The minimum reload s-t path/trail/walk problems
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Transcript of The minimum reload s-t path/trail/walk problems
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
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
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
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
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
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
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
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
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
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).
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
Minimum reload s-t walk
t
s
1
2
3
v1v2
4,1,1 132312 rrr
All cases can be solved in polynomial time !
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
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
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
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
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.
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)
)( ,)(),(),(
76169875
75348713
xxxCxxxCxxxCxxxC
C3
C6
C4
C5
Theorem 1 (Proof)
literal x7
||3||11
1,22,1
CLLM
Mrr
Every other entries of R are set to 1
C6
Theorem 1 (Proof)
C3
C4
C5
||3||11 CL
t
s
Theorem 1 (Proof)
)( 7534 xxxC
Theorem 1 (Proof)3x 5x 7x
)( 7534 xxxC
Theorem 1 (Proof)3x 5x 7x
FxTxFx
7
5
3
falseisC4
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.
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
t
s
Non-approximation (Proof)
r1,2 = r2,1 = M
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( )(
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()()( )(
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
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
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.
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??
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
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.
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
),(),(),(),(
32173215
43126531
xxxCxxxCxxxCxxxC
5C
7C
1C
2C
x3
Reload costs = M
NP-completeness (Proof)
(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.
(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.
(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.
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.
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??
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
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
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
Thanks for your attention!!