The minimum reload s-t path/trail/walk problems

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The minimum reload s-t path/trail/walk problems
Current Trends in Theory and Practice of Comp.
Science, SOFSEM09
L. Gourvès, A. Lyra, C. Martinhon, J. Monnot
Špindleruv Mlýn / Czech Republic
2
Topics

• 1. Motivation and basic definitions
• 2. Minimum reload s-t walk problem
• 3. Paths\trails with symmetric reload costs
• Polynomial and NP-hard results.
• 4. Paths\trails with asymmetric reload costs
• Polynomial and NP-hard results.
• 5. Conclusions and open problems

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Some applications involving reload costs
1. Cargo transportation network when
the colors are used to denote route subnetworks
2. Data transmission costs in large
communication networks when a color
specify a type of transmission 3. Change of
technology when colors are associated
to technologies etc
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Basic Definitions

• Paths, trails and walks with minimum reload costs

b
c
5
1
1
R
1
1
1
1
1
5
a
s
t
Reload cost matrix
d
5
Basic Definitions

• Minimum reload s-t walk

b
c
5
1
1
R
1
1
1
1
1
5
a
s
t
Reload cost matrix
d
c(W)
3
6
Basic Definitions

• Minimum reload s-t trail

b
c
5
1
1
R
1
1
1
1
1
5
a
s
t
Reload cost matrix
d
c(W) c(T)
3
4
7
Basic Definitions

• Minimum reload s-t path

b
c
5
1
1
R
1
1
1
1
1
5
a
s
t
Reload cost matrix
d
c(W) c(T) c(P)
3
4
5
8
Basic Definitions

• Symmetric or asymmetric reload costs

rij ? rji
rij rji
for colors i and j
or

• Triangle inequality (between colors)

1
2
rij rjk rik
y
z
x
for colors 1,2,3
3
w
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Basic Definitions
NOTE Paths (resp., trails and walks) with reload
costs generalize both properly edge-colored
(pec) and monochromatic paths (resp., trails and
walks).
rij 0, for i j and rii 1
?
s
t
pec s-t path cost of the minimum reload s-t
path is 0
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Basic Definitions
NOTE Paths (resp., trails and walks) with reload
costs generalize both properly edge-colored
(pec) and monochromatic paths (resp., trails and
walks).
rij 1, for i j and rii 0
?
s
t
monochomatic s-t path cost of the min.
reload s-t path is 0
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Minimum reload s-t walk
s
1
v2
2
v1
3
t
c
Minimum reload s-t walk in G Shortest s0-t0
path in H
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Minimum reload s-t walk
s
1
v2
2
v1
3
t
All cases can be solved in polynomial time !
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Minimum symmetric reload s-t trail
Symmetric R
1
y
v
2
1
x
z
c
a) Neighbourhood of v in G
b) Weighted non-colored subgraph G(v)
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Minimum symmetric reload s-t trail
Symmetric R
1
y
v
2
1
x
z
c
a) Neighbourhood of v in G
b) Weighted non-colored subgraph G(v)
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Minimum symmetric reload s-t trail
Symmetric R
1
y
v
2
1
x
z
c
a) Neighbourhood of v in G
b) Weighted non-colored subgraph G(v)
Minimum symmetric reload s-t trail
Minimum perfect matching
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Minimum symmetric reload s-t trail
Symmetric R
1
y
v
2
1
x
z
c
a) Neighbourhood of v in G
b) Weighted non-colored subgraph G(v)
The minimum symmetric reload s-t trail can be
solved in polynomial time !
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NP-completeness
Theorem 1 The minimum symmetric reload st path
problem is NP-hard if c 3, the triangle
inequality holds and the maximum degree of Gc is
equal to 4.
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Theorem 1 (Proof)

• Reduction from the (3, B2)-SAT (2-Balanced
  3-SAT)
• Each clause has exactly 3 literals
• Each variable apears exactly 4 times (2
  negated and 2 unnegated)

xi is false
xi is true
Gadget for clause Cj
Gadget for literal xi
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Theorem 1 (Proof)
C4
C3
C6
C5
literal x7
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Theorem 1 (Proof)
C4
C3
C6
C5
Every other entries of R are set to 1
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Theorem 1 (Proof)
s
t
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Theorem 1 (Proof)
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Theorem 1 (Proof)
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Non-approximation
Theorem 2 In the general case, the minimum
symmetric reload st path problem is not
approximable at all if c 3, the triangle
inequality holds and the maximum degree of Gc is
equal to 4.
We modify the reload costs, so that OPT(Gc)0
I is satisfiable. OPT(Gc) gtM I
is not satisfiable.
In this way, to distinguish between OPT(Gc)0 or
OPT(Gc) M is NP-complete, otherwise PNP!
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Non-approximation
Theorem 2 In the general case, the minimum
symmetric reload st path problem is not
approximable at all if c 3, the triangle
inequality holds and the maximum degree of Gc is
equal to 4.
r1,2 r2,1 M
r2,2 0
Proof
r1,3 r3,1 0
r1,1 0
r2,3 r3,2 0
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Non-approximation (Proof)
s
r1,2 r2,1 M
t
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Non-approximation
Theorem 3 If , for every i,j the
minimum symmetric reload st path problem is not
-approximable for every if
c 3, the triangle inequality holds and the
maximum degree of Gc is equal to 4.
r1,2 r2,1 M
r2,2 1
Proof
r1,3 r3,1 1
r1,1 1
r2,3 r3,2 1
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Non-approximation
Theorem 3 If , for every i,j the
minimum symmetric reload st path problem is not
-approximable for every if
c 3, the triangle inequality holds and the
maximum degree of Gc is equal to 4.
Proof
It is NP complete to distinguish between
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NP-Completeness

• Corollary 4
• The minimum symmetric reload st path problem is
  NP-hard if c 4, the graph is planar, the
  triangle inequality holds and the maximum degree
  is equal to 4.

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Corollary 4 (Proof)
c
c
b
a
f
a
b
d
r1,2 r2,1 M
d
c
c
c
b
b
a
a
f
a
b
d
d
d
r3,4 r4,3 M
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Some polynomial cases
Theorem 5 Consider Gc with c2 colors. Further,
suppose that the reload cost matrix R satisfies
the triangle inequality. Then, the minimum
symmetric reload st path problem can be solved
in polynomial time.
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Some polynomial cases
Theorem 5 Consider Gc with c2 colors. Further,
suppose that the reload cost matrix R satisfies
the triangle inequality. Then, the minimum
symmetric reload st path problem can be solved
in polynomial time.
If the triangle ineq. does not hold??
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Some polynomial cases

• The minimum toll cost st path problem may be
  solved in polynomial time.
• ? ri,jrj , for colors i and j and ri,i 0

toll points
s
s
t
0
auxiliar vertex and edge
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NP-completeness
Theorem 6 The minimum asymmetric reload st
trail problem is NP-hard if c 3, the triangle
inequality holds and the maximum degree of Gc is
equal to 4.
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NP-completeness (Proof)

• Reduction from the (3, B2)-SAT (2-Balanced
  3-SAT)
• Each clause has exactly 3 literals
• Each variable apears exactly 4 times (2
  negated and 2 unnegated)

True
False
Clause graph
Variable graph
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NP-completeness (Proof)
x3

Reload costs M
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Non-approximation
Theorem 7 (a) In the general case, the minimum
asymmetric reload st trail problem is not
approximable at all if c 3, the triangle
inequality holds and the maximum degree of Gc is
equal to 3.
(b) If , for every i,j the minimum
asymmetric reload st trail problem is not
-approximable for every if c 3,
the triangle inequality holds and the maximum
degree of Gc is equal to 3.
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Non-approximation
Theorem 7 (a) In the general case, the minimum
asymmetric reload st trail problem is not
approximable at all if c 3, the triangle
inequality holds and the maximum degree of Gc is
equal to 3.
(b) If , for every i,j the minimum
asymmetric reload st trail problem is not
-approximable for every if c 3,
the triangle inequality holds and the maximum
degree of Gc is equal to 3.
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Non-approximation
Theorem 7 (a) In the general case, the minimum
asymmetric reload st trail problem is not
approximable at all if c 3, the triangle
inequality holds and the maximum degree of Gc is
equal to 3.
(b) If , for every i,j the minimum
asymmetric reload st trail problem is not
-approximable for every if c 3,
the triangle inequality holds and the maximum
degree of Gc is equal to 3.
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A polynomial case
Theorem 8 Consider Gc with c2 colors. Further,
suppose that the reload cost matrix R satisfies
the triangle inequality. Then, the minimum
asymmetric reload st trail problem can be
solved in polynomial time.
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A polynomial case
Theorem 8 Consider Gc with c2 colors. Further,
suppose that the reload cost matrix R satisfies
the triangle inequality. Then, the minimum
asymmetric reload st trail problem can be
solved in polynomial time.
If the triangle ineq. does not hold??
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Conclusions and Open Problems
Polynomial time problems NP-hard problems
s-t walk
s-t trail
s-t trail
s-t path
s-t path
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Conclusions and Open Problems
Problem 1
Input Let be 2-edge-colored graph and 2
vertices
Question Does the minimum symmetric reload s-t
path problem can be solved in
polynomial time?
Note If the triangle ineq. holds Yes!
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Conclusions and Open Problems
Problem 2
Input Let be 2-edge-colored graph and 2
vertices
Question Does the minimum asymmetric reload s-t
trail problem can be solved in
polynomial time?
Note If the triangle ineq. holds Yes!
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Thanks for your attention!!