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The minimum reload The minimum reload s-ts-t path/trail/walk problems path/trail/walk problems
Current Trends in Theory and Practice of Comp. Science, SOFSEM09
L. Gourvès, A. Lyra, C. Martinhon, J. Monnot
Špindlerův Mlýn / Czech Republic
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Topics
1. Motivation and basic definitions2. 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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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
Some applications involving reload costs
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Basic Definitions Paths, trails and walks with minimum reload costs
s t 5
5
111
11
1
1
Reload cost matrix
R =a
bc
d
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Basic Definitions Minimum reload s-t walk
s t 5
5
111
11
1
1
c(W)
Reload cost matrix
R =
3
a
bc
d
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Basic Definitions Minimum reload s-t trail
s t 5
5
111
11
1
1
c(W) ≤ c(T)
Reload cost matrix
R =
3 4
a
bc
d
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Basic Definitions Minimum reload s-t path
s t 5
5
111
11
1
1
c(W) ≤ c(T) ≤ c(P)
Reload cost matrix
R =
3 4 5
a
bc
d
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Basic Definitions• Symmetric or asymmetric reload costs
rij ≠ rji
• Triangle inequality (between colors)
zy
w
x1 2
3
rij ≤ rjk + rik
for colors “i” and “j”rij = rji or
for colors 1,2,3
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Basic DefinitionsNOTE: Paths (resp., trails and walks) with reload costs generalize both properly edge-colored (pec) and monochromatic paths (resp., trails and walks).
s t
rij = 0, for i j and rii = 1≠
pec s-t path cost of the minimum reload s-t path is 0
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s t
rij = 1, for i j and rii = 0≠
monochomatic s-t path cost of the min. reload s-t path is 0
Basic DefinitionsNOTE: Paths (resp., trails and walks) with reload costs generalize both properly edge-colored (pec) and monochromatic paths (resp., trails and walks).
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Minimum reload s-t walk
Minimum reload s-t walk in G Shortest s0-t0 path in H
t
s
1
2
3
v1v2
4,1,1 132312 rrr
c
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Minimum reload s-t walk
t
s
1
2
3
v1v2
4,1,1 132312 rrr
All cases can be solved in polynomial time !
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z
yv 1
2
x
1
a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v)
212r
zvxv yv
212r
212r
212r
211r
211r
0 0
00 0
c
0
0 0 0Symmetric R
Minimum symmetric reload s-t trail
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z
yv 1
2
x
1
a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v)
212r
zvxv yv
212r
212r
212r
211r
211r
0 0
00 0
c
0
0 0 0Symmetric R
Minimum symmetric reload s-t trail
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z
yv 1
2
x
1
a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v)
212r
zvxv yv
212r
212r
212r
211r
211r
0 0
00 0
c
Minimum symmetric reload s-t trail Minimum perfect matching
0
0 0 0Symmetric R
Minimum symmetric reload s-t trail
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z
yv 1
2
x
1
a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v)
212r
zvxv yv
212r
212r
212r
211r
211r
0 0
00 0
c
0
0 0 0Symmetric R
The minimum symmetric reload s-t trail can be solved in polynomial time !
Minimum symmetric reload s-t trail
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NP-completeness
Theorem 1
The minimum symmetric reload s–t 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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xi is false
Gadget for literal xi
Gadget for clause Cj
xi is true
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)
Theorem 1 (Proof)
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)( ,)(),(),(
76169875
75348713
xxxCxxxCxxxCxxxC
C3
C6
C4
C5
Theorem 1 (Proof)
literal x7
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||3||11
1,22,1
CLLM
Mrr
Every other entries of R are set to 1
C6
Theorem 1 (Proof)
C3
C4
C5
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||3||11 CL
t
s
Theorem 1 (Proof)
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)( 7534 xxxC
Theorem 1 (Proof)3x 5x 7x
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)( 7534 xxxC
Theorem 1 (Proof)3x 5x 7x
FxTxFx
7
5
3
falseisC4
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We modify the reload costs, so that:
OPT(Gc)=0 I is satisfiable.
OPT(Gc) >M I is not satisfiable.
In this way, to distinguish between OPT(Gc)=0 or
OPT(Gc) ≥M is NP-complete, otherwise P=NP!
Non-approximationTheorem 2In the general case, the minimum symmetric reload s–t path problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 4.
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Non-approximationTheorem 2In the general case, the minimum symmetric reload s–t path problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 4.
Proof: r1,2 = r2,1 = M
r1,3 = r3,1 = 0
r2,2 = 0
r1,1 = 0
r2,3 = r3,2 = 0
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t
s
Non-approximation (Proof)
r1,2 = r2,1 = M
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Theorem 3If , for every i,j the minimum symmetric reload s–t 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: r1,2 = r2,1 = M
r1,3 = r3,1 = 1
r2,2 = 1
r1,1 = 1
r2,3 = r3,2 = 1
Non-approximation
1ijr)2( )(npO )(np
LOM np )2( )(
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Theorem 3If , for every i,j the minimum symmetric reload s–t 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:
1ijr)2( )(npO )(np
LOM np )2( )(
Non-approximation
It is NP –complete to distinguish between
LOGOPTandLGOPT npcc )2()()( )(
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Corollary 4: The minimum symmetric reload s–t
path problem is NP-hard if c ≥ 4, the graph is planar, the triangle inequality holds and the maximum degree is equal to 4.
NP-Completeness
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a b
d
c
ab
d
c
f
a b
d
c
a
b
d
c
fd’
c’
a’ b’
r3,4 = r4,3 = M
Corollary 4 (Proof):
r1,2 = r2,1 = M
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Some polynomial cases
Theorem 5
Consider Gc with c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality.
Then, the minimum symmetric reload s–t path problem can be solved in polynomial time.
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Some polynomial cases
Theorem 5
Consider Gc with c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality.
Then, the minimum symmetric reload s–t 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 s–t path
problem may be solved in polynomial time.
∀ ri,j=rj , for colors i and j and ri,i =0
s ts0
auxiliar vertex and edge
toll points
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NP-completeness
Theorem 6
The minimum asymmetric reload s–t 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)
Variable graph Clause graph
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)
False True
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),(),(),(),(
32173215
43126531
xxxCxxxCxxxCxxxC
5C
7C
1C
2C
x3
Reload costs = M
NP-completeness (Proof)
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(b) If , for every i,j the minimum asymmetric reload s–t trail problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 3.
Non-approximationTheorem 7
(a) In the general case, the minimum asymmetric reload s–t trail problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 3.
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(b) If , for every i,j the minimum asymmetric reload s–t trail problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 3.
Non-approximation
1ijr)2( )(npO )(np
Theorem 7
(a) In the general case, the minimum asymmetric reload s–t trail problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 3.
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(b) If , for every i,j the minimum asymmetric reload s–t trail problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 3.
Non-approximation
1ijr)2( )(npO )(np
Theorem 7
(a) In the general case, the minimum asymmetric reload s–t trail problem is not approximable at all 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 c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality.
Then, the minimum asymmetric reload s–t trail problem can be solved in polynomial time.
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A polynomial case
Theorem 8
Consider Gc with c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality.
Then, the minimum asymmetric reload s–t trail problem can be solved in polynomial time.
If the triangle ineq. does not hold??
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Conclusions and Open ProblemsPolynomial time
problemsNP-hard problems
s-t walk
s-t trail
s-t path
)3()3)(().( cGRAsym c)( RSymmetric
)2(.)().( cineqRAsym
casesallIn
.)()2( ineqc
)3)(().( cGRSym
.)()3()4)(().(
ineqcGRSym c
)4)((.)(
)4()().(
c
c
Gineq
cplanarGRSym
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Conclusions and Open Problems
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?
cG
Note: If the triangle ineq. holds Yes!
Problem 1
)(, cGVts
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Conclusions and Open Problems
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?
cG
Note: If the triangle ineq. holds Yes!
Problem 2
)(, cGVts
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Thanks for your attention!!