Bin Paking via
Disrepany of Permutations
F. Eisenbrand, D. Palv
olgyi & T. Rothvo
Cargese Workshop 2010
Bin Paking
Input:
I
Items with sizes s ; : : : ; sn 2 [0; 1℄
Pak items into minimum number of bins of size 1.
1
1
Goal:
0
1
si
bin 1
bin 2
input
Bin Paking
Input:
I
Items with sizes s ; : : : ; sn 2 [0; 1℄
Pak items into minimum number of bins of size 1.
1
1
Goal:
0
1
si
bin 1
bin 2
input
Bin Paking
Input:
I
Items with sizes s ; : : : ; sn 2 [0; 1℄
Pak items into minimum number of bins of size 1.
1
1
Goal:
0
1
si
bin 1
bin 2
input
Bin Paking
Input:
I
Items with sizes s ; : : : ; sn 2 [0; 1℄
Pak items into minimum number of bins of size 1.
1
1
Goal:
0
1
si
bin 1
bin 2
input
Bin Paking
Input:
I
Items with sizes s ; : : : ; sn 2 [0; 1℄
Pak items into minimum number of bins of size 1.
1
1
Goal:
0
1
si
bin 1
bin 2
input
Bin Paking
Input:
I
Items with sizes s ; : : : ; sn 2 [0; 1℄
Pak items into minimum number of bins of size 1.
1
1
Goal:
0
1
I
si
bin 1
bin 2
First Fit Dereasing [Johnson '73℄: AP X input
11
9
OP T
+4
Bin Paking
Input:
I
Items with sizes s ; : : : ; sn 2 [0; 1℄
Pak items into minimum number of bins of size 1.
1
1
Goal:
0
1
I
I
si
bin 1
bin 2
input
First Fit Dereasing [Johnson '73℄: AP X OP T + 4
Asymptoti PTAS [de la Vega & Luker '81℄ :
AP X (1 + ")OP T + O(1=" ) in time O(n) f (")
11
9
2
Bin Paking
Input:
I
Items with sizes s ; : : : ; sn 2 [0; 1℄
Pak items into minimum number of bins of size 1.
1
1
Goal:
0
1
si
bin 1
bin 2
input
First Fit Dereasing [Johnson '73℄: AP X OP T + 4
Asymptoti PTAS [de la Vega & Luker '81℄ :
AP X (1 + ")OP T + O(1=" ) in time O(n) f (")
I Asymptoti FPTAS [Karmarkar & Karp '82℄:
AP X OP T + O(log n) in poly-time
I
I
11
9
2
2
Bin Paking
Input:
I
Items with sizes s ; : : : ; sn 2 [0; 1℄
Pak items into minimum number of bins of size 1.
1
1
Goal:
0
1
si
bin 1
bin 2
input
First Fit Dereasing [Johnson '73℄: AP X OP T + 4
Asymptoti PTAS [de la Vega & Luker '81℄ :
AP X (1 + ")OP T + O(1=" ) in time O(n) f (")
I Asymptoti FPTAS [Karmarkar & Karp '82℄:
AP X OP T + O(log n) in poly-time
I Strongly NP-hard even if < si < ! 3-Partition
I
I
11
9
2
2
1
4
1
2
The Gilmore Gomory LP relaxation
I
Feasible patterns:
P = fp 2 f0; 1gn j sT p 1g
I
Gilmore Gomory LP relaxation:
X
min xp
Xp2P
p xp p2P
xp
1
0 8p 2 P
The Gilmore Gomory LP relaxation - Example
1
si
0:44 0:4 0:3 0:26
input
The Gilmore Gomory LP relaxation - Example
1
si
input
0:44 0:4 0:3 0:26
X
min xp
p2P
01
0
1 0 0 0 1 1 1 0 0 0 1 01
1
C
B
B
0 1 0 0 1 0 0 1 1 0 0 1C x B1C
B
1C
0 0 1 0 0 1 0 1 0 1 1 1A
A
0 0 0 1 0 0 1 0 1 1 1 1
x
1
0
The Gilmore Gomory LP relaxation - Example
1
si
input
0:44 0:4 0:3 0:26
X
min xp
p2P
01
0
1 0 0 0 1 1 1 0 0 0 1 01
1
C
B
B
0 1 0 0 1 0 0 1 1 0 0 1C x B1C
B
1C
0 0 1 0 0 1 0 1 0 1 1 1A
A
0 0 0 1 0 0 1 0 1 1 1 1
1=2
1=2
x
0
1=2
1
The Gilmore Gomory LP relaxation
I
Gilmore Gomory LP relaxation:
P
min
x
P p2P p
p2P p xp
xp
1
0 8p 2 P
Modied Integer Roundup Conjeture
OP T dOP Tf e + 1
I
I
True, if # of dierent item sizes 7 [Seb}o, Shmonin '09℄
Best known general bound: OP T OP Tf + O(log n)
2
The Gilmore Gomory LP relaxation
I
Gilmore Gomory LP relaxation:
P
min
x
P p2P p
p2P p xp
xp
1
0 8p 2 P
Modied Integer Roundup Conjeture
OP T dOP Tf e + 1
I
I
I
True, if # of dierent item sizes 7 [Seb}o, Shmonin '09℄
Best known general bound: OP T OP Tf + O(log n)
Additive integrality gap := OP T
OP Tf
2
The Gilmore Gomory LP relaxation
I
Gilmore Gomory LP relaxation:
P
min
x
P p2P p
p2P p xp
xp
1
0 8p 2 P
Modied Integer Roundup Conjeture
OP T dOP Tf e + 1
I
I
I
True, if # of dierent item sizes 7 [Seb}o, Shmonin '09℄
Best known general bound: OP T OP Tf + O(log n)
Additive integrality gap := OP T
OP Tf
2
Question
Is additive gap onstant (if
1
4
< si <
1
2
)?
Bek's Conjeture
3-Permutations Conjeture [Bek℄
Given any 3 permutations on n symbols, one an olor the
symbols with red and blue, suh that in any interval of any of
those permutations, the number of red and blue symbols diers
by O(1).
permutation 1:
4
6
1
5
7
2
8
3
permutation 2:
7
8
2
5
3
4
1
6
permutation 3:
2
1
6
4
8
5
3
7
Bek's Conjeture
3-Permutations Conjeture [Bek℄
Given any 3 permutations on n symbols, one an olor the
symbols with red and blue, suh that in any interval of any of
those permutations, the number of red and blue symbols diers
by O(1).
permutation 1:
4
6
1
b
5
7
2
8
3
permutation 2:
7
8
2
5
3
4
1
b
6
permutation 3:
2
1
6
4
8
5
3
7
b
Bek's Conjeture
3-Permutations Conjeture [Bek℄
Given any 3 permutations on n symbols, one an olor the
symbols with red and blue, suh that in any interval of any of
those permutations, the number of red and blue symbols diers
by O(1).
permutation 1:
4
6
1
b
5
7
2
b
8
3
permutation 2:
7
8
2
b
5
3
4
1
b
6
permutation 3:
2
1
6
4
8
5
3
7
b
b
Bek's Conjeture
3-Permutations Conjeture [Bek℄
Given any 3 permutations on n symbols, one an olor the
symbols with red and blue, suh that in any interval of any of
those permutations, the number of red and blue symbols diers
by O(1).
permutation 1:
4
6
1
b
5
7
2
b
8
3
permutation 2:
7
8
2
b
5
3
b
4
1
6
permutation 3:
2
1
6
4
8
5
3
7
b
b
b
b
b
Bek's Conjeture
3-Permutations Conjeture [Bek℄
Given any 3 permutations on n symbols, one an olor the
symbols with red and blue, suh that in any interval of any of
those permutations, the number of red and blue symbols diers
by O(1).
permutation 1:
4
b
6
1
b
5
7
2
b
8
3
permutation 2:
7
8
2
b
5
3
b
4
b
1
6
permutation 3:
2
1
6
4
8
5
3
7
b
b
b
b
b
b
Bek's Conjeture
3-Permutations Conjeture [Bek℄
Given any 3 permutations on n symbols, one an olor the
symbols with red and blue, suh that in any interval of any of
those permutations, the number of red and blue symbols diers
by O(1).
permutation 1:
4
b
6
1
b
5
b
7
2
b
8
3
permutation 2:
7
8
2
b
5
b
3
b
4
b
1
6
permutation 3:
2
1
6
4
8
5
3
7
b
b
b
b
b
b
b
Bek's Conjeture
3-Permutations Conjeture [Bek℄
Given any 3 permutations on n symbols, one an olor the
symbols with red and blue, suh that in any interval of any of
those permutations, the number of red and blue symbols diers
by O(1).
permutation 1:
4
b
6
b
1
b
5
b
7
2
b
8
3
permutation 2:
7
8
2
b
5
b
3
b
4
b
1
6
permutation 3:
2
1
6
4
8
5
3
7
b
b
b
b
b
b
b
b
b
Bek's Conjeture
3-Permutations Conjeture [Bek℄
Given any 3 permutations on n symbols, one an olor the
symbols with red and blue, suh that in any interval of any of
those permutations, the number of red and blue symbols diers
by O(1).
permutation 1:
4
6
b
1
b
5
b
7
b
2
b
8
3
permutation 2:
7
b
8
2
b
5
b
3
b
4
b
1
6
permutation 3:
2
1
6
4
8
5
3
7
b
b
b
b
b
b
b
b
b
b
b
Bek's Conjeture
3-Permutations Conjeture [Bek℄
Given any 3 permutations on n symbols, one an olor the
symbols with red and blue, suh that in any interval of any of
those permutations, the number of red and blue symbols diers
by O(1).
permutation 1:
4
6
b
1
b
5
b
7
b
2
b
8
b
3
permutation 2:
7
b
8
b
2
b
5
b
3
b
4
b
1
6
permutation 3:
2
1
6
4
8
5
3
7
b
b
b
b
b
b
b
b
b
b
b
b
Bek's Conjeture
3-Permutations Conjeture [Bek℄
Given any 3 permutations on n symbols, one an olor the
symbols with red and blue, suh that in any interval of any of
those permutations, the number of red and blue symbols diers
by O(1).
permutation 1:
4
6
b
1
b
5
b
7
b
2
b
8
b
3
permutation 2:
7
b
8
b
2
5
3
4
1
6
permutation 3:
2
1
6
4 8 5 3
di red/blue O(1)
7
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
Bek's Conjeture
3-Permutations Conjeture [Bek℄
Given any 3 permutations on n symbols, one an olor the
symbols with red and blue, suh that in any interval of any of
those permutations, the number of red and blue symbols diers
by O(1).
permutation 1:
4
6
b
1
b
5
b
7
b
2
b
8
b
3
permutation 2:
7
b
8
b
2
5
3
4
1
6
permutation 3:
2
1
6
4 8 5 3
di red/blue O(1)
7
I
b
b
b
b
b
b
b
b
b
b
b
b
b
W.l.o.g. onsider intervals that start at beginning
b
b
b
Disrepany theory
I
Set system S = fS ; : : : ; Sm g; Si [n℄
1
Disrepany theory
I
I
Set system S = fS ; : : : ; Sm g; Si [n℄
Coloring : [n℄ ! f 1; +1g
1
Disrepany theory
I
I
I
Set system S = fS ; : : : ; Sm g; Si [n℄
Coloring : [n℄ ! f 1; +1g
1
Disrepany
dis(S ) = nmin
max j(S )j:
!f g S 2S
where (S ) = Pi2S (i).
:[ ℄
1
Disrepany theory
I
I
I
Set system S = fS ; : : : ; Sm g; Si [n℄
Coloring : [n℄ ! f 1; +1g
1
Disrepany
dis(S ) = nmin
max j(S )j:
!f g S 2S
:[ ℄
1
where (S ) = Pi2S (i).
Known results:
p
I n sets, n elements: dis(S ) = O( n) [Spener '85℄
Disrepany theory
I
I
I
Set system S = fS ; : : : ; Sm g; Si [n℄
Coloring : [n℄ ! f 1; +1g
1
Disrepany
dis(S ) = nmin
max j(S )j:
!f g S 2S
:[ ℄
1
where (S ) = Pi2S (i).
Known results:
p
I n sets, n elements: dis(S ) = O( n) [Spener '85℄
I Every element in t sets:pdis(S ) < 2t [Bek & Fiala '81℄
Conjeture: dis(S ) O( t)
Matrix disrepany
I
Matrix A
dis(A) := x2fmin; gn kAx
01
1
2
A ( ;:::;
1
2 )k1
Matrix disrepany
I
Matrix A
dis(A) := x2fmin; gn kAx
01
1
2
A ( ;:::;
1
2 )k1
0
1 1 01
A = 0 1 1A
i
1 0 1
S
dis(S ) = 2 dis(A)
i
set S
Matrix disrepany
I
Matrix A
dis(A) := x2fmin; gn kAx
01
1
2
A ( ;:::;
1
2 )k1
0
1 1 01
A = 0 1 1A
i
1 0 1
S
dis(S ) = 2 dis(A)
I
i
Linear disrepany:
lindis(A) := y2max; n x2fmin; gn kAx
[0 1℄
01
Ayk1
set S
Overview
O(1)
O(1)
3-Partition
gap
Linear
dis. of
3-monotone
matries
Disrepany of
3 permutations
O(1)
Overview
O(1)
O(1)
3-Partition
gap
Linear
dis. of
3-monotone
matries
Disrepany of
3 permutations
O(1)
I A is 3-monotone, if
I olumns are monotone inreasing
I entries are 2 f0; : : : ; 3g
00 1 01
B0 2 1C
A=B
2 2 1C
A
3 2 2
Overview
O(1)
O(1)
3-Partition
gap
Linear
dis. of
3-monotone
matries
Disrepany of
3 permutations
O(1)
I A is 3-monotone, if
I olumns are monotone inreasing
I entries are 2 f0; : : : ; 3g
Theorem
00 1 01
B0 2 1C
A=B
2 2 1C
A
Bek's Conjeture ) 3-Partition gap is O(1)
3 2 2
Overview
O(1)
3-Partition
gap
O(1)
Linear
dis. of
3-monotone
matries
Disrepany of
3 permutations
O(1)
I A is 3-monotone, if
I olumns are monotone inreasing
I entries are 2 f0; : : : ; 3g
Theorem
00 1 01
B0 2 1C
A=B
2 2 1C
A
Bek's Conjeture ) 3-Partition gap is O(1)
3 2 2
Redution: Gap
! LinDis
Lemma
(A) O(1)
3
O(1)
I Sort items: s s : : : sn
Suppose lindis
Then the
for any 3-monotone matrix
-Partition gap is
1
2
.
A.
Redution: Gap
! LinDis
Lemma
(A) O(1)
A
3
O(1)
I Sort items: s s : : : sn
I y be opt. frational 3-Partition sol., B pattern matrix
Suppose lindis
Then the
for any 3-monotone matrix
-Partition gap is
1
2
01 0 11
B1 1 0 C
B=B
0 1 1C
A
0 1 1
.
.
Redution: Gap
! LinDis
Lemma
(A) O(1)
A
3
O(1)
I Sort items: s s : : : sn
I y be opt. frational 3-Partition sol., B pattern matrix
I Add up rows 1; : : : ; i to obtain row i for new matrix A.
Append row (3; : : : ; 3)
Suppose lindis
Then the
for any 3-monotone matrix
-Partition gap is
1
.
2
01 0 11
01 0 11
B
2 1 1C
C
B
C
B
1
1
0
B
C
B
B=
!
A = B2 2 2C
A
0 1 1
A
2 3 3C
0 1 1
3 3 3
.
Redution: Gap
! LinDis
Lemma
(A) O(1)
A
3
O(1)
I Sort items: s s : : : sn
I y be opt. frational 3-Partition sol., B pattern matrix
I Add up rows 1; : : : ; i to obtain row i for new matrix A.
Append row (3; : : : ; 3)
I A is 3-monotone ) x 2 f0; 1gm : kAx Ayk1 = O(1)
Suppose lindis
Then the
for any 3-monotone matrix
-Partition gap is
1
.
2
01 0 11
01 0 11
B
2 1 1C
C
B
C
B
1
1
0
B
C
B
B=
!
A = B2 2 2C
A
0 1 1
A
2 3 3C
0 1 1
3 3 3
.
Redution: Gap
! LinDis
Lemma
(A) O(1)
A
3
O(1)
I Sort items: s s : : : sn
I y be opt. frational 3-Partition sol., B pattern matrix
I Add up rows 1; : : : ; i to obtain row i for new matrix A.
Append row (3; : : : ; 3)
I A is 3-monotone ) x 2 f0; 1gm : kAx Ayk1 = O(1)
I Due to last row: 1T x = 1T y O(1)
Suppose lindis
Then the
for any 3-monotone matrix
-Partition gap is
1
.
2
01 0 11
01 0 11
B
2 1 1C
B
C
C
B
1
1
0
B
C
B
B=
!
A = B2 2 2C
A
0 1 1
A
2 3 3C
0 1 1
3 3 3
.
Redution: Gap
! LinDis
Lemma
(A) O(1)
A
3
O(1)
I Sort items: s s : : : sn
I y be opt. frational 3-Partition sol., B pattern matrix
I Add up rows 1; : : : ; i to obtain row i for new matrix A.
Append row (3; : : : ; 3)
I A is 3-monotone ) x 2 f0; 1gm : kAx Ayk1 = O(1)
I Due to last row: 1T x = 1T y O(1)
I Bi y = 1 ) Ai y = i ) Ai x = i O(1)
I Due to ith row: x reserves i O(1) slots for items 1; : : : ; i
01 0 11
01 0 11
B
2 1 1C
C
B
C
B
1
1
0
B
C
B
B=
!
A = B2 2 2C
A
0 1 1
A
2 3 3C
0 1 1
3 3 3
Suppose lindis
Then the
for any 3-monotone matrix
-Partition gap is
1
2
.
.
Redution: Gap
! LinDis (2)
input items V slots provided by x
1
1 2
2
i
n
...
...
...
...
2
1
i
0
n
1
i O(1)
slots
Redution: Gap
! LinDis (2)
input items V slots provided by x
1
1 2
2
i
n
...
...
...
...
2
1
i
0
n
1
i O(1)
slots
Redution: Gap
! LinDis (2)
input items V slots provided by x
1
1 Bx
1
2
i
n
I
...
...
...
...
2
B2 x
i
Bi x
n
Bn x
Bipartite graph G = (V [_ U; E ) with (i; j ) :, si sj
Redution: Gap
! LinDis (2)
input items V slots provided by x
1
1 Bx
1
V0
2
i
n
I
I
...
...
...
...
2
B2 x
N (V 0 )
i
Bi x
n
Bn x
i O(1)
Bipartite graph G = (V [_ U; E ) with (i; j ) :, si sj
Halls Marriage Theorem:
There is a V -perfet mathing i
for any V 0 V , Pv2N V deg(v) jV 0j
(
0
)
slots
Redution: Gap
! LinDis (2)
input items V slots provided by x
1
1 B x + O(1)
1
V0
2
i
n
...
...
...
...
2
B2 x
N (V 0 )
i
Bi x
n
Bn x
i slots
Bipartite graph G = (V [_ U; E ) with (i; j ) :, si sj
Halls Marriage Theorem:
There is a V -perfet mathing i
for any V 0 V , Pv2N V deg(v) jV 0j
I x + O(1) extra bins is feasible (osts OP Tf + O(1))
I
I
(
0
)
Overview
O(1)
O(1)
3-Partition
gap
Linear
dis. of
3-monotone
matries
Disrepany of
3 permutations
O(1)
Overview
O(1)
O(1)
3-Partition
gap
Linear
dis. of
3-monotone
matries
Disrepany of
3 permutations
O(1)
Redution: LinDis
Lemma
Let A be 3-monotone.
I
I
! Perm.Dis.
Bek's Conjeture
)
Let x 2 [0; 1℄n be given.
Goal: Find y 2 f0; 1gn with Ax Ay
lindis
(A) = O(1)
.
Redution: LinDis
Lemma
Let A be 3-monotone.
I
I
! Perm.Dis.
Bek's Conjeture
)
lindis
(A) = O(1)
Let x 2 [0; 1℄n be given.
Goal: Find y 2 f0; 1gn with Ax Ay
Theorem (Lovasz, Spener & Vesztergombi '86)
There is always a submatrix B of A with
lindis(A) 2 dis(B ):
I
I
Intuitively: Worst ase is x 2 f0; gn
It suÆes to show: dis(A) = O(1)
1
2
.
Redution: LinDis
0
A=B
0
1
2
2
3
3
3
0
0
1
1
1
1
2
1
1
1
2
2
3
3
0
0
0
0
0
2
2
1
C
A
! Perm.Dis. (2)
Redution: LinDis
I
! Perm.Dis. (2)
Write A = B + B + B with B i 1-monotone
1
0
A=B
0
1
2
2
3
3
3
0
0
1
1
1
1
2
1
1
1
2
2
3
3
0
0
0
0
0
2
2
2
1 0
C
A =B
3
0
1
1
1
1
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
1 0
C
A+B
0
0
1
1
1
0
0
0
0
0
10
1 1
0
0
0
1
1
0
0
0
0
0
11
1 0
C
A+B
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
1
C
A
| {z } | {z } | {z }
B1
=
1 1
B2
=
B3
=
Redution: LinDis
I
I
! Perm.Dis. (2)
Write A = B + B + B with B i 1-monotone
Column order of B i indues permutation i
1
0
A=B
0
1
2
2
3
3
3
0
0
1
1
1
1
2
1
1
1
2
2
3
3
0
0
0
0
0
2
2
2
1 0
C
A =B
3
0
1
1
1
1
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
1 0
C
A+B
0
0
1
1
1
0
0
0
0
0
10
1 1
0
0
0
1
1
0
0
0
0
0
11
1 0
C
A+B
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
1
C
A
| {z } | {z } | {z }
B1
=
1 = (3; 1; 2; 4)
1 1
B2
=
2 = (1; 3; 4; 2)
B3
=
3 = (1; 3; 2; 4)
! Perm.Dis. (2)
Redution: LinDis
I
I
I
Write A = B + B + B with B i 1-monotone
Column order of B i indues permutation i
Let : [n℄ ! f1g be oloring that's good for ; : : : ; .
1
2
3
1
dis(A) kAk1
0
A=B
0
1
2
2
3
3
3
0
0
1
1
1
1
2
1
1
1
2
2
3
3
0
0
0
0
0
2
2
1 0
C
A =B
triangle ineq
0
1
1
1
1
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
X
3
i
|kB {zk1} = O(1)
i=1
1 0
C
A+B
3
O(1)
0
0
1
1
1
=
0
0
0
0
0
10
1 1
0
0
0
1
1
0
0
0
0
0
11
1 0
C
A+B
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
1
C
A
| {z } | {z } | {z }
B1
=
1 = (3; 1; 2; 4)
1 1
B2
=
2 = (1; 3; 4; 2)
B3
=
3 = (1; 3; 2; 4)
Open problem (1)
O(1)
O(1)
3-Partition
gap
Linear
dis. of
3-monotone
matries
Disrepany of
3 permutations
O(1)
Open problem (1)
O(1)
O(1)
Linear
dis. of
3-monotone
matries
3-Partition
gap
?
Disrepany of
3 permutations
O(1)
Open problem (2)
I
I
Dene
(n) := max dis(A) j
Example:
A 2 [0; 1℄nn ;
A has monotone olumns
0
0:1 0:0 0:51
A = 0:4 0:7 0:9A
0:5 0:9 1:0
Open problem (2)
I
I
Dene
(n) := max dis(A) j
A 2 [0; 1℄nn ;
A has monotone olumns
0
0:1 0:0 0:51
A = 0:4 0:7 0:9A
Example:
0:5 0:9 1:0
Lemma
For
any Bin Paking instane
OP T
I
OP Tf + O(log n) (n):
We an prove (n) O(log n).
Open problem (2)
I
I
Dene
(n) := max dis(A) j
A 2 [0; 1℄nn ;
A has monotone olumns
0
0:1 0:0 0:51
A = 0:4 0:7 0:9A
Example:
0:5 0:9 1:0
Lemma
For
any Bin Paking instane
OP T
I
OP Tf + O(log n) (n):
We an prove (n) O(log n).
Question
Is (n) = O(1)?
The end
Thanks for your attention
I
Bin Paking via Disrepany of Permutations
(F. Eisenbrand, D. Palvolgyi, T. Rothvo - to appear in
SODA'11;
)
http://arxiv.org/abs/1007.2170