Bin packing problem two approximation

International Journal in Foundations of Computer Science & Technology (IJFCST), Vol.5, No.4, July 2015
DOI:10.5121/ijfcst.2015.5401 1
BIN PACKING PROBLEM: TWO APPROXIMATION
ALGORITHMS
AbdolahadNoori Zehmakan
Department of Computer Science, Sharif University of Technology, Tehran, Iran
ABSTRACT
The Bin Packing Problem is one of the most important optimization problems. In recent years, due to its
NP-hard nature, several approximation algorithms have been presented. It is proved that the best
algorithm for the Bin Packing Problem has the approximation ratio 3/2 and the time orderO(n),
unlessP=NP. In this paper, first, a -approximation algorithm is presented, then a modification to FFD
algorithm is proposed to decrease time order to linear. Finally, these suggested approximation algorithms
are compared with some other approximation algorithms. The experimental results show the suggested
algorithms perform efficiently.
In summary, the main goal of the research is presenting methods which not only enjoy the best theoretical
criteria, but also perform considerably efficient in practice.
KEYWORDS
Bin Packing Problem, approximation algorithm, approximation ratio, optimization problems, FFD (First-
Fit Decreasing)
1. INTRODUCTION
The Bin Packing Problem has several applications, including filling containers, loading trucks
with weight capacity constraints, creating file backups in removable media and technology
mapping in Field-programmable gate arraysemiconductor chip design. Unfortunately, this
problem is NP-hard therefore many approximation algorithms [1,2,3,4,5] have been suggested.
In computer science and operational research, approximation algorithms are used to find
approximate solutions to optimization problems. Approximation algorithms are often associated
with NP-hard problems. They are also increasingly used for problems where exact polynomial-
time algorithms are known but too expensive due to the input size. The quality and ability of an
approximation algorithm depend on its approximation ratio and time order. For some
approximation algorithms, it is possible to prove certain properties about the approximation of the
optimal result. A ρ-approximation algorithm A is defined to be an algorithm for which it been
proven that the value of the approximate solution A(x) to an instance x will not be more (or less,
depending on the situation) than a factor ρ times the value, OPT, of an optimum solution.
In the classical one-dimensional Bin Packing Problem, a list of items = { , . . . , }, each with a
size ∈ 0,1 , is given and we are asked to pack them into minimum number of unit-
capacity bins.
Many variations of this problem is proposed, such as 2D and 3D bin packing [6,7,8,9,10], with
item fragmentation [11], fragile objects [12,13], extendable bins [14] packing by cost [3] and
variable size bin packing [15]. In this paper, the original and off-line version of the problem is
considered, due to its applications and importance.

2
Simchi-Levi in [16] proved that the FF (First-Fit) and BF (Best-Fit) algorithms, two of the
foremost approximation algorithms for the Bin Packing Problem, have an absolute worst-case
ratio of 7/4. He also proved that the FFD and BFD algorithms have an absolute worst-case ratio
of 3/2. Zhang and Cai in [17] provided a linear time constant space off-line approximation
algorithm with absolute approximation ratio of 3/2. Their algorithm depends on two kind of
active and extra bins and follows a simple but exact procedure. In 2003, Rudolf and Florian in
[18] presented an approximation algorithm for the BPP with a linear running time and an absolute
approximation factor of 3/2. As mentioned, it is proven that the best algorithm for the Bin
Packing Problem has the approximation ratio of 3/2 and the time order of , unless =
[16].
In [20] Martel defined the asymptotic approximation ratio instead of the approximation ratio and
proved his proposed algorithm has a 4/3 asymptotic approximation ratio. Furthermore, in [20] the
method of Martel was expanded and a 5/4 asymptotic approximation algorithm was suggested.
In this paper two new approximation algorithms are presented. The first algorithm works based
on a kind of sorting and after classification items into 4 ranges tries to choose the best matching
between them. The second algorithm is a time improved version of FFD. In this algorithm, we try
to decrease FFD time order while maintaining the instructive qualities of FFD and its
performance.
Finally, the two suggested algorithms are compared with two approximation algorithms [17,18],
and FFD. Experimental results show the two suggested algorithms perform much better than the
others.
The reminder of this paper is organized as follows. In section 2, two suggested algorithms are
presented. Furthermore, it is proved that the approximation factor of the first algorithm is 3/2.
Then in sections 3 the experimental results and computational analysis are discussed. Finally, in
section 4 conclusions of the results are drawn and some methods for enhancing previous
algorithms are suggested.
2. THE PROPOSED ALGORITHMS
In this section, two proposed algorithm A1 and A2 are discussed. Algorithm A1 utilizes ranging
technique and classifies inputs into 4 ranges. It will be proved that this algorithm's approximation
ratio is 3/2. Furthermore, a new linear version of FFD algorithm is presented.
2.1. The Proposed Algorithm A1
The algorithm tries to create output bins which are at least 2/3 full. It is proved that in this
condition the approximation ratio of the algorithm is 3/2.
As mentioned, in this algorithm inputs are classified into 4 ranges (0- ), ( -
.
), (
.
- ) and ( -1)
called , , and , respectively.
In first step, items are put in separate output bins, then and are sorted. We try to match
any item in with the biggest possible item in . Obviously, after that this step, some items
will be remained in and . We match items with each other and add
| |
to ! −
#$% &'( (The number of used bins). In next step, we try to match items with items. Finally,
items are matched with each other.

3
Definition1: ) is the number of bins in OPT solution and )∗
is the number of bins in the
proposed algorithm.
Lemma1: If at least size of each output bin is full, the approximation ratio is at least .
Proof: consider the worst condition that all output bins are completely full in OPT solution.
Suppose that W is the sum of input items. In this condition:
≥ , & ∗
≤
,
⇒ ∗
/ ≤
3
2
∎
Theorem1: The proposed algorithm A1 is a -approximation algorithm.
Proof: Based on the algorithm in first step, all items are put in separated bins and obviously at
least 2/3 size of these output bins are full. After that, some items are matched with some
items. Definitely, in this step at least 2/3 size of output bins are also full since a item is at least
1/3 and a item is at least 1.5/3. Consequently, their sum is at least 2/3.
In next step, items are matched with each other 2 by 2 and put in separated bins. At least 2/3
size of these bins are full since an item is at least 1/3. After that the rest of items with
items are matched. Now there are two cases:
Case1:
4567 48
| 6|
>

4
Case2:
4567 48
| 6|
≤
, : The sum of all items which remain in this step.
,: : The sum of all S items.
| 2| : The number of all items which are remain in this step.
We claim all output bins are more than fill in this step. According on the algorithm, at first we
match some items with some items. Obviously the output bins in this step are more than
full because a item is no more than and we close a bin when it does not have enough space for
a S item. After that, two configurations are possible:
C1: If there are just some items left we put all of them into separate bins therefore the number
of output bins is | 2|. Consequently the average of the output bins equals
4567 48
| 6|
that is more
than based on case1 assumption.
C2: If there are only some items, the output bins in this step are also more than full because a
item is at most .
In case2, the bins that have some S items like case1 are at least full. Therefore we only consider
the bins which have only one item. We claim that in the OPT solution these items are also
associated separate bins because:
On one hand, they cannot be matched with the items and with the items because a bin does
not have enough space for an item and an item or for two items. On the other hand, if a
item (primary item) is matched with a item in the OPT solution, in the suggested
algorithm it will be matched with a item or its complement (meaning the item matched
with it in the OPT solution) is matched with another item (second item). The second
item is bigger than the primary item since the items are sorted. Therefore, the primary
item can be put in every bin that the second has been put (in this condition the algorithm has
been performed better than OPT solution until now).
Based on the mentioned reasons and discussions, for any output bin in the proposed algorithm
which is less than full, there is a bin in the OPT solution that its used capacity is equal or less
than it. Furthermore, all other bins are more than full. In conclusion, based on the lemma1 the
approximation ratio of the suggested algorithm is . ∎
2.2. The Proposed Algorithm A2
As mentioned, the second proposed algorithm is based on the Firs-Fit Decreasing algorithm. In
FFD, the items are packed in order of non-decreasing size, and next item is always packed into
the first bin in which it fits; that is, we first open bin1 and we only start bin k+1 when the current
item does not fit into any of the bins 1, … , <.
In the algorithm A2, we consider 10 classes of bins and 10 ranges of items and in any step we
check at most one bin in each class. The order of choosing items and checking the bins classes are
considered completely intelligently. A pseudocode of the algorithm A2is shown.

5
Obviously, the running time of the algorithm A2 is (n is the number of input items) since for
making decision about each item the algorithm at most spend 10 time-unit for checking 10 classes
of bins.
We also can make the algorithm more efficient and consider the Scale Parameter r that shows the
number of ranges and bins classes in the algorithm. This parameter can be chosen based on the
number of inputs. For instance, if the number of inputs is 10 =
is reasonable choose ( = 10
instead of ( = 10.
3. COMPUTATIONAL RESULTS
In this section, at first the computational results of two suggested algorithms and three other
algorithms are presented, and it is shown that the proposed algorithms perform considerably
much more efficient. Furthermore, we compare the algorithm A1 with the Algorithm A2 from an
application point of view and their utilization in variant fields and stipulations.
In this section, the two proposed algorithms are compared with two other approximation
algorithms [18, 19] which are the only algorithms have the best possible approximation ratio.
This comparison has been drawn based on all standard instances for BPP from OR-LIBRARY
[21]. We define Ratio as the proportion of the proposed algorithm solution to the OPT solution.
Obviously, ratiohas a direct relationship with algorithm’s approximation ratio. Consequently,
ratio is utilized as a factor for measuring approximation algorithms’ performances.
As mentioned, the standard instances in OR-LIBRARY are used for simulations. Each set of
instances contains 20 instances for the Bin Packing Problem. The two proposed algorithm have
been compared with the Guochuan's algorithm [17], and the Berghammer's algorithm [18] based
on the 8 set of instances. The results of these comparisons for bp1, bp2, bp3, bp4, bp5, bp6, bp7
and bp8 are shown in Fig1, Fig2, Fig3, Fig4, Fig5, Fig6, Fig7, Fig8, respectively.

Figure 1. The ratios of the algorithms for the set
Figure 2. The ratios of the algorithms for the set problems of instance bp2
The ratios of the algorithms for the set problems of instance bp1
6

Figure 4. The ratios of the algorithms for the set problems of
7

The diagrams show the two
algorithms. As mentioned, the
the best possible approximation factor. Furthermore, the algorithm
acceptable than the algorithm
similarity between performances of
The results are measured for 20 instances in any class, but for simplification of understanding
the points corresponding to an algorithm are joined by a line.
In Fig9, the average of the simulations results is shown for four mentioned algorithms for the
all sets of instances. This diagram shows that the proposed algorithm
performs more efficiently. After that, the suggested algorithm
performance. Therefore, two suggested algorithms are completely superior to two other ones,
in practice.
two suggested algorithms perform much better than
algorithms. As mentioned, the two other algorithms are only approximation algorithms with
the best possible approximation factor. Furthermore, the algorithm A1 performance is more
acceptable than the algorithm A2. Another interesting point in the experimental results is the
ween performances ofGuochuan's algorithm, and the Berghammer
the points corresponding to an algorithm are joined by a line.
f the simulations results is shown for four mentioned algorithms for the
all sets of instances. This diagram shows that the proposed algorithm A1 in all instances
performs more efficiently. After that, the suggested algorithm A2 has much better
. Therefore, two suggested algorithms are completely superior to two other ones,
8
suggested algorithms perform much better than two other
other algorithms are only approximation algorithms with
performance is more
. Another interesting point in the experimental results is the
Berghammer's algorithm.
f the simulations results is shown for four mentioned algorithms for the
in all instances
has much better
. Therefore, two suggested algorithms are completely superior to two other ones,

Figure 9. The average of ratios for the 4 algorithms based on the all instances
In Fig 10, the experimental results of the
based on the all sets of instances.
Figure 10. The average of ratios for two suggested algorithms and FFD based on the all instances
The results show that the two suggested algorithms perform much better than
bp5, bp6, bp7, and bp8, but the
bp3 and bp4. It seems their performances are very similar in average. We claim that the
suggested algorithms are more effective and efficient than
order are similar, but FFD is an on
algorithm) while the algorithm A1
superior to FFD because it is a linear time algorithm while the running time of
even in worst-case .
We drew the conclusion that the algorithms
criteria, but also execute better than other ones
The average of ratios for the 4 algorithms based on the all instances
In Fig 10, the experimental results of the two suggested algorithms and FFD algorithm are shown
based on the all sets of instances.
The average of ratios for two suggested algorithms and FFD based on the all instances
suggested algorithms perform much better than FFD
bp5, bp6, bp7, and bp8, but the FFD algorithm performances are more acceptable in bp1, bp2,
bp3 and bp4. It seems their performances are very similar in average. We claim that the
suggested algorithms are more effective and efficient than FFD. The algorithm A1 and
is an on-line space algorithm (it means that it save all bins during the
algorithm) while the algorithm A1 use much less space. Furthermore, the algorithm
is a linear time algorithm while the running time of FFD
We drew the conclusion that the algorithms A1 and A2 not only enjoy the best possible theoretical
than other ones in practice, but a natural question which comes up
9
algorithm are shown
The average of ratios for two suggested algorithms and FFD based on the all instances
FFD algorithm in
algorithm performances are more acceptable in bp1, bp2,
bp3 and bp4. It seems their performances are very similar in average. We claim that the two
and FFD time
line space algorithm (it means that it save all bins during the
use much less space. Furthermore, the algorithm A2 is also
FFD is >$?
not only enjoy the best possible theoretical
, but a natural question which comes up

10
is that "Which algorithm should be used in practice, A1 or A2?". The answer is that it depends. In
the following paragraphs we try to clarify this point.
Firstly, obviously if the important factor is accuracy, AlgorithmA1 is the better one, but if the
significant criterion is speed, AlgorithmA2 will be the choice inasmuch as AlgorithmA1 shows
better performance based on the aforementioned outputs; on the other hand, AlgorithmA2 is a
linear time algorithm. Another point which can be taken into consideration is that AlgorithmA1 is
a constant-space one while the second one is not. Therefore, if space order is a noteworthy factor,
we should exploit AlgorithmA1.
Needless to say, if the input items are almost sorted, the algorithm A1 performs a lot better, but if
the number of input items is significantly high or they are distributed homogenously, the
algorithm A2 will be the option In that AlgorithmA1 needs to sort the items, and the algorithm A2
is much more flexible and is able to use Scale Factor. The aforementioned computational results
confirm this claim because the number of items in the instances increases from bp1 to bp8.
If the number of S (small) items is considerable, Algorithm A1 performs more efficiently. On the
other hand, if the number of L (large) items is high, the second one is the right choice. Moreover,
the state that nearly all items are relevant to the ranges M1 and M2 (are medium) forces the user
to utilize the algorithm A2.
For instance, in packing trucks and ships when the goods are small, we use the first one, but in the
state that they are large enough by considering the capacity unit in the ship or truck, the choice is
second one. Furthermore, in assigning tasks to machines in machine scheduling problem if the
durations of different tasks are approximately equal with each other, the second algorithm
executes better.
Consider the problem of placing computer files with specified sizes into memory blocks of fixed
size. For example, recording all of a computer's music where the length of the pieces to be
recorded are the weights and the bin capacity is the amount of time that can be sorted on an audio
(say 80 minutes). If we want to save the information for a long time, it is better to use the first
algorithm to amplify the accuracy, but if we want to rewrite the information several times, using
the second one is a rational solution. If all items are similar in size, for instance all of them are
songs, probably AlgorithmA1 works acceptably.
Table 1 tries to summarize the aforementioned discussions regarding the application of the
algorithms A1 and A2 in different situations.
Table 1. Choosing between algorithms 1 and 2 based on different factors and condition.
Factor/Condition Algorithnm1 Algorithm2
Accuracy Yes
Speed Yes
Space Yes
Sorted Items Yes
High Number of Items Yes
Homogenous Distribution of
Items
Yes
Majority by S Items Yes
Majority by L Items Yes
Majority by M Items Yes

11
3. CONCLUSIONS
Two approximation algorithms A1, and A2 were proposed in this paper. It was proved that the A1
approximation ratio is . After that we observed the results of experimental simulations and
analyzed them. Based on the results, we can claim that the two proposed algorithms in this article
are the best presented approximation algorithms for the Bin Packing Problem, in theory and in
practice until now.
In future researches, the focus on Scaling Factor r can enhance the algorithm A2 more and more.
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Bin packing problem two approximation

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