on s-t paths and trails in edge-colored graphs
DESCRIPTION
On s-t paths and trails in edge-colored graphs. L. Gourvès, A. Lyra * , C. Martinhon, J. Monnot, F. Protti. Lagos 2009 (RS, Brazil ) *[email protected]. Topics. Applications Basic Definitions Paths and trails in G c without PEC closed trails Monochromatic s-t paths - PowerPoint PPT PresentationTRANSCRIPT
On s-t paths and trails in edge-colored graphs
L. Gourvès, A. Lyra* , C. Martinhon, J. Monnot, F.
ProttiLagos 2009 (RS, Brazil)
Topics
Applications Basic Definitions Paths and trails in Gc without PEC
closed trails Monochromatic s-t paths Conclusions and future directions
Topics
Applications Basic Definitions Paths and trails in Gc without PEC
closed trails Monochromatic s-t paths Conclusions and future directions
Some Applications using edge colored graphs
1. Computational Biology
2. Criptography (when a color specify a type of transmission)
3. Social Sciences (where a color represents a relation between 2 individuals)
etc
Some Bibliography
D. Dorniger, On permutations of cromossomes, In Contributions of General Algebra, 5, 95-103, 1987. D. Dorniger, Hamiltonian circuits determining the order of cromossomes, In Disc. App. Math., 5, 159-168, 1994. P. Pevzner, Computational Molecular Biology: An Algorithmic Approach, MIT Press, 2000.
Topics
Applications Basic Definitions Paths and trails in Gc without PEC
closed trails Monochromatic s-t paths
Basic Definitions
Properly edge-colored (PEC) path between « s » and « t »
t
source destination
2 3
s
4
(without node repetitions!!)
1
Basic Definitions
Properly edge-colored (PEC) trail between « s » and « t »
t
source destination
2 3
s
4
1
(without edge repetitions!!)
Basic Definitions
Properly edge-colored (PEC) closed trail.
5
start
2 3
x
4
1
(without edge repetitions!!)
How to find a PEC s-t Path?
Basic Definitions
tstart
u
s
q
v
pdest.
color 1
color 2
color 3pa
v’ v’’ u’ u’’
va vb
v1 v2
q’ q’’
qa qb
q1 q3
p’ p’’
pb pc
p1 p2 p3
ua ub uc
u1 u2 u3
s t
(a) 3-edge colored graph (b) non-colored (Edmond-Szeider) graph
PEC s-t Paths in colored Graphs (Szeider’s Algorithm, 2003)
color 1
color 2
color 3pa
v’ v’’ u’ u’’
va vb
v1 v2
q’ q’’
qa qb
q1 q3
p’ p’’
pb pc
p1 p2 p3
ua ub uc
u1 u2 u3
s t
(a) 3-edge colored graph
PEC s-t Paths in colored Graphs (Szeider’s Algorithm, 2003)
tstart
u
s
q
v
pdest.
(b) non-colored (Edmond-Szeider) graph
tstart
u
s
q
v
pdest.
color 1
color 2
color 3pa
v’ v’’ u’ u’’
va vb
v1 v2
q’ q’’
qa qb
q1 q3
p’ p’’
pb pc
p1 p2 p3
ua ub uc
u1 u2 u3
s t
(a) 3-edge colored graph
PEC s-t Paths in colored Graphs (Szeider’s Algorithm, 2003)
(b) non-colored (Edmond-Szeider) graph
Equivalence between paths and trails (trail-path graph)
s t1
32
Graph Gyx
X’
X’’
y’
y’’
yx
X’
X’’
y’
y’’
PEC s-t trail ρ
Equivalence between paths and trails (trail-path graph)
s t1
32
s’1’’
1’
t’
2’’
2’
3’
3’
Graph H
We have a properly edge colored trail in G we have a properly edge colored path in H
Theorem: Abouelaoualim et al., 2008
Graph G
PEC s-t trail ρ
PEC s-t path ρ’
Topics
Applications Basic Definitions Paths and trails in Gc without PEC
closed trails Monochromatic s-t paths
2 vertex/edge disjoint PEC s-t paths
Theorem: Abouelaoualim et al., 2008
NP-complete, for general Gc Even for Ω = (n2) colors, where |V|=n
s t
Open problem (Abouelaoualim et al., 2008)
2 vertex/edge disjoint PEC s-t paths in Gc with no PEC cycles or PEC closed trails.
s t
18
PEC s-t paths with no PEC closed trails
Theorem 1: To find 2 PEC s-t paths with length at most L> 0 is NP-complete.
Even for graphs with maximum vertex degree
equal to 3.
19
PEC s-t paths with bounded length
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)
20
NP-completeness (Proof)
21
NP-completeness (Proof)
)(),(),(),( 5214421363225321 xxxCxxxCxxxCxxxC
22
NP-completeness (Proof)
Lc = 14n - 1
Lv = 14mn+2m-14n+1L = 14mn + 2m + 2
Lc
Lc
Lc
Lv
NP-completeness (Proof)
23
Maximum degree 3!
Edge-colored graphswith no PEC closed trails
Theorem 2: Find a PEC s-t trail visitingall vertices x ∈ V(Gc) exactly f(x) times,
with fmin(x) ≤ f(x) ≤ fmax(x)
s t
bc
de
a
x = a, f(a)=2
Solved in polynomial
time!
Edge-colored graphswith no PEC closed trails
Finding a PEC s-t trail passing by a vertex v is NP-complete in general edge-colored graphs. (Chou et al., 1994)
Finding a PEC s-t trail visiting A ⊆V \{s,t} is polynomial time solvable in 2-edge-colored complete graphs. (Das and Rao, 1983)
26
Theorem 2 (proof)
The idea is to construct the trail-path graph and the Edmonds-Szeider Graph associated to the trail-path graph.
yx
X’
X’’
y’
y’’
fmin(x) = 1fmax(x) = 2
fmin(y) = 1fmax(y) = 3
y’’’
27
Theorem 2 (proof)
x
x’a x’’b
x’1 x’2 x’3
x1 x2 x3
color 1
color 2
color 3
x’a x’’b
x’1 x’2 x’3
x1 x2 x3
Subgraph H’x associated to x ∈
S’(x)
Subgraph Hx associated to x ∈
S(x)
Corollary 3: A shortest (resp. longest) PEC s-t
trail visiting vertices x of Gc at least fmin(x) times (resp. at most fmax(x) times)
Edge-colored graphswith no PEC closed trails
Solved in polynomial time!
Corollary 3 (proof)
Compute the minimum perfect matching (resp. maximum perfect matching ) M in Hm.
y’a y’’b
y’1 y’2 y’3
y1 y2 y3
x’a x’’b
x’1 x’2 x’3
x1 x2 x3
0 0 0
0 0
0 0 0
00
1
1 1
111 1
Edge-colored graphswith no PEC closed trails
Theorem 4:The determination of a PEC s-t trail visiting all edges of E’ ⊆ E(Gc)is solved in polynomial time.
s
bc
a
e f
d t
E’ = {ab, bc, ca, df, fe}
Polynomial case (proof):
Construct the trail-path graph, Construct an associated modified
Edmonds-Szeider graph. xy ∈ E’
yx
x1
x2
y1
y2
Hx1
Hx2
Hy1
Hy2axy
bxy
PEC closed trails are allowed!
32
Edge-colored graphswith no PEC cycles
1
32
54
76
33
Theorem 5To find a PEC s-t trail passing by a vertex v is NP-complete.
Edge-colored graphswith no PEC cycles
Surprisingly, finding a PEC s-t path passing by a subset A={v1,..., vk} is
polynomial time solvable!
34
Theorem 5 (proof) Use the problem Path Finding Problem in D
ts x
s tb
xc
da
Theorem 5 (proof)
35
G’(v)
G’’(e)),( uve
Theorem 5 (proof)
36
Without incoming arcs at sWithout outgoing arcs at t
Theorem 5 (proof)
37
Topics
Applications Basic Definitions Paths and trails in Gc without PEC
closed trails Monochromatic s-t paths
NP-completeness
Theorem 6:Find 2 vertex disjoint monochromatic s-t paths with different colors in Gc is NP-complete.
tstart s dest.
The edge disjoint case is trivial
Theorem 6 (proof)
14u
24u
34u
4q
4w
12u
22u
32u
2q
2w
11u
21u
31u
2q
2w
)( , )( , )( , )( 6324421362125321 xxxcxxxcxxxcxxxc
c1
c3
c2
c413u
23u
33u
3q
3w2b
2a
Theorem 6 (proof)
1b
1a
2q
2w
mq
mw
1q
1w
2b
2a
nb
na
s
t
Conclusions and future directions
We deal with PEC and monochromatic s-t paths and trails on c-edge colored graphs.
Future directions: Given Gc without PEC cycles, is there a
polynomial algorithm to find two PEC s-t paths?
If Gc is planar, to find two monochromatic vertex disjoint s-t paths is NP-complete?
Thanks for your attention!
You can download this presentation athttp://www.ic.uff.br/~alyra
My email: [email protected]