A new lower bound for bin-packing problem with
general conflicts graph
Mohamed MAIZA
Christelle GUERET
Laboratoire des Mathématiques Appliquées
Ecole Militaire Polytechnique
BP 17 Bordj El Bahri, Alger, ALGERIE
[email protected]
Département Automatique-Productique
Ecole des Mines de Nantes
Nantes, FRANCE
[email protected]
for the problem without and with conflicts and we used it for
developed our new heuristics.
Abstract— We propose a new lower bound for the one
dimensional bin-packing problem with conflicts. The conflicts are
represented by a graph whose nodes are the items, and adjacent
items cannot be packed into the same bin. The lower bound is
based on an iterative search of maximal cliques in the conflict
graph using Johnson's heuristic. At each step we delete from the
graph the items of the last clique found, together with small items
that could eventually be packed with them. The lower bound
corresponds to the sum of the bins required at each step.
Keywords:
Cutting
and
Packing,
Networks,Transportation and Logistics
Graphs
Notation
:
Bin capacity
:
Set of item.
; :
Index of items and bins respectively
Weight or size of item i
:
Degree of item i
:
Residual capacity of bin j
and
Set of items constituted a maximal-click
I.
INTRODUCTION
;
The bin-packing problem with conflicts called BPPC is
one of problems often encountered in daily life; it consist to
determining the minimal number of identically bins needed to
store a set of items with heights is less than the capacity of
bins, and where some of this items are incompatible with each
other, therefore cannot be packed together in the same bin.
0
Conflicts graph of
;
0
vertex and
edges
Extended conflicts graph,
II.
EXISTING LOWERS BOUND
We first note that any lower bound for the BPP is a valid
lower bound also for the BPPC. Analogously, any lower bound
for the VCP is also a lower bound for the BPPC.
This problem occur in a number of industrial and
transportation contexts where one wishes to optimally affects
different types of items to an expensive bins. Obvious
examples of these contexts are: loading flammable and
explosive items, backup of a several types of files,
telecommunications signals affectation. So, each item has some
space requirement, which is met by the bins.
In Section II.A we briefly describe lower bounds from the
BPP, the VCP and the BPPC literature, while in Section II.B
and II.C, we present litterature lower bounds for the BPPC. In
the following, we denote by Lx both the procedure used to
produce a lower bound, and the lower bound value itself
(where x changes accordingly to the procedure). Similarly, we
denote by Ux both the procedure used to produce an upper
bound, and the upper bound value itself.
This problem is a variation of the classical one
dimensional bin-packing problem where there are many
solution approaches available in literature.
A. Continious lower bound
Many lower bounding techniques have been proposed in
the BPP literature. The simplest technique corresponds to the
so called continuous lower bound (denoted as
in the
Whilst, a few work carried out on this problem with
conflicts constraint. We presented some best known heuristics
1
high weight have systematically high degree in G0 and they are
favored to be first included in the clique, even when they do
not belong to large clique of G0.
following), computed as the rounding-up of the sum of the
weights divided by the bin capacity
=∑
(1)
It’s why the improvement of Muritiba & al. focused on
finding a maximal click K on initially conflict graph G using
Johnson’s heuristic, and enlarge this click by added items in
V\K from extended conflict graph G0 obtained by updating E to
E0 (we called this strategy “G2 search click”). In [1] this
strategy leads to larger clicks than those obtained by applying
the Johnson’s algorithm directly on graph G0.
B. Gendreau’s lower bound
The Vertex Coloring Problem (VCP in literature) has
produced lower bounding techniques. A simple idea consists in
determining a maximum clique, whose cardinality naturally
gives a valid lower bound value for the VCP. Gendreau & al.
[4] developed a valid BPPC bound by heuristically computing
a maximum clique on the extended conflict graph G0 using the
greedy algorithm of Johnson [2]. This heuristic method
initializes the clique with the vertex of maximum cardinality,
then it enlarges the clique by iteratively adding new vertices: at
each iteration it considers the remaining vertices according to a
non-increasing degree order, and adds to the clique the first
vertex which can fit. This simple lower bound is denoted as
LMC in the following.
III.
In this section, we have described our lower bound in
order to better evaluate the performance of methods.
A. First idea
Continuous lower bound for the problem without
conflicts can be expressed by :
In conflicts case, the cardinal of maximal cliques | | in
conflicts graph represents a valid lower bound of the problem,
but this last give a bad bounding for weak value density case.
In this why, Gendreau & al. [6] and Muritiba & al. [1]
proposed lowers bounds (called GLB and MLB respectively in
follows) based in the two previous lower bound, expressed by :
One can note that, when applied to the BPPC, any BPP
lower bound is arbitrarily bad (it is enough to consider an
instance with bin capacity 1 and items weights << 1) and also
any VCP lower bound is arbitrarily bad (by considering an
instance with E = ;). It is thus more convenient to focus on
bounds tailored to the BPPC structure. One of these bounds
was proposed by Gendreau & al. [6], and is denoted as
constrained packing lower bound (LCP in the following).
GLB = MLB = | | +
1
(3)
∑
\ \
(4)
I\K is a set of items and possibly fractional
Where I
way items assigned to clique-bins (bins which are initialized
for each item of clique K) by solving a transportation problem
(called TP) that takes into account both the weights and the
given conflicts.
All the remaining items (or fractions of items) than cannot
fit in the | | initialized bins are stored in a new set V1, on
which
is applied. A valid lower bound is finally obtained
by computing
1
1
where
1 is the continuous lower bound for
remaining items V1 expressed by :
In LCP a maximal clique K of graph G0 is computed by
means of Johnson's algorithm as in LMC. Then a bin is
initialized for each item in the clique, and the items in V \ K are
assigned to these bins (possibly in a fractional way), by solving
a transportation problem that takes into consideration both the
weights and the given conflicts.
LCP = | | +
OUR LOWER BOUND
The difference between GLB and MLB lies in the
strategy of maximal cliques computing, the first uses the G1
technique, whereas the second uses the G2 technique. However,
GLB and MLB are the same second term of equation (2) which
is defined by the following TP :
(2)
C. Muritiba’s improvement lower bound
Origin set Io is the weights set of items I \ K ; destination
set Id is a set of residual capacity clique-bins where it adding a
dummy clique-bin b0 of infinite residual capacity ; and
are follows : For
1,
is equal to 1
transportation costs
if item i is compatible and enough fit with the item of clique
bin j, and to ∞ otherwise. For the added bin j = 0,
for
.
all
Muritiba & al. [1] provided an improvement of
Gendreau’s lower bound heuristic, this improvement affect
essentially the computing of the maximal click.
First, Gendreau & al. [6] finding a maximal click directly
on extended conflict graph G0 using Johnson’s heuristic (we
called this strategy “G1 search click”). In this sense, items of
2
bin and removed them from V. Let rmax be the largest
residual capacity of created bins;
The resolution of TP made an attempt to find a set of
items and fractional items that fits all residual clique-bins as
much as possible with favorite initially clique-bins than last
one added. In otherwise, all items which can’t be fit in initially
clique-bins, we will find in the adding bin b0. Therefore, a set
U defined in equation (4) is exactly the set of items and portion
of items assigned to the adding bin b0.
Step 2 : Define a set
of items with weights
greater than rmax and remove this items from initial set V. If
I1 is empty then go to step 6 ;
Step 3 : Apply TP with Io
V and Id
bk U b0 and
define a values
B. Iterative lower bound
From (3), we can see that all items
greater than the biggest residual clique-bin
∑
\ with weights
where :
/
are assigned to clique-bin b0 because they have infinite value of
transportation cost by construction. Therefore, this set denoted
will be treated like items without conflicts through the
second term of (3). On this way, we have developed our new
lower bound based on iterative method (called ILB), where in
each iteration t we computed a partial lower bound LBt using
of items for used in
equation (5) and we moved back a set
the next iteration.
wi
Step 5 : Put
|
∑
Step 6 : Apply TP with Io
define a values
and redefine a set V by
V and
and
∑
Step 7 : Put
IV.
| |
, end of procedure.
COMPUTATIONAL RESULTS
Lowers bound has been coded in C++ and run on a
Pentium IV 2.8 GHz processor with 512 MB of RAM, under a
Windows XP operating system.
(5)
\
Where,
| |
and go to step 1 ;
At each iteration t, our modification consists in separated
and preserved a set of items for the next iteration, any time
with the set of m unit item of zero degree denoted which
represents the remaining space of iteration t, where m is an
Euclidean division rest of sum of items of clique-bin b0 , by
capacity of bins C. Therefore, the used set in the next iteration
defined by :
|
Step 4 : Define a set I2 of y ‐unit items with weights
1 and degrees di 0 ;
0, and a set
We considered two data-sets of 8 classes proposed in
literatures by Faulkenauer [3] for BPP without conflicts which
are widely recognized as being rather difficult [6], each classes
contains 20 different instances. The first data-set includes four
classes of bin capacities C = 150 and items with integer
weights uniformly distributed in interval [20; 100]. The
number of items n for each class is respectively, 120, 250, 500
and 1000. The second data-set called the triplet bin-packing
instances where each bin cannot be filled with more than three
items, includes also four classes of bin capacities C = 100 and
items with integer weights uniformly distributed in interval
[25; 50] with one decimal digit. The number of items n for
each class is respectively, 60, 120, 249 and 501.
Step 1 : Determine a maximal clique K on G0 V; E0
using Johnson’s heuristic and assign each item of K to a new
A conflicts graph G was generated with ten different
density for each instances of the 8 described classes, by using
the following description : For each vertex of G which
K is a set of maximal click items computed by means of
I \K is a
Johnson’s heuristic directly in G’ (Vt; E’t) and I
set of items and possibly fractional way items assigned to
clique-bins Kt by solving TP. Iterations stopped when
becomes empty, and the final lower bound LB(I) is :
∑
(6)
C. Algorithm
Here is step by step description of our lower bound
procedure:
Step 0 : Initialization of lower bound LB
I;
of items V
3
represents an item i, an uniform value vi on [0; 1] is assigned
and for each used density d = 0%; 10%; :::; 90%, an edge (i; i’)
is created if
.
Among the three lowers bound, the deviation of lower
bound dev LB is expressed by :
7
where Best is a constant for all lowers bound
represents the best known heuristic results.
Figure 1 – Lowers bound deviation results
Figure 1 indicates the deviation of lowers bound from the
best one. For each density, we calculate the average lowers
bound for ten instances and the deviation of each lower bound
from the best one, computed by expression (7).
We can see from this figure that iterative lower bound
ILB gives the best lower bound for each positive density value,
where the deviation is always equal to zero and smaller than
Gendreau’s lower bound GLB and Muritiba’s lower bound
MLB.
This result was obtained for the first data-set instances
(class1 to class4). Whilst, for the second dataset (triplet binpacking) the results have not visualized because all methods
gives the same lower bound for each density value, and the
deviations is equal to zero.
4
Our lower bound is guaranteed to always perform as good
as that of Muritiba & al. In practice, it provides significant
improvements at a very low cost in terms of running times.
Let, this result indicate that the iterative lower bound,
provides a significant improvement over with the bounds by
Gendreau & al. and Muritiba & al., when compared on the new
instances, especially when the density of the conflict graph lies
between 20% and 60%. This may be explained by :
This method gives in worst case the same values as
Muritiba by construction, and it runs in a small number of
iterations. It is effective when the conflicts between items are
generated randomly and independently of the elements in
question. Moreover, this is a case in practically problem.
On Muritiba’s instances, the conflict graphs are generated
in such a way that vertex which has a high weight is likely to
be incompatible with many more items. Therefore, the clique
found using Johnson’s heuristic is likely to be composed of
such items, and most other items are likely to be mutually
compatible.
[1]
The iterative method stops in this case at the first
iteration. When the density is high, all methods converge to the
same value, because the first term (the cardinal of a clique)
becomes bigger and dominates the value obtained; when the
density is smaller, the second term (size bounding) dominates,
because the cardinal of a clique becomes smaller.
V.
[2]
[3]
[4]
[5]
[6]
CONCLUSION
[7]
We presented in this paper, a new lower bound for the
BPPC problem. This bound is an iterative version of bounds
proposed by Gendrau & al. and Muritiba & al. based in
iterative runs of Gendreau and Muritiba lower bound.
[8]
5
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