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Communications and Network, 2018, 10, 1-10
http://www.scirp.org/journal/cn
ISSN Online: 1947-3826
ISSN Print: 1949-2421
DOI:
10.4236/cn.2018.101001 Dec. 13, 2017 1 Communications and Network
Some Complexity Results for the k-Splittable
Flow Minimizing Congestion Problem
Chengwen Jiao, Qi Feng, Weichun Bu
College of Science, Zhongyuan University of Technology, Zhengzhou, China
Abstract
In this paper, we mainly consider the complexity of the k-splittable flow min
i-
mizing congestion problem. We give some complexity results. For the k-
splittable
flow problem, the existence of a feasible solution is strongly NP-hard. When
the
number of the source nodes is an input, for the uniformly exactly k-
splittable
flow problem, obtaining an approximation algorithm with performance ratio
better
than 2 is NP-hard. When k is an input, for single commodity k-
splittable flow
problem, obtaining an algorithm with performance ratio better than
( )
512+
is NP-
hard. In the last of the paper, we study the relationship of minimizing
congestion and minimizing number of rounds in the k-splittable flow pro
b-
lem. The smaller the congestion is, the smaller the number of rounds.
Keywords
k-Splittable Flow, Minimize Congestion, Minimize Number of Rounds,
Complexity
1. Introduction
In the traditional multi-commodity flow problems, flow being sent from the source
nodes to the destination nodes may travel on large number of paths through the
network. This effect is undesired or even forbidden in many applications. Such as
in the modern broadband communication networks, namely the multiple protocol
label-switched networks (MPLS), data packets are gathered under a single label
with the aim to limit the routing tables and to increase the quality of data trans-
mission. The label-switched paths (LSPs) must support the routing of data traffic
between different terminal nodes, from one endpoint to another. An important fea-
ture of MPLS is its ability to set up traffic engineering mechanism (MPLS-TE). It
can control the structure of the traffic for each customer by setting restriction on
How to cite this paper:
Jiao, C.W.,
Feng,
Q
. and Bu, W.C. (2018)
Some Complexity
Results for the k
-Splittable Flow Minimiz-
ing Congestion Problem
.
Communications
and Network
,
10
, 1-10.
https:
//doi.org/10.4236/cn.2018.101001
Received:
November 27, 2017
Accepted:
December 10, 2017
Published:
December 13, 2017
Copyright © 201
8 by authors and
Scientific
Research Publishing Inc.
This work
is licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
C. W. Jiao et al.
DOI:
10.4236/cn.2018.101001 2 Communications and Network
the number of routes the customer used. In order to preserve the QoS require-
ment, the demand of each customer must be satisfied. Using a single path would
possibly increase the congestion of the network, while, on the other hand, a large
number of LSPs would decrease the performance of the protocol, and the inter-
mediate situation is considered in the MPLS network.
In 2002, Baier [1] proposed the k-splittable flow problem. Given a directed
graph
( )
, ,,G VEuc=
with arc capacities
0
e
u>
and arc cost
0,
e
c eE> ∀∈
.
A set of commodities is denoted by
L
, each commodity
lL
∈
has a certain
amount of demand
l
d
to transmit from a source node
l
s
to a destination
node
l
t
. The number of paths each commodity can use is restricted, that is
commodity
l
can only use at most
l
k
paths to transmit the flow. If each
commodity uses exactly
l
k
paths and the flow of each path is the same value,
l
l
dk
, this problem is called uniformed exactly k-splittable flow problem. If
1,
l
k lL
= ∀∈
, this problem is in fact the unsplittable flow problem (UFP) which
was introduced by Kleinberg [2]. If
,
l
k E lL≥ ∀∈
, it is in fact the traditional
flow problem.
Kleinberg [2] introduced the following optimization versions of the unsplitta-
ble flow problem. Minimum congestion: find the smallest value
1
α
≥
such that
there exists an unsplittable flow such that violates the capacity of any edge at
most by a factor
α
. Minimum number of rounds: partition the set of commod-
ities into a minimum number of subsets. Maximum routable demand: find a feasi-
ble unsplittable flow for a subset of demands maximizing the sum of demands in
the subset. In this paper, we are mainly interested in the minimum congestion
problem and minimum number of rounds problem.
For the unsplittable flow problem, Erlebach
et al
. [3] proved that for arbitrary
ε
, obtaining an approximation algorithm for minimizing congestion and cost
with performance ratio better than
()
2 ,1
ε
−
is NP-hard. The unsplittable flow
problem is much easier if all commodities share a common single source node.
However, the resulting single-source unsplittable flow problem still remains strongly
NP-hard. Researches gave some approximation algorithms for this problem (see
references [4] [5] [6]).
As for the k-splittable flow problem, researchers generalize the above optimi-
zation versions and there are a lot of studies on the related problems. Baier
et al
.
[7] solved the maximum Single- and Multi-commodity k-splittable flow problem
using approximation algorithms. The authors proved that the maximum single-
commodity k-splittable flow problem is NP-hard in the strong sense for directed
graphs. Koch
et al
. [8] studied the single commodity maximum k-splittable flow
problem. It is proved that when k is a constant, this problem is strongly NP-hard
and obtaining an approximation algorithm with performance ratio better than
( )
1kk+
is NP-hard. While when k is not a constant, obtaining an approxima-
tion algorithm with performance ratio better than 5/6 is NP-hard. Kolliopoulos
[9] studied the approximation algorithms for the single source 2-splittable flow
problem using rounding down strategy and Salazar
et al.
[10] considered the single
C. W. Jiao et al.
DOI:
10.4236/cn.2018.101001 3 Communications and Network
source k-splittable flow problem using rounding up strategy. Truffot
et al.
[11]
[12] [13] and Gamst
et al.
[14] [15] design branch-and-price exact algorithms for
the k-splittable flow problem. Palel [16] used a randomized rounding approach to
solve the k-splittable flow problem.
By the current researches, we can see that for the k-splittable flow problem,
there is little research on the optimization version of minimizing congestion, es-
pecially for the complexity of this problem. In this paper, we give some com-
plexity results for the minimizing congestion k-splittable flow problem. For the
k-splittable flow problem, the existence of a feasible solution is strongly NP-hard.
When the number of the source nodes is an input, for the uniformly exactly
k-splittable flow problem, obtaining an approximation algorithm with perfor-
mance ratio better than 2 is NP-hard. When k is an input, for the single com-
modity k-splittable flow problem, obtaining an algorithm with performance ra-
tio better than
( )
512+
is NP-hard. We also study the relationship between
minimum congestion and minimum number of rounds.
2. Some Complexity Results
In this section, we will give three complexity results for the minimizing conges-
tion k-splittable flow problem. These results are given by the following Theo-
rems.
Theorem 1: For the k-splittable multi-commodity transmission problem, the
existence of a feasible solution is strongly NP-hard.
Proof: Given an instance of the 3-partition problem with 3
n
elements,
,1 3
i
a in≤≤
.
3
1
n
i
i
a nd
=
= ⋅
∑
. For arbitrary
i
, we suppose that
42
i
d ad<<
.
Next construct a simple directed graph
G
, see Figure 1. The network has 2
nodes, one is source node
s
, and the other is sink node
t
. There are 3
n
parallel
directed edges from
s
to
t
with edge capacities
,1 3
i
a in≤≤
, respectively.
n
com-
modities from
s
to
t
with same demands
d
need to be transmitted. Each commod-
ity can use at most 3 paths.
If the 3-partition has a solution, that is, the 3
n
elements can be partitioned in-
to
n
sets. Each set has 3 elements and the total value of the 3 elements is exactly
d
. For each commodity, it can use the 3 edges corresponding to one of the
n
sets
to transmit flow. The demands of the
n
commodities can be transmitted through
the 3
n
edges of
G
. Thus there is a 3-splittable flow with congestion 1 to satisfy
the demands of the
n
commodities in the network. On the other hand, if there is
a feasible 3-splittable flow satisfying the demands in the network, since each edge
Figure 1. Network
G
obtained by 3-partition instance.
C. W. Jiao et al.
DOI:
10.4236/cn.2018.101001 4 Communications and Network
capacity is less than
d
/2 and larger than
d
/4 and the number of paths each com-
modity can use is at most 3, we have that each commodity must use exactly 3
paths to transmit the demand and the total flow value of the paths is
d
. In this
way, we obtain a feasible solution of the 3-partition instance. Partite the 3
n
ele-
ments into
n
sets, the elements in each set corresponding to the transmitting
paths of some commodity and the total value of the 3 elements are exactly
d
.
Since the 3-partition problem is strongly NP-hard, by the above discussion,
we know that the existence of a feasible solution of the k-splittable flow problem
is also strongly NP-hard.
For the uniformly exactly k-splittable flow problem, by referring the construc-
tion strategy used in [2] to prove some complexity result of the unsplittable flow
problem, we can obtain the following theorem.
Theorem 2: When the number of the source nodes
N
is an input, for the un-
iformly exactly k-splittable flow minimizing congestion problem, obtaining an
approximation algorithm with performance ratio better than 2 is NP-hard.
Proof: Given an instance of a 3-D matching. Denote set
{ }
12
,,,
n
A aa a=
,
set
{ }
12
,,,
n
B bb b=
, set
{ }
12
,,,
n
C cc c=
. A set
I ABC⊆××
is given with
Im
=
. Suppose that all the 3
n
elements in
,,ABC
appear in
I
. Further, if
there is a subset
MI⊆
with
Mn=
and all the elements in
,,ABC
appear
in
M
, that is each element in
,,ABC
appears exactly one time in
M
, we say that
there is a 3-D matching in
I
, and so the 3-D matching instance has a solution.
Next we construct a directed graph
G
and an instance
I′
of the k-splittable
flow problem. There are
N
source nodes in
G
, namely
12
,,,
N
SS S
. Suppose
, 1, ,
i
a Ai n∈=
, appears
i
t
times in
I
, that is there are
i
t
triples contains
i
a
in
I
.
For each
( )
,,
i jh
u abc=
, there is only one node in
G
corresponding to
j
b
and there are
1N−
nodes in
G
corresponding to
h
c
, denote them by
12 1
,,,
N
hh h
cc c
−
. For
i
a
, there are
1
i
t−
sink nodes
1
12
,,,
i
t
ii i
TT T
−
and
( )
( )
11
i
Nt−−
virtual nodes in
G
. For
( )
,,
i jh
u abc=
, there are two nodes
,
uu
xy
in
G
. The construction of the subgraph of
G
corresponding to
u
is showed
in Figure 2.
There are directed edges
() ( ) ( )
12 1
,,,,, ,
j j Nj
Sb S b S b
−
,
( )
,
ju
bx
,
( )
,
uu
xy
,
( )
,
Nu
Sx
,
( ) ( ) ( )
12 1
,,,,,,
N
uh uh uh
yc yc yc
−
in
G
. For
1, 2, , 1
i
rt= −
, there are
1N−
edge-disjoint paths going through the virtual nodes from
u
y
to the sink
node
r
i
T
, respectively. The edge capacities of
( )
( ) ( )
, ,, , ,
ju uu Nu
bx xy S x
are all
1N−
and the capacities of the remaining edges are 1. There are two classes of
commodities, one is called
C
-class commodity, the other is called
T
-class com-
modity. The
C
-class commodity is from
p
S
to
p
h
c
with demand 1,
1, ,
hn=
,
1, , 1pN= −
. Each
C
-class commodity can only use one path to
transmit flow. The
T
-class commodity is from
N
S
to
r
i
T
with demand
1N−
,
1, ,in=
,
1, , 1
i
rt= −
. Each
T
-class commodity can use
1N−
paths to trans-
mit flow.
If there is a 3-D matching in
I
, we can prove that there exists a uniformly ex-
actly k-splittable flow in
G
that satisfying the demands of all commodities with
C. W. Jiao et al.
DOI:
10.4236/cn.2018.101001 5 Communications and Network
Figure 2. The subgraph of
G
corresponding to triple
( )
,, .
i jh
u abc=
congestion value 1. Suppose the 3-D matching in
I
is
M
. For each triple
( )
,,
i jh
abc
in
M
, there is a path from
p
S
to
p
h
c
with flow 1,
1, , 1pN= −
. That is the
C
-class commodity can be satisfied by the subgraphs corresponding the triples in
M
. For the
T
-class commodity, they can be satisfied by the subgraphs correspond-
ing to the triples in
\
IM
. Since
i
a
appears
i
t
times in
I
, while in
M
, it appears
exactly once, the remaining
1
i
t−
times appears in
\IM
, for
1, , 1
i
rt= −
,
there are
1N−
paths from
N
S
to
r
i
T
with flow 1.
If there is a uniformly exactly k-splittable flow in
G
satisfying all commodities,
there is a 3-D matching in
I
. Suppose the triples that satisfying the
C
-class com-
modity be
M
, we will prove that
M
is in fact a 3-D matching in
I
. The number of
T
-class commodity is
( )
1
1
n
i
i
t mn
=
−=−
∑
and each
T
-class commodity needs
1N−
paths with flow 1 from source node
N
S
to sink node
r
i
T
. Each transmit-
ting path of both
C
-class commodity and
T
-class commodity contains exactly
one edge of kind of
( )
,
uu
xy
. Since the number of this kind of edges is
m
and
the capacities of these edges are
1N−
, the
C
-class commodity can only use
( )
m mn n− −=
−
( )
,
uu
xy
class edge which corresponding to
n
triples, that is
M
. For
1, , 1pN= −
, the commodity from
p
S
to
p
h
c
can only use one triple
containing
i
a
(the remaining
1
i
t−
triples containing
i
a
are used by
T
-class
commodity), that is all the elements in
A
appear in
M
. Since the total number of
incoming edges with capacity 1 of
j
b
is
1N−
and the total demands of the
C
-class commodity which must going through the nodes corresponding to ele-
ments in set
B
are
( )
1Nn−
, the triples in
M
contains all the elements in
B
.
Furthermore, since the number of triples in
M
is
n
, there is only one triple con-
tains
, 1, ,
j
bj n=
. Since the demand of the sink node
p
h
c
is 1 and all the de-
mands of the
C
-class commodity are satisfied, the triples in
M
must contain all
the elements in
C
. Similarly, since the number of triples in
M
is
n
, there is exact-
ly one triple in
M
containing
, 1, ,
h
ch n=
. By the above discussion, we prove
that
M
is indeed a 3-D matching of
I
.
Since the maximum flow from the source node to the sink node in
G
is 1, if
C. W. Jiao et al.
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10.4236/cn.2018.101001 6 Communications and Network
there is no 3-D matching, it is easy to see that there is no uniformly exactly
k-splittable flow in
G
that satisfying the capacity constraints. When
2N=
, the
capacity value of all the edges in
G
is 1. When there is no 3-D matching in
I
, the
congestion values of
G
is at least 2. Since the 3-D matching problem is a
NP-hard problem, for the uniformly exactly k-splittable flow problem, obtaining
an approximation algorithm with congestion value less than 2 is NP-hard.
Theorem 3: When k is an input, for the single commodity k-splittable flow
minimizing congestion problem, obtaining an approximate algorithm with per-
formance ratio better than
δ
is NP-hard, in which
( )
512
δ
= +
.
Proof: We prove this theorem by SAT reduction, referring the construction
strategy used in Baier [1] to prove some NP-hard problem. Given an instance of
SAT, the variable set is
{ }
12
,,,
n
xx x
and there are
m
subsets of
{ }
11
,,, ,,
nn
x xx x
¬¬
, namely
1,,
m
CC
. Next we construct a directed graph
G
, see Figure 3, the demand of the single commodity from source node
s
to sink
node
t
is
1k
δ
−+
. Next we will prove that if there is a feasible solution of the
SAT problem, that is there existing a 0-1 assignment to
1
,,
n
xx
such that each
subset
, 1, ,
j
Cj m=
, containing at least one element with value 1, the conges-
tion value of
G
is 1. Otherwise, if there is no feasible solution of the SAT prob-
lem, the congestion value of
G
is at least
δ
.
There are three steps to construct the directed graph
G
. Step 1, construct
n
-parallel chains connected by
1n+
directed edges. The
n
parallel chains cor-
respond to variables
, 1, ,
i
xi n=
, respectively. For each parallel chain, denote
one of it by true chain, the other by false chain. The number of edges in each
chain is equal to
42
n+
. The capacities of all the current constructed edges are
δ
. Step 2, we continue to construct edges for
G
using the
m
subsets
1
,,
m
CC
.
For each
1, ,jm=
, there are two nodes, namely
,
jj
uv
, in
G
corresponding
to subset
j
C
. The new directed edges are
( )
( )
11
, , , , 1, , 1
jj
su v u j m
+
= −
and
()
,
m
vt
. For each subset
j
C
, if
ij
xC
¬
∈
, two adjacent edges in the true chain of
i
x
are corresponding to
j
C
only. Two new directed edges are constructed:
j
u
is connected to the first node of the first edge and the second node of the
Figure 3. The graph
G
constructed by an instance of the SAT problem.
C. W. Jiao et al.
DOI:
10.4236/cn.2018.101001 7 Communications and Network
first edge is connected to
j
v
. Similarly, if
ij
xC∈
, two adjacent edges in the
false chain of
i
x
are corresponding to
j
C
only. Two new directed edges are
constructed:
j
u
is connected to the first node of the first edge and the second
node of the first edge is connected to
j
v
. Since each chain contains
42n+
di-
rected edges and the number of elements in each subset
j
C
is at most 2
n
, Step
2 can be implemented successfully. If there are more than one subsets contain the
same variable, these adjacent edges corresponding to each subset are arranged by
the decreasing order of the subset index. Step 3, add
2
k−
directed edges from
s
to
t
. The capacities of all the new constructed edges in Step 2 and Step 3 are
equal to 1 (see Figure 3).
If there is a feasible solution of the SAT problem, that is there existing a 0 - 1
assignment to
1
,,
n
xx
such that each subset
, 1, ,
j
Cj m
=
, contains at least
one element with value 1. By the construction of Step 2, we know that there are
two adjacent edges corresponding to this 1-value element, denote the first edge
by the key edge of
j
C
. From
s
, followed by
j
u
, the key edge of
j
C
, and node
, 1, ,
j
vj m=
, we can obtain a
-st
path with flow value 1, denote this path by
1
P
. From
s
, followed by the chains not used by
1
P
, we can obtain another
-st
path with flow value
δ
. Together with the
2k−
edges from
s
to
t
, we obtain
-kst
paths with flow value
1k
δ
−+
and the congestion of
G
is 1. If there is a
k-splittable flow satisfying the demand with congestion value 1, there is a feasi-
ble solution of the SAT problem. Since the total capacity of the directed edges
going out from
s
is
1k
δ
−+
and the maximum capacity of the
-st
paths is
δ
,
furthermore, the commodity can use at most
k
paths, according the above rea-
sons, we know that there must be
1 -k st−
paths with flow 1 and one
-st
path
with flow
δ
. Denote
P
be the path from
s
followed by
1
u
. Assign the variables
corresponding to the chains that path
P
transformed to 1. Since the congestion is
1,
P
and the path with flow
δ
must be edge-disjoint. Then the path
P
must go
through
11
,, , ,
mm
uv u v
and the key edges of
1
,,
m
CC
. And so for each
, 1, ,
j
Cj m
=
, there must be an element with value 1, and a feasible solution of
the SAT is obtained.
Now suppose there is no feasible solution of the SAT problem, the paths from
s
with length larger than 1 must not be edge-disjoint (otherwise, it contradicts to
the hypothesis), the congestion of any k-splittable flow satisfying the demand is
at least
( )
1
δδ
+
or
δ
. Since the SAT problem is NP-hard, for the k-splittable
single commodity problem with
k
as an input, obtaining an approximation algo-
rithm with congestion less than
δ
is NP-hard.
3. Minimum Number of Rounds
In this section, we consider the minimum number of rounds of the k-splittable
flow problem. The demand
l
d
of commodity
l
can be transformed in mul-
tiple times and the total number of the transmission paths cannot be larger than
l
k
. The objective of this problem is to transform all commodities but using least
times of the network. Minimizing number of rounds reduce the total transmis-
C. W. Jiao et al.
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10.4236/cn.2018.101001 8 Communications and Network
sion cost in some extent. In the k-splittable flow problem, there is some relation-
ship between minimum congestion and minimum number of rounds and we have
the following Theorem.
Theorem 4: Given a network
G
and a commodity set
L
, the commodities in
L
share a common source node
s
. Suppose
max min
du≤
. If for arbitrary feasible flow
x
that satisfies all the demands
12
,,,
L
dd d
, there exists a k-splittable flow
y
that
satisfies the demands and the path restrictions, and for arbitrary
eE∈
,
ee
yu
α
≤⋅
, we have that for all the commodities with demand not larger
min
uq
can be transformed in
α
rounds, in which
q
α
≥
.
Proof: Copy the network
G
in
α
times, denoting them by
1,,GG
α
,
respectively. For each
( )
e EG∈
, its capacity in
, 1, ,
j
Gj
α
=
is defined by
e
u
α
. Adding a super source node
S
and
L
super sink nodes
1
,,
L
TT
. For
each
1, ,j
α
=
, connect the super source node
S
to the source node
s
in
j
G
,
the sink nodes
i
t
is connected to
, 1, ,
i
Ti L=
. The capacities of all the new
edges are
+∞
. The construction of the new network is showed in Figure 4, de-
note it by
G′
. In
G′
, there are
L
commodities with demand
i
d
and path
restriction number
i
k
from source node
S
to the sink node
, 1, ,
i
Ti L=
, re-
spectively. Denote the commodity set in
G′
by
L′
. Suppose
x
is a feasible flow
in
G
satisfying all the commodities in
L
. By the construction of
G′
, we can ob-
tain a feasible flow
x′
from
x
in
G′
satisfying all the commodities in
L′
. For
1, ,
j
α
=
, the flow value
()
xe
′
of edge
( )
e EG
′
∈
is
e
x
α
. Denote the
set
*
L
be the commodities in
′
G
with demands not larger than
min
uq
. It is
easily to see that we can obtain a feasible flow
*
x
from
x′
satisfying
*
L
.
Since
min
uq
is not larger than the minimum capacity
min
u
α
of
G′
, by the
theorem hypothesis, there is a k-splittable flow
y
in
G′
satisfying the demands
and path restrictions of commodities in
*
L
. Further, for any
1, ,j
α
=
,
( )
j
e EG∈
, we have that
() () ( ) ()( ) ( )
*xe
yexexe xeue
ααα
α
′
≤⋅ ≤⋅ =⋅ ≤ ≤
That is in
y
, the flow value of each edge in
j
G
is not larger than the capacity
of the original graph
G
. Since in flow
y
, the paths from
S
to
i
T
can be transformed
Figure 4. The construction of
G'
.
C. W. Jiao et al.
DOI:
10.4236/cn.2018.101001 9 Communications and Network
into paths from
s
to
i
t
in
G
, and the number of paths is not larger than
i
k
, and
then the commodities in
L
with demands not larger than
min
/uq
can be
transformed into
α
rounds. The transmission paths and their path values in
round
j
are corresponding to the transmission paths with their flow values in
y
in
, 1, ,
j
Gj
α
=
.
In inference [10], for the minimizing congestion k-splittable flow problem,
there is a theorem as follows.
Theorem 5 [10]: For the single source k-splittable flow problem, suppose the
number of paths each commodity can use is all equal to
k
. For any feasible flow
x
that satisfying all the demands
1,,
L
dd
, there exists a k-splittable flow
y
that
satisfies the demands and the path restrictions, and for arbitrary
, ∈eE
( ) ( )
max
2
21
≤+
−
d
k
ye xe
kk
.
In Theorem 5, it is not required that
max min
du≤
, similar to the proof in Theo-
rem 4, we can have that for the commodities with demands between in
(
]
min
0,qu⋅
,
if we require these commodities be transmitted in
r
rounds, as long as the fol-
lowing inequality holds:
2 11 1
21
kq
k rk
⋅+ ⋅≤
−
By the above inequality, we know that when
2k=
, all the commodities with
demands not larger than
min
2
3u⋅
can be transmitted in 2 rounds. For the general
2k≥
, all the commodities with demands not larger than
min
u
can be trans-
mitted in 3 rounds.
4. Conclusion
In this paper, we give some complexity results of minimizing congestion in the
k-splittable flow problem. The two approximation algorithms in [9] and [10] are all
relied on the unsplittable flow algorithms. In the future, we will design approxi-
mation algorithms for minimizing congestion of the k-splittable flow problem
using new strategies.
Acknowledgements
The work is supported by the National Natural Science Foundation of China
under Grant No.11701595.
References
[1] Baier, G. (2003) Flows with Path Restrictions. Ph.D. Thesis, Technische Universitat
Berlin, Berlin.
[2] Kleinberg, J. (1996) Single-Source Unsplittable Flow.
Proceedings of the
37
th An-
nual Symposium on Foundations of Computer Science
, Burlington, 14-16 October
1996, 68-77. https://doi.org/10.1109/SFCS.1996.548465
[3] Erlebach, T. and Hall, A. (2002) NP-Hardness of Broadcast Scheduling and Inap-
C. W. Jiao et al.
DOI:
10.4236/cn.2018.101001 10 Communications and Network
proximability of Single-Source Unsplittable Min-Cost Flow.
Proceedings of the
13
th
ACM-SIAM Symposium on Discrete Algorithms
, San Francisco, 6-8 January 2002.
[4] Dinitiz, Y., Garg, N. and Goemans, M.X. (1999) On the Single Source Unsplittable
Flow Problem.
Combinatorica
, 19, 1-25. https://doi.org/10.1007/s004930050043
[5] Kolliopoulos, S.G. and Stein, C. (2001) Approximation Algorithms for Sin-
gle-Source Unsplittable Flow.
SIAM Journal on Computing
, 31, 919-946.
https://doi.org/10.1137/S0097539799355314
[6] Skutella, M. (2002) Approximating the Single-Source Unsplittable Min-Cost Flow
Problem.
Mathematical Programming
, 91, 493-514.
https://doi.org/10.1007/s101070100260
[7] Baier, G., Kohler, E. and Skutella, M. (2005) On the k-Splittable Flow Problem.
Al-
gorithmica
, 42, 231-248. https://doi.org/10.1007/s00453-005-1167-9
[8] Koch, R., Skutella, M. and Spenke, I. (2005) Approximation and Complexity of
k-Splittable Flows. In: Erlebach, T. and Persinao, G., Eds.,
Approximation and On-
line Algorithms
,
WAOA
2005, Springer, Berlin, 244-257.
[9] Kolliopoulos, S.G. (2005) Minimum-Cost Single-Source 2-Splittable Flow.
Informa-
tion Processing Letters
, 94, 15-18. https://doi.org/10.1016/j.ipl.2004.12.009
[10] Salazar, F. and Skutella, M. (2006) Single-Source k-Splittable Min-Cost Flows.
Op-
erations Research Letters
, 37, 71-74. https://doi.org/10.1016/j.orl.2008.12.004
[11] Truffot, J. and Duhamel, C. (2008) A Branch and Price Algorithm for the
k-Splittable Maximum Flow Problem.
Discrete Optimization
, 5, 629-646.
https://doi.org/10.1016/j.disopt.2008.01.002
[12] Truffot, J., Duhamel, C. and Mahey, P. (2005) Using Branch-and-Price to Solve
Multicommodity k-Splittable Flow Problem.
Proceedings of International Network
Optimization Conference
(
INOC
), Lisbonne, 20-23 March 2005.
[13] Truffot, J., Duhamel, C. and Mahey, P. (2007) k-Splittable Delay Constrained
Routing Problem: A Branch and Price Approach. 6
th International Workshop on
Design and Reliable Communication Networks
(
DRCN
), La Rochelle, 7-10 October
2007. https://doi.org/10.1109/DRCN.2007.4762284
[14] Gamst, M., Jensen, P.N., Pisinger, D. and Plum, C. (2010) Two- and Three-Index
Formulations of the Minimum Cost Multicommodity k-Splittable Flow Problem.
European Journal of Operational Research
, 202, 82-89.
https://doi.org/10.1016/j.ejor.2009.05.014
[15] Gamst, M. and Petersen, B. (2012) Comparing Branch-and-Price Algorithms for the
Multicommodity k-Splittable Maximum Flow Problem.
European Journal of Oper-
ational Research
, 217, 278-286. https://doi.org/10.1016/j.ejor.2011.10.001
[16] Pawel, M.B. (2017) A Randomized Rounding Approach to a k-Splittable Multi-
commodity Flow Problem with Lower Path Flow Bounds Affording Solution Quali-
ty Guarantees.
Telecommunication Systems
, 64, 525-542.
https://doi.org/10.1007/s11235-016-0190-2
