi (−1)k+1+i

det(Mi) = det(M) − 2k (1 − yk) .

Using again Cramer’s rule,

b =det(Mb)

det(M)= 1 +

1 − yk

1y1

+k∑

i=2

yi−1

yi+ yk − k

.

By a similar analysis we can prove that the other conditions of (10) also hold. As anexample we present only the inequality

z =det(Mz)

det(M)=

12

det(M) − 2k−1 yk−1y1

det(M)=

1

2+

yk − 1

2y1

(

1y1

+k∑

i=2

yi−1

yi+ yk − k

) <1

2,


where

Mz =

−1 − y1 0 . . . 0 0 −1 − y1

−1 − y2 2 y2

y1. . . 0 0 −1 − y2

......

. . ....

......

−1 − yk 2 yk

y1. . . 2 yk

yk−10 −1 − yk

−1 2y1

. . . 2yk−1

2yk

−1

0 1 . . . 1 1 0

.

The validity of the other inequalities can be shown easily.Lemma 6. The solution of system (11) is an optimal solution of the linear program

(10).Proof. Consider the solution vector v of (11). Because of Lemma 5 it is a feasible

solution of (10). Let v be a basis solution. The objective function contains onlyb, which has a positive coefficient, satisfying the optimum criterion. So we get thestatement of the lemma.

4. Getting the lower bound. In this section we deal with the optimal choiceof the variables y1, y2, . . . , yk. To choose them in an optimal way, we have to solvethe following nonlinear optimization problem:

max

{

1 +1 − yk

1y1

+k∑

i=2

yi−1

yi+ yk − k

}

(15)

1

2≥ y1 > · · · > yk > 0.

It does not seem to be easy to solve (15) analytically for arbitrary fixed k, but itcan be done numerically by global optimization tools. Details can be found in [1]. Inthis paper we analyze (15) for the case when k → ∞.

Lemma 7. The optimal solution of (15) converges to the maximum of the function

f(x) := 1 +1 − x

x + 1 + ln(

1x

)

− ln (2)

in the interval(

0, 12

]

if k → ∞.Proof. We distinguish two cases.Case 1. In this case we assume that y2 ≤ y1

2 . Then 2 ≤ y1

y2. Using y1 ≤ 1

2 , y2 ≤14, yk ≤ y2, 4 ≥ 2 + yk, and applying the inequality between the arithmetic and

geometric means for the numbers y2

y3, y3

y4, . . . ,

yk−1

yk, yk in the denominator of (15), we

obtain the upper approximation (see [14])

1 +1 − yk

1y1

+k∑

i=2

yi−1

yi+ yk − k

≤ 1 +1 − yk

4 +k∑

i=3

yi−1

yi+ yk − k

≤ 1 +1 − yk

2 + yk − k + (k − 1) k−1√

yk

.


1.15

1.2

1.25

1.3

1.35

0 0.1 0.2 0.3 0.4 0.5

x

Fig. 1. Graph of the function f(x) in the interval`

0, 12

˜

.

Using that limk→∞ (k − 1) k−1√

x− k = ln (x)− 1 and ln(x) < −54 if x ≤ 1

4 , we obtainthat

limk→∞

1 +1 − yk

1y1

+k∑

i=2

yi−1

yi+ yk − k

≤ 1 +1 − yk

yk + 1 + ln (yk)< 1.

Case 2. Suppose that y2 > y1

2 . From this we get 2y2 (1 − 2y1) ≥ y1 (1 − 2y1), andso

(16)1

y1+

y1

y2≥ 2 +

1

2y2.

Using (16) and y1 ≤ 12 and then applying again the inequality between the arithmetic

and geometric means for the numbers 12y2

, y2

y3, . . . ,

yk−1

ykin the denominator of (15),

we obtain the upper approximation

1 +1 − yk

1y1

+k∑

i=2

yi−1

yi+ yk − k

≤ 1 +1 − yk

2 + 12y2

+k∑

i=3

yi−1

yi+ yk − k

≤ 1 +1 − yk

2 + yk − k + (k − 1) k−1

√

12yk

.

Since limk→∞ (k − 1) k−1

√

12x

− k = ln(

1x

)

− ln(2) − 1, we get that

limk→∞

1 +1 − yk

1y1

+k∑

i=2

yi−1

yi+ yk − k

≤ 1 +1 − yk

yk + 1 + ln(

1yk

)

− ln (2).

Since 1−xx+1+ln( 1

x)−ln(2)

≥ 0 if x ∈ (0, 12 ], we get that the maximum of the limit comes

from Case 2, which proves the statement of the lemma.


Fig. 2. Graph of the function W−1(x) in the interval [−0.3, 0).

Here we used the fact that, if we compute the maximum of f(x) in the giveninterval, then we can construct a series of values y1, y2, . . . , yk which gives an optimalsolution for (15). The construction sets y1 to 1

2 and defines the geometrical series

yi = 12(2yk)

i−1

k−1 , i = 2, . . . , k, which ends with yk.Lemma 8. The maximum of the function f(x) is 1− 1

W−1(−2

e3 )+1≈ 1.3871 in the

interval(

0, 12

]

, where W−1 (x) is the real branch of the Lambert W function for whichW (x) ≤ −1 holds.

Proof. The proof is straightforward by taking the derivative of f(x) and investi-gating the function analytically.

5. Conclusions. In this paper we discussed improved lower bounds for the c-repacking version of the classical online bin packing problem. Note, however, thatthese bounds are valid for c-repacking fully dynamic bin packing algorithms as well.Namely, when deletions are allowed, our construction can be applied in the followingway. We first give the list L0, then insert L1, and after deleting this, we insert L2, etc.By examining the list L0 and the L0L1, L0L2, . . . , L0Lk configurations, we obtain thesame lower bound with similar arguments as above.

REFERENCES

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Lower Bound for the Online Bin Packing Problem with Restricted Repacking

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