FINDING TWO DISJOINT PATHS IN A NETWORK WITH NORMALIZED
-MIN-SUM OBJECTIVE FUNCTION
A
ferent lengths. The objective of this problem is to find
disjoint paths such that the total length of the paths is minimized, where the th edge-length is associated with the th
path. They showed that all four versions of the G-MINSUM -path problem are strongly NP-complete even for
[10].
In [12], we considered the problem of finding two disjoint paths such that the length of the shorter path is minimized (named the MIN-MIN 2-path problem). We showed
that all four versions of the MIN-MIN 2-path problem are
strongly NP-complete[12]. In the same paper, we also
showed there does not exist any polynomial-time approximation algorithm with a constant approximation bound for
any of these four versions of the MIN-MIN 2-path problem
. In this paper we consider a generalized
unless
and
be two disjoint
weighted 2-path problem. Let
paths from to in a given graph , and a non-negative
value. Define
ABSTRACT
, a graph
Given a number with
and two nodes and in , we consider the problem
and
from to such
of finding two disjoint paths
with
that
is minimized. The paths may be node-disjoint
or edge-disjoint, and the network may be directed or undirected. This problem has applications in reliable communication. We show that all four versions of this problem are NP-complete. Then we give an approximation of
, and show that this bound is best possible for directed
graphs. For acyclic directed graphs, we also give a pseudopolynomial-time algorithm for finding optimal solutions.
!#"%$'&(*)+,-.0/213!,"%$'&(*)+,4
!,"%$'&(*)+,-.65
!#"4$'&7*)8#9(
B
ACD9
;:= <
A
.
and
such
Our objective is to find two disjoint paths
is minimized. Graph may be directed
that
or undirected, and the paths may be node-disjoint or edgedisjoint. We call this problem the -MIN-SUM 2-path
and ,
problem. According to the relative values of
this problem can be treated as having two versions. One is
to minimize
A reliable telecommunication network, which is modeled
, is designed in such a way that
by a graph
multiple connections exist between every pair of communicating nodes. Usually, paths are selected according to
an objective function. Each edge in is assigned a non, which reflects the resource and/or pernegative length
formance associated with the edge, such as cost, distance,
of a path is defined as the
latency, etc. The length
sum of the lengths of its edges. To avoid single point of failure, the paths may be node-disjoint or edge disjoint, and the
network may be directed or undirected. Thus, a problem of
finding disjoint paths have four versions.
Various problems of finding optimized disjoint paths
have been investigated.
between two nodes and in
Ford and Fulkson gave gave a polynomial-time algorithm
for finding two paths with minimum total length (called
MIN-MAX 2-Path Problem), using the algorithm of finding minimum weighted network flows [7]. Suurballe and
Tarjan provided different treatment, and presented algorithms that are more efficient [18] [19]. Li et al. proved
that all four versions of the problem of finding two disjoint paths such that the length of the longer path is minimized (called the MIN-MAX 2-Path Problem) are strongly
NP-complete [9]. They also considered a generalized MINSUM problem (which we call the G-MIN-SUM -Path
Problem) assuming that each edge is associated with dif-
>?
"
!;,"%
!@#
G <+,LM
K
K
G <'N ,-O H QPSRMTVU%!@#-.W;!@# HXM/01 PZY\['U%!;,-*H@!;# H.X]
and the other is to minimize
G <V^ # * LIP_Y`[aU4!;# H@!;# .XO/0K1 PRMT(U%!@# W;!@# XVb
We name the former as the c -MIN-SUM 2-path
problem, and the latter as the : -MIN-SUM 2-path problem. It is clear that if de , the c -MIN-SUM 2-path
:
problem, and the -MIN-SUM 2-path problem degener-
A
466-807
G <+,-%* H I!;#-H=/J1!;,4
Introduction
B
EF
KEY WORDS
network, graph, reliability, shortest path, disjoint paths,
NP-complete, approximation
1
Bing Yang , S. Q. Zheng and Enyue Lu
Department of Computer Science, University of Texas at Dallas, Richardson, TX 75083-0688, USA
Mathematics and Computer Science Department, Salisbury University, Salisbury, MD 21801, USA
ates to the MIN-MAX 2-path problem and MIN-MIN 2path problem, respectively, which are NP-complete for all
, both degenerate to the MINfour versions. But if
SUM 2-path problem, which is polynomial-time solvable.
Investigation on the -MIN-SUM 2-path problem is of theoretical interest: what are the computational complexities
?
of the -MIN-SUM 2-path problem with
Jf
A
342
ghij
Since
prove that for directed graphs there does not exist any
polynomial-time approximation algorithm guaranteeing an
approximation bound smaller than
unless
.
Finally, give pseudo-polynomial-time algorithms for finding optimal disjoint paths in acyclic directed graphs.
!;#-H=/ 1H!;,4 D 1 1H!;,LH=/ !;# H
an -MIN-SUM problem with I5 can be transformed
. Hence, it is sufficient to only
into a problem with e: . We call the c consider the problem with MIN-SUM 2-path problem (resp. -MIN-SUM 2-path
f the normalized c -MIN-SUM
problem) with 2-path problem (resp. normalized : -MIN-SUM 2-path
;:= <
2
α +-MIN-SUM
α=0
α <= 1
> *
c
MIN-MAX
normalized α - -MIN-SUM
α >= 1
α=1
α >= 1
α=1
NP-Completeness
if
Two paths are said to be edge-disjoint in
they have no edge in common, while they are said to be
node-disjoint if they have no intermediate node in common. Clearly, node-disjoint paths are also edge-disjoint,
but the converse is not true. There are four versions of the
-MIN-SUM 2-path problem:
normalized
problem). The relationship among these 2-path problems
is shown in Figure 1.
α - -MIN-SUM
MIN-SUM
MIN-MIN
node-disjoint paths in directed graphs (ND-D 2-path
problem);
normalized α +-MIN-SUM
α <= 1
α=0
edge-disjoint paths in directed graphs (ED-D 2-path
problem);
Figure 1. Relations among various 2-path problems.
Kc
-c
c
c
:
2.1
Kc
Edge-Disjoint Paths in Directed Graphs
The Partition Problem is defined as following ([8]):
INSTANCE: A set of integers: QUESTION: Is there a subset such that
jU 41%141 XVb
dF ]U 74141%1a$8X
c
The partition problem is a well-known NP-complete
problem[1]. We prove that the ED-D version of the nor-MIN-SUM 2-path problem is NP-complete by
malized
reducing the partition problem to it. Let .
to denote a partition of , where
We use ! "$# % ! # .
and , we
For a partion problem instance construct a direct graph in a way shown in Figure 2.
We call a section corresponding to , shown in Figure
3, a block and denote it by & .
-c
F (H +;
7L2U
X
+ jU B X
U 4141%1a X
c
c
edge-disjoint paths in undirected graphs (ED-UD 2path problem).
All graphs considered in this paper are simple graphs,
i.e. graphs without self-loops and parallel edges between
any pair of nodes. We show that all these versions are NPcomplete by reducing the partition problem ([1]) to the de-MINcision problem corresponding to the normalized
SUM 2-path problem.
c
:
node-disjoint paths in undirected graphs (ND-UD 2path problem);
Kc
-MIN-SUM
In this paper, we consider normalized
. As indicated in Figure 1,
2-path problem with
-MIN-SUM 2-path
the algorithmic issues related to the
have to be addressed by considerproblem with -MIN-SUM 2-path problem. Reading the normalized
ers should bear in mind that the conclusions on the gen-MIN-SUM 2-path problem and
-MIN-SUM 2eral
path problem can be drawn from the combination of the
-MIN-SUM 2-path problem and
results on normalized
-MIN-SUM 2-path problem. We would like
normalized
-MINto mention that our results on the normalized
SUM 2-path problem are reported in [13]. It turns out that
the properties of these two normalized 2-path problems are
quite different.
-MIN-SUM
Apart from its theoretical interest, this
2-path problem has important applications in survival network design. In many real-world network applications,
there requires to provide two disjoint (node-disjoint or
edge-disjoint) paths between two nodes to guarantee reliability. One is called primary route and the other is called
a secondary route. In case there is a failure on the primary route, the traffic can be quickly switched to the secondary route. To build a path, the service provider normally
charges a fee proportional to the length of the path. The
service provider sometimes will give price discount to the
shorter path, as times the normal price. There are similar cases in other areas, like retail business. For example,
a shoe store will have “buy one and get 2nd one with 50%
off” policy for business promotion. The one with 50% off
has to have smaller or equal value as the other one.
In this paper, we first show that all four versions
-MIN-SUM 2-path problem are NPof the normalized
complete. We show that using MIN-SUM 2-path so-MIN-SUM solulutions to approximate normalized
. Then, we
tions achieves an approximation bound
D d
? :
EF
g
:= <
Lemma 1 Any two edge-disjoint paths from to in
define a unique partition of .
343
0
0
0
0
.....
0
0
0
.....
s
c1
0
c2
0
0
t
cn
0
c2
0
g
Figure 2. Graph
0
ci
Figure 5. Two paths: (a)
0
0
0
0
0
0
0
ci
0
ci
0
7 , and (b) O
0
0
0
#
ci
0
0
0
0
ci
0
(H +;
7
+
7
Z5
0
0
(b)
0
ci
0
0
0
c i+1
0
c i+1
0
0
.
0
0
ci
0
0
ci
0
0
c i+1
0
0
(d)
0
0
0
0
ci
0
0
c i+1
0
0
0
0
0
0
0
0
0
0
c i+1
0
0
ci
0
c i+1
0
0
(g)
4
c i+1
(f)
0
0
0
0
0
ci
0
0
ci
(e)
(
0
0
of , we construct
and
Proof: Given corresponding to and , respectively, as follows.
is represented by dashed edges, and
by
In Figures,
dotted edges.
The edges from to & are shown in Figure 5, with
Figure 5 (a) and (b) corresponding to # and #
, respectively.
, we have four cases, as shown
If # , " is in
in Figure 6 (a), (b), (g) and (h), depending on if " , we have four cases, as
or not. If # ,
shown in Figure 6 (c), (d), (e) and (f), depending on if is in or not. Clearly, these include all possible cases.
As an example, Figure 7 shows two edge-disjoint
,
paths constructed from of and .
This way, we get two unique edge-disjoint paths
and
and
such that
% .
M
c i+1
0
7H @ U O4 H(41%141a X
K !;,-H !;# H0
ci
0
0
(c)
of Lemma 2 Any partition defines two unique edge-disjoint paths
and
from
and
to in such that
% .
c i+1
0
ci
0
(b)
+ .
0
(a)
ci
0
Figure 4. Edge-disjoint paths to go through block &
7
0
0
(a)
7
0
0
0
#
0
U %141413 X
O
& Proof: In , any two edge-disjoint paths going through
a block & must be in only one of the two cases shown in
Figure 4. Thus, one and only one edge with cost is used
in the two paths. Count the non-zero edges used in each
.
path, we obtain a partition of 0
(b)
(a)
0
Figure 3. Block
0
c1
0
ci
0
0
c1
0
c1
0
0
0
constructed from .
0
s
c1
0
cn
0
0
0
s
.....
c1
0
0
0
0
(h)
Figure 6. Cases for the proof of Lemma 2.
:8
0
:8
0
0
0
0
0
0
0
0
0
s
eU ( ( OX
7L2U 44 X
+2U WM X
K
!;#. ;! # 4
0
c1
0
c2
0
c3
0
c4
0
0
0
c1
0
c2
0
c3
0
c4
Figure 7. An example.
344
t
c5
0
0
c5
c
Corollary 2 All four versions of the normalized
-MINSUM 2-path problem on planar graphs are NP-complete.
We define the two paths constructed by the method in
the proof of Lemma 2 as partition defined paths.
9
U O% HM41%1413 X
Lemma 3 For the graph constructed from any par ,
tition instance with
-MIN-SUM 2-path problem has the opthe normalized
timal value
if and only if the partition problem has
a feasible solution.
Proof: The theorem follows from the fact that graph
is planar.
Proof: The proof consists of two parts.
Part I: We show that if the partition problem for
has a feasible solution, then the corresponding graph
has two edge-disjoint paths with normalized
. Let the feasible partition be
MIN-SUM cost
and . Then
% . From
and
corLemma 2, we can find 2 edge-disjoint paths
with
. Then
responds to and
.
is the normalized
-MINWe show that
SUM cost. Let and be any two edge-disjoint paths
in . From the proof in Lemma 1 we have,
. Assume
, then . So , . Hence
,
which
achieves
its minimum
.
at value
Part II: We show that if has two edge-disjoint
-MIN-SUM cost
then
paths with normalized
and
be two
there is a feasible solution for . Let
. Then, it
disjoint paths in such
that
to make
to
requires
achieve its minimum value
. If such two paths exist
in , we can use the method of Lemma 1 to map
and
to sets and to obtain a a partition with
%
.
% -MIN-SUM
We say that an instance of the normalized
2-path problem is feasible if for which a feasible solution
exists. We say that there exists an approximation algorithm
with bounded (worst-case) error for the normalized
MIN-SUM 2-path problem if there exists a polynomialtime algorithm such that for the feasible instance space
of := < 1 c
7
+
;:= < 1
>
3
Kc
Kc
Kc
F ; =: <
!;, 6!;# !;, =/ 1H!;, := < 1
c
1
g
!;, /
!;, !;: # S5f!;, f!;, !;, 5D ;:=< < c
!;, +/J1
!;, :=< 1 6/ c 1
I
1
;:= < 1
c
!;,. 5!;# H
!;#-HKj!;,4K ;:=<
!;,-H+/Ji1%!@# H
1
K
(
Theorem 1 The ED-D version of the normalized Kc -MIN-
!;,-.=/JS1H!@# % !;# =/JS1H!@# Q
(1)
# is the solution produced by algorithm ,
# is an optimal solution, and is a constant. The
next theorem shows that all four versions of the normalized
-c -MIN-SUM 2-path problem are approximatable.
Theorem 2 For all four versions of the normalized Kc MIN-SUM 2-path problem, there exists a polynomial-time
approximation algorithm with error bound :=< .
!;# , be the optiProof: Let and , !;# 5
where
mal solution of the MIN-SUM 2-path problem, which
is polynomial-time
solvable [7] [18]). We show that
.
is an optimal solution for MIN-SUM 2As
path problem,
:=:=<< # ;:= <
@! # +/ !;# ;! , =/ !@# Hb
And as !@# -5J!;# ,
!;#-H=/ !@# H J!;# +/ !;# Q901!;# Wb
SUM 2-path problem is NP-complete.
Proof: Obviously, this problem belong to the NP class.
Lemmas 1-3 show that the partition problem is polynomial-MIN-SUM 2time reducible to edge-disjoint, directed
path problem.
Kc
Then,
Corollary 1 The ND-D, ED-UD and ND-UD versions of
-MIN-SUM 2-path problem are NPthe normalized
complete.
c
Proof: Since paths used in the proof of previous lemmas
are also node-disjoint, the node-disjoint normalized
MIN-SUM 2-path problem for directed graphs is also NPcomplete. Since the proofs of the previous lemmas also apply undirected graph , edge-disjoint and node-disjoint
-MIN-SUM 2-path problem for undirected
normalized
graphs are also NP-complete.
How about special graphs? The following result indi-MIN-SUM 2-path problem is
cates that the normalized
not easier if graphs are restricted to planar graphs.
Kc
c
Approximation Analysis
!@# =/ 1H!;, 1M !;, +/ !@# ;8/Q; 1H !@# : b
5 ;:=1M< !;, 8/ !;, ;=/
@ 1 1M !;, =/ !;# ;
Hence,
c
:=:=
: << :=< : ^ : !
"^ ;:= <
We now establish the tightness of our approximation
bound.
345
2
UO(HX ,
7* %( H % * HX , and
2. set UO
3. length assigned to edges 7 and * , and
length assigned to O , H and all edges in
.
Figure 9 shows a general graph and its corresponding .
Theorem 3 With respect to using a MIN-SUM 2-path solu-MIN-SUM solution,
tion to approximate a normalized
for all four versions of the normalized
the error bound
-MIN-SUM 2-path problem is tight.
; := <
-c
1. set
Kc
- Proof: Let , , and be defined the same way as
in the proof of Theorem 2. Consider the graph shown in
Figure 8(a). For this graph, the optimal MIN-SUM solu-MIN-SUM solution are shown in
tion and the optimal
Figure 8(b) and (c), respectively.
c
t1
s1
M/2 + 1
1
1
0
1
t
s
1
0
M/2 + 1
s
1
t2
s2
t
M
(a)
(b)
(a)
Figure 9. (a) Graph
M/2 + 1
1
1
M/2 + 1
s
1
t
M/2 + 1
1
1
M/2 + 1
s
1
M
M
(b)
(c)
:=:=: << < :=< := t
;:=<
;:= <
approaching )
b
!;, H@!;, ; # # U]@74%H4#9'*(X
!;, W;!@# @ U];(%4H4#9'*(X
I!;,LH=/J1!;, % # U]L/ K*9 X
h!;# +/ 1W!;# # UVL/J9VX
which leads to
Proof: We use the same technique Li used in [9] - the
polynomial-time reduction from the DP problem. The
DP problem asks if there exists two paths from nodes
to nodes
in a given directed graph . The problem was proven NP-complete by Fortune, Hopcroft and
Wyllie in [11]. The polynomial-time reduction is carried
and four distinct nodes
out as follows. For
, create a new graph with
dL/ 9
O%(H% *
%
, -% H !@#-*=/J1W!;# H 9
!;# =/J1W!;# L / ;:= <
.
:= <
Theorem 4 For the two directed versions of the nor-MIN-SUM 2-path problem, if there exists a
malized
polynomial-time approximation algorithm with an error
, then P = NP.
bound smaller than
Suppose that there is a polynomial-time approximation al. Then algogorithm with error bound smaller than
in such that
rithm can find
c
9
*
c
c
Hence, the error bound
is tight for the two (edgedisjoint and node-disjoint) directed versions. Obviously,
this example can be used the two undirected versions.
Theorem 3 states that using optimal MIN-SUM so-MIN-SUM solutions,
lutions to approximate optimal
is the best possible error bound. A question arises:
Is there any polynomial-time approximation algorithm for
-MIN-SUM with a smaller error bound?
the normalized
The following theorem partially answers this question.
;:= <
H
(as
0Q!;, / 1 !;# K
% H 4 * 7 * 4
0d=/
7 H % *
j9
H
, * c
, * /J
Clearly,
. (b) Graph
Then the question of “whether or not there exist two
disjoint paths from to and to in ” is equivalent
to the question of “whether or not there exist two disjoint
from to in with paths
”. That is because any two disjoint paths from to
in include edges
. Thus,
and
(resp.
and
) are
either
and
(resp.
and
in the same path, or
) are in the same path. In the former case, and there exist two disjoint paths from to and to .
and there exist disjoint paths from
In the latter case, be the optimal paths
to and
to . Let
-MIN-SUM 2-path problem on .
for the normalized
Then, we have
Figure 8. An example.
g
1 L/J9 D9(
6L/J Hb
Then this algorithm can answer the question of “whether or
from to with
not there exist two disjoint paths
”. We are led to conclude that
algorithm can solve DP problem in polynomial time. It
.
cannot be true unless
J!@#-.M/Z1 !;# 4
> d * 346
a /Z
9
QF
# * 4
%! ! ! ! 3( 1%141 C from
*(% to in , let and be the set
of horizontal and vertical edges in respectively. De 41%1413 (resp. f
fine f
% % 41%141a % ) as the sequence of distinct first
(resp. second) labels of nodes in C (resp. ).
By a straightforward extension of the results of [15], we
know that there exist two node-disjoint paths K
14141 and! ! 14141 from to (from 1 to after relabeling) in if and only
if there exists a path from 74 to ! ! ! ! such that
a( 0 O 1%141 . That14is,1%1 - and and
correspond to the
vertical edges and horizontal edges in , respectively. In
the example of Figure 10(a), there are two node-disjoint
paths -j "! "# and d 9 %$ &# in
corresponding path in is '
. The
*# 74# 4 $
#
!
#
*9 , as shown in
Figure 10(b).
Then our goal is to find such that
/ X
U7 8 PZ Y`( [
Pseudo-Polynomial-Time Algorithm for
Acyclic Directed Graphs
c
-MIN-SUM 2-path problem on
For the special case
acyclic directed graphs, there exists a pseudo-polynomialtime algorithm to find optimal solutions. The algorithm
is obtained by extending the Li’s algorithm [9] to find two
disjoint paths on acyclic directed graphs for MIN-MAX objectiveh.
4.1
The Node-Disjoint Case
For this case, we adopt a technique used in a pseudopolynomial-time algorithm of [9]. Given an acyclic diand source and destination ,
rected graph
we can relabel nodes with number 1 to ! ! to ensure that
! !
in satisfies
,
and
any edge #
[15] (it is assumed that
; otherwise we can add
by and ). After
a node and replace relabeling, we transform graph into an acyclic directed
, whose nodes are arranged as a ! ! ! !
graph
array, as follows:
!
#
, and
unless
or
! #
and !
#
and Then we assign the lengths to edges in as follows:
, and
.
Figure 10 shows an example of and its corresponding .
e
C 1
d 0
U 8 + JHX jj U 8
8 X
KU X
!; + 8 D 1H!; !; Q!@ 1
1
2
1
2
5
1
7
G <1,2>
<1,3>
<1,4>
1
α
<4,1>
<4,2>
2α
<4,3>
α
<5,1>
α
<5,2>
2α
<2,5>
2
α
<6,2>
<2,6>
1
2
1
2α
<3,6>
<3,5>
α
1
<4,5>
1
α
<4,6>
1
1
1
1
<5,3)
α
<6,1>
1
<2,4>
α
α
1
α 2
<3,4>
<3,2>
α
<6,3>
<5,4>
α
<5,6>
α
<6,4>
1
<6,5>
Figure 10. (a) An acyclic graph
do
% P_Y`[ 8(
. (b) Graph
+ 8'H:.+/ PR T3+7K/j 1
D(*
has no label then stop (no path
exists in );
if
that
else select a label
to
is minimized; trace the path back to
get the optimal path .
<6,6>
(b)
;
let is a vertical edge then
if
on ;
update label
is a horizontal edge then
if
on ;
update label
(label update procedure: given new label
, if there is no existing label in the
el
ement of the label array, add
into the array as the element; if there exists a
in the element, replace the
).
if
element with
1
α
<3,1>
<1,6>
1
<2,3>
1
<1,5>
1
1
<2,1>
;(%( α
1
1 2 3( !@#"4
4.+/
%
I!;5.8/
5.8/ %
/ M'H6.+/ %
5.8/
+ / MH6.+/ +aW
.8/ 8aW7.+/ + H-. / 8aW8.+/ C9
6
α
<1,1>
0.8/
Q7
_. D(41%1413
/ 741%141 (a)
α
8'H.8/ with
;
label
do
for
do
for
for each node adjacent to
1
1
We use the similar “scanning and labeling”
algorithm[9] to keep multiple labels at each node.
The labels for a node are represented by
,
where and are the total vertical and horizontal distances
from
to the current node along the path through
previous node
. Different from [9], we use an
array index by to store these labels, and the size of the
. The algorithm is as follows:
array is
3
1
4
) *,+ .
From the proof in [9], the above algorithm covers all
to
. Thus,
valid non-redundant paths from
the is the optimal path.
and
of a node
in
the first
We call
and second label of the node, respectively. Given a path
347
h(*
%
G
[7] L.R. Ford and D.R. Fulkerson, Flows in Networks,
Princeton, 1962.
Complexity Analysis:
Let be sum of all edge lengths in . The scan! !
time, as there are ! ! nodes and
ning stage takes
each node has at most ! ! neighbors. The update procedure takes constant time. The final optimization stage takes
! !
time. So the total time complexity is
.
! !
space.
And the algorithm takes
9 G 4.2
G [8] A. Itai, Y. Perl and Y. Shiloach, “The Complexity of
finding Maximum Disjoint Paths with Length Constraints”, Networks 12 (1982) 277-286.
/ G
[9] C. L. Li, S. T. McCormick, and D. Simchi-Levi, “The
Complexity of Finding Two Disjoint Paths with MinMax Objective Function”, Discrete Appl. Math. 26(1)
(1990), 105-115.
The Edge-Disjoint Case
For this case, we directly apply a technique of [9] that transforms the acyclic edge-disjoint case to the acyclic nodedisjoint case, and then apply the algorithm presented in
the previous section. First, we transform the given graph
to a corresponding directed line-graph [2],
! ! nodes and
! ! edges. This can
which contains
!
!
time. Then, finding two node-disjoint
be done in
! !
time. For more
paths in can be done in
details, refer to [9].
E *g
5
/ G
[10] C. L. Li, S. T. McCormick, and D. Simchi-Levi,
“Finding Disjoint Paths with Different Path-Costs:
Complexity and Algorithms”, Networks, 22 (1992),
653-667.
[11] S. Fortune, J. Hopcroft, and J. Wyllie, “The directed
subgraph homeomorphism problem”, Theoret. Comput. Sci., 10 (1980), 111-121.
[12] B. Yang, S.Q. Zheng and S. Katukam, “Finding Two
Disjoint Paths in a Network with Min-Min Objective
Function”, Proceedings of the 15th IASTED International Conference on Parallel and Distributed Computing and Systems, (2003), 75-80.
Concluding Remarks
We have shown that for all four versions of the normal-MIN-SUM 2-path problem one cannot obtain
ized
polynomial-time algorithms for finding optimal solutions,
unless P = NP. We showed that MIN-SUM solutions can be
used as good approximations of optimal normalized
MIN-SUM solutions. We also showed that for acyclic directed case, pseudo-polynomial-time algorithms exist for
-MIN-SUM solutions. We
finding optimal normalized
is the best possible approximation bound
showed that
-MIN-SUM solutions in directed graphs.
for normalized
is also the optimal apA challenging question is if
proximation bound for undirected graphs. Polynomial-time
algorithms may exist for finding optimal normalized
MIN-SUM 2-path solutions or achieving smaller approximation bound for graphs of some special properties.
-c
:
[13] B. Yang, S.Q. Zheng and E. Lu, “Finding Two Disjoint Paths in a Network with Normalized
-MinSum Objective Function”, to appear in Proceedings
of the 16th International Symposium on Algorithms
and Computation, (ISAAC 2005).
Kc
; := <
c
Kc
; := <
[14] Y. Liu, D. Tipper, “Successive Survivable Routing for
Node Failures”, IEEE Globecom (2001) 2093-2097.
[15] Y. Perl and Y. Shiloach, “Finding Two Disjoint Paths
between Two Pairs of Vertices in a Graph”, J. ACM,
25(1) (1978), 1-9.
Kc
[16] Y. Shiloach, A polynomial solution to the undirected
two paths problem. J. ACM 27(3) (1980) 445-456.
References
[17] M.R. Garey, D.S. Johnson and L. Stockmeyer, “Some
Simplified NP-Complete Graph Problems”, Theoret.
Comput. Sci. 1 (1976) 237-267.
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