Paths and Trails in Edge Colored Graphs Latin-American on Theoretical Informatics Symposium LATIN 2008 Abouelaoualim, K. Das, L. Faria, Y. Manoussakis,

Paths and Trails in Edge Paths and Trails in Edge Colored Graphs Colored Graphs

Latin-American on Theoretical Informatics Symposium LATIN 2008

Abouelaoualim, K. Das, L. Faria, Y. Manoussakis, C. Martinhon, R. Saad

Buzios-RJ - Brazil

Topics

1. Motivation and basic definitions2. Properly edge-colored s-t path/trail

and extensions 3. NP-completeness 4. Approximation Algorithms for

associated maximization problems5. Some instances solved in

polynomial time6. Conclusions and open problems

2k

1k

1. Computational Biology

when the colors are used to denote a sequence of chromosomes;

2. Cryptography

when a color specify a type of transmission;

3. Social Sciences

where a color represents a relation between 2 individuals;

etc

Some Applications using edge colored graphs

Basic Definitions Prop. edge-colored path between « s » and

« t »

t

source destination

2 3

s

4

(without node repetitions!!)

1

Basic Definitions

Prop. edge-colored trail between « s » and « t »

t

source destination

2 3

s

4

(without edge repetitions!!)

1

Basic Definitions Properly edge-colored cycle passing by

« x »

5

start

2 3

x

4


1

Basic Definitions Prop. edge-colored closed trail passing by

« x »

5

start

2 3

x

4


1

Basic Definitions

Almost prop. edge-colored cycle passing by « x »


5

start

2 3

x

4

1

Basic Definitions Almost properly edge-colored closed trail

passing by « x »


5

start

2 3

x

4

1

How to find a properly edge-colored s-t path?

source destination

2 3

s

4

1

2-edge-colored graph G

t

source destination

2 3

s

4

1


Graph G’

blue red

3’’

s

2’’

3’

4’’4’

1’

t

1’’

2’

t

We find a perfect matching (if possible) !!


source destination

2 3

s

4

1


Graph G’

blue red

3’’

s

2’’

3’

4’’4’

1’

t

1’’

2’

t


t

a pec s-t path in G G’ contains a perfect matchingTherem: Jensen&Gutin[1998]

tstart

u

s

q

v

pdest.

color 1

color 2

color 3pa

v’ v’’ u’ u’’

va vb

v1 v2

q’ q’’

qa qb

q2 q3

p’ p’’

pb pc

p1 p2 p3

ua ub uc

u1 u2 u3

s t

(a) 3-edge colored graph (b) non-colored graph

Properly s-t path in edge-colored graphs (Szeider’s Algorithm – [2003])

color 1

color 2

color 3pa

v’ v’’ u’ u’’

va vb

v1 v2

q’ q’’

qa qc

q2 q3

p’ p’’

pb pc

p1 p2 p3

ua ub uc

u1 u2 u3

s t



tstart

u

s

q

v

pdest.

color 1

color 2

color 3pa

v’ v’’ u’ u’’

va vb

v1 v2

q’ q’’

qa qc

q2 q3

p’ p’’

pb pc

p1 p2 p3

ua ub uc

u1 u2 u3

s t



tstart

u

s

q

v

pdest.

tstart

u

s

q

v

pdest.

color 1

color 2

color 3pa

v’ v’’ u’ u’’

va vb

v1 v2

q’ q’’

qa qb

q2 q3

p’ p’’

pb pc

p1 p2 p3

ua ub uc

u1 u2 u3

s t



Our results:

Lemma: Consider a c-edge-colored graph G, and an arbitrary pec trail T between « s » and « t ». Further, suppose that at least one node in T is visited 3 times or more. Then, there exists another pec trail T’ where no nodes are visited more than 2 times

s x ty

Cycles or closed trails passing by x Almost cycles or closed trails passing by y

a b

How to find a prop. edge-colored s-t trail?

Equivalence between paths and trails

s t1

32

Graph G

pec trail P

yx

X’

X’’

y’

y’’

yx

X’

X’’

y’

y’’

Equivalence between paths and trails

s t1

32

s’1’’

1’

t’

2’’

2’

1’

1’

Graph GGraph H

pec trail P pec path P’

Theorem: We have a pec s-t trail in G we have a pec s’-t’ path in H

Shortest properly edge-colored s-t Path

destination


Graph G’

blue reds

2’’

3’

4’’4’

1’ 1’’

2’

source

2 3

s

4

1 t

1

1

1

1

1

1

1

1

1

1

0

0

0

0

t

3’’

Find a minimum perferct matching (if it exists)!

Shortest properly edge-colored s-t trail

Algorithm: Shortest prop. edge-colored s-t Trail

1. Construct H=(V’,E’) associated to G2. Find a short. pec path P (if possible) between « s’ » and « t’ » in H3. Return trail T in G, and size(T)=size(P)/3

Input: A 2-edge colored graph G=(V,E), and 2 nodes s,t in VOutput: A shortest prop. edge-colored trail T between « s » and « t ».

Construction of H yx

X’

X’’

y’

y’’

yx

X’

X’’

y’

y’’Hxy

Existence of prop. edge-colored closed trails

Theorem: Let G a c-edge colored graph, such that every vertex of G is incident with at least two edges of different colors. Then either G has a bridge, or G has a prop. edge-colored closed trail.

1

32

Algorithm: Delete all bridges and all nodes adjacent to edges of the same color

54

76

1

3

5

7

pec closed trail 1,2,3,1,5,7,6,4,1

Longest prop. edge-colored path in graphs with no pec cycles

destination


source

2 3

s

4

1 t

destination


source

2 3

s

4

1 t

Graph G’

blue reds

2’’

3’

4’’4’

1’ 1’’

2’1

1

1

1

1

1

1

1

1

0

0

0

0

t

3’’

Find a maximum perfect matching (if it exists)!

Longest prop. edge-colored path in graphs with no pec cycles

Longest pec trail in graphs with no pec closed trails

s x ty

Cycles or closed trails passing by x(not possible !!)

Almost cycles or closed trails passing by y

We can visit node « y » several times !!

FACT: Node « y » can be visited at most times!

2

1nd

s x ty

Cycles or closed trails passing by x(not possible !!)

Almost cycles or closed trails passing by y

We can visit node « y » several times !!

FACT: Node « y » can be visited at most times!

2

1nd



yx

X1

X2

Xd

Y1

Y2

Yd

yx

X1

X2

Xd

Y1

Y2

Yd

2

1nd

Construction of H

Theorem: We have a pec s-t trail in G we have a pec s’-t’ path in H

s

k-Properly Vertex Disjoint Path problem

Input: Given a 2-edge colored graph G, a const. k and nodes s,t V.

Question: Does G contains k pec vertex disjoint paths between « s » and « t »?

t k-PVDP

Without node repetitions !!

s

k-Properly Edge Disjoint Trails problem

Question: Does G contains k pec edge disjoint trails between « s » and « t »?

t k-PEDT

Without edge repetitions !!


s



t k-PEDT


Input: Given a 2-edge-colored graph G, a const. k and nodes s,t V.

s



t k-PEDT


Input: Given a 2-edge-colored graph G, a const. k and nodes s,t V.

s



t k-PEDT



s




t k-PEDT


NP-Completeness

u v

Fortune, Hopcroft, Wylie [1980]

Directed cycle problem - DC

Input: A digraph D=(V,A) and a pair of nodes u,v V

Output: Does exist a vertex disjoint circuit passing by « u » and « v » ?

Output: Does exist an arc disjoint Circuit passing by « u » and « v » ?

Theorem: DC problem is NP-Complete

u vDirected Closed-Trail problem - DCT

NP-Completeness

Theorem: Both 2-PVDP and 2-PEDT problems are NP Complete on arbitrary 2-edge-colored graphs.

Reduction: DC problem 2-PVDP

Reduction: DCT problem 2-PEDT

Lemma: DCT problem is NP-Complete.

Proof : (sketch)

1.

2.

3.

0. Both 2-PVDP and 2-PEDT are in NP

Both 2-PVDP and 2-PEDT in c-edge colored graphs

)( 2nc

)( 2nO

)( 2nO

Theorem: Both 2-PVDP and 2-PEDT problems are NP-Complete even for graphs with colors

s

t2-edge-coloredgraph G

Complete graph Kn

with colorsx

GKH n

)( 2n

Additional color

The k-PVDP is NP-Complete in graphs with no pec cycles

l

k

l

CB

1

SAT k-AVDP

2-edge-colored graph G=(V,E)(with no pec cycles) and 2 nodes s,t є V

True assignments for B k-Vertex Disjoint s-t Paths in G


)()()( 321321321 xxxxxxxxxB Example:

Variable x1

t2

s2

s1

t3

s3

t11 2 3t2s2

t3

s1 t1

s3

Variable x2

4

6

5

t3

s1 t1

s2

s3

t2

Variable x3

11

7

8

910

t2

s2

s1

t3

s3

t1

1

2 3

6

4

5

7

89

10

11



Variable x1

t2

s2

s1

t3

s3

t11 2 3t2s2

t3

s1 t1

s3

Variable x2

4

6

5

t3

s1 t1

s2

s3

t2

Variable x3

11

7

8

910

t2

s2

s1

t3

s3

t1

1

2 3

6

4

5

7

89

10

11

s

t



Variable x1

t2

s2

s1

t3

s3

t11 2 3t2s2

t3

s1 t1

s3

Variable x2

4

6

5

t3

s1 t1

s2

s3

t2

Variable x3

11

7

8

910

t2

s2

s1

t3

s3

t1

1

2 3

6

4

5

7

89

10

11

s

tfalsex

falsex

truex

3

2

1



Variable x1

t2

s2

s1

t3

s3

t11 2 3t2s2

t3

s1 t1

s3

Variable x2

4

6

5

t3

s1 t1

s2

s3

t2

Variable x3

11

7

8

910

t2

s2

s1

t3

s3

t1

1

2 3

6

4

5

7

89

10

11

s

tfalsex

falsex

truex

3

2

1

t2

s2

s1 t1

t2

s2

s1 t1

NP-Completeness in graphs with no pec cycles

Grid G(x)

t

s s

t

k-PEDT is also NP-complete !!

Both 2-PVDP and 2-PEDT in c-edge with no properly edge-colored cycles (closed trails)

Theorem: The 2-PVDP (2-PEDT) problem is NP-Complete for graphs with no pec cycles (closed trail) and c=Ω(n) colors

s

t2-edge-coloredgraph Gb

GKH n

Additional colora

c

d

e

Kn with n-1 colors



s


GKH n

Additional colora

c

d

e

Kn with n-1 colors



s


GKH n

Additional colora

c

d

e

Kn with n-1 colors



s


GKH n

Additional colora

c

d

e

Kn with n-1 colors



s


GKH n

Additional colora

c

d

e

Kn with n-1 colors

Approximation Algorithm for the MPEDT

Greedy-ED Procedure

1. S Ø

2. Repeat

Find an pec shortest trail T between « s » and « t »;

S S E(T);

E E - E(T);

Until (no pec s-t trails are found)

Theorem: The Greedy-ED has performance ratio equal tofor the MPEDT problem

mO /1

Approximation Algorithm for the MPVDP

Greedy-VD Procedure

1. S Ø

2. Repeat

Find a pec shortest path P between « s » and « t »;

S S E(P);

V V - V(P);

Until (no pec s-t paths are found)

Theorem: The Greedy-VD has performance ratio equal to for theMPVDP problem

nO /1

0T

1T

2/kT

s t

0T

3T

2T

1T

2T

3T

2/kT

Greedy solution ZH = 1

Approximation ratio for MPEDT

1)( 0 kTE 2/,...,1,2)( kiforkTE i mk

0T

1T

2/kT

s t

0T

3T

2T

1T

2T

3T

2/kT

Greedy solution ZH = 1


Optimum solution Opt = k/2

1)( 0 kTE 2/,...,1,2)( kiforkTE i mk

0T

1T

2/kT

s t

0T

3T

2T

1T

2T

3T

2/kT


1)( 0 kTE 2/,...,1,2)( kiforkTE i mk

Approximation ratio =

m

OkkOpt

ZH 12

2/

1

tstart

u

s

q

v

pdest.

color 1

color 2

color 3pa

v’ v’’ u’ u’’

va vb

v1 v2

q’ q’’

qa qb

q2 q3

p’ p’’

pb pc

p1 p2 p3

ua ub uc

u1 u2 u3

s1


s2

t1

t2

Some Polynomial Cases: we have no (almost) pec cycles passing by « s » or « t ».

tstart

u

s

q

v

pdest.

color 1

color 2

color 3pa

v’ v’’ u’ u’’

va vb

v1 v2

q’ q’’

qa qb

q2 q3

p’ p’’

pb pc

p1 p2 p3

ua ub uc

u1 u2 u3

s1


s2

t1

t2

Some Polynomial Cases: we have no (almost) pec cycles passing by « s » or « t ».

Open Problems and Future Diretions

Input: Given a c-edge-colored complete graph , and vertices s,t of

Open question: Maximize the number of edge-disjoint pec s-t paths in is in P?

Future work: What about the performance ratio of both MPVDP and MPEDT problems in graphs with no pec cycles (closed trails)?

cnK

cnKcnK

Input: Given a 2-edge-colored graph with no pec cycles, vertices s,t V(G) and a fixed k 2.Question: Does G contains k pec vertex disjoint paths between « s » and « t »?

Thanks for your attention!!

Documents

Paths and Trails in Edge Colored Graphs Latin-American on Theoretical Informatics Symposium LATIN 2008 Abouelaoualim, K. Das, L. Faria, Y. Manoussakis,