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 Definitions

NOTE: 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 Definitions

NOTE: 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

v1

v2

4,1,1,1 13222312 rrrr

c

Minimum reload s-t walk

t

s

1

2

3

v1

v2

4,1,1,1 13222312 rrrr

All instances 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 literal apears exactly 4 times (2 negated and 2 unnegated)

Theorem 1 (Proof)

)( ,)(

),(),(

76169875

75348713

xxxCxxxC

xxxCxxxC

C3

C6

C4

C5

Theorem 1 (Proof)

literal x7

||3||11

1,22,1

CL

KM

Mrr

Every other entries of R are set to 1

C6

Theorem 1 (Proof)

C3

C4

C5

||3||11 CK

t

s

Theorem 1 (Proof)

)( 7534 xxxC

Theorem 1 (Proof)

3x 5x 7x

)( 7534 xxxC

Theorem 1 (Proof)

3x 5x 7x

Fx

Tx

Fx

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-approximation

Theorem 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.

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.

What happens 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 literal apears exactly 4 times (2 negated and 2 unnegated)

False True

),(),(

),(),(

32173215

43126531

xxxCxxxC

xxxCxxxC

5C

7C

1C

2C

x3

Reload costs = M

NP-completeness (Proof)

||6||15 CK

(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

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.

What happens if the triangle ineq. does not hold??

Conclusions and Open Problems

Polynomial time problems

NP-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)(().(

ineq

cGRSym 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!!

Basic Definitions Paths/trails and walks with minimum reload costs

s t 5

5

111

11

1

1

c(W) ≤ c(T) ≤ c(P)

Reload cost matrix

R =

3 4 5

a

bc

d

Niteroi – RJ (Brazil)

Itacoatiara’s beach Piratininga’s beach

You are welcome!!

Non-approximation

Theorem 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

Niteroi – RJ (Brazil)

Itacoatiara’s beach Piratininga’s beach

You are welcome!!

Minimum reload s-t walk

t

s

1

2

3

v1

v2

112 r

123 r

413 r

1s

2s

3s

0s

1t

2t

3t

11v

21v

31v

12v

22v

32v

0t

(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.