Tree Spanners for Bipartite Graphs and

Probe Interval Graphs

Andreas Brandstädt1, Feodor Dragan2, Oanh Le1, Van Bang Le1, and Ryuhei Ueha

ra3

1 Universität Rostock

2 Kent State University

3 Komazawa University

Tree Spanners for Bipartite Graphs and

Probe Interval Graphs

Andreas Brandstädt1, Feodor Dragan2, Oanh Le1, Van Bang Le1, and Ryuhei Ueha

ra3

1 Universität Rostock

2 Kent State University

3 Komazawa University

Tree Spanner

Spanning tree T is a tree t-spanner iff

dT (x,y) ≦t dG (x,y)

for all x and y in V.

G T

xy

xy

Tree Spanner

Spanning tree T is a tree t-spanner iff

G T

dT (x,y) ≦ t dG (x,y)for all {x,y} in E.

Tree Spanner

Spanning tree T is a tree 6-spanner.

G T

Tree Spanner

G admits a tree 4-spanner (which is optimal). Tree t-spanner problem asks

if G admits a tree t-spanner for given t.

G T

Applications in distributed systems and communication networks

synchronizers in parallel systems topology for message routing

there is a very good algorithm for routing in trees

in biology evolutionary tree reconstruction

in approximation algorithms approximating the bandwidth of graphs

Any problem related to distances can be solved approximately on a complex graph if it admits a good tree spanner

G

7-spanner for G

Known Results for tree t -spanner general graphs [Cai&Corneil’95]

a linear time algorithm for t =2 (t=1 is trivial) tree t -spanner is NP-complete for any t 4≧　　 ( NP-completeness of ⇒ bipartite graphs for t 5)≧ tree t -spanner is Open for t=3

Known Results for tree t -spanner chordal graphs [Brandstädt, Dragan, Le & Le ’02]

tree t -spanner is NP-complete for any t 4≧ tree 3-spanner admissible graphs [a Number of Authors]

cographs, complements of bipartite graphs, interval graphs, directed path graphs, split graphs, permutation graphs, convex bipartite graphs, regular bipartite graphs, distance-hereditary graphs

tree 4-spanner admissible graphs AT-free graphs [PKLMW’99], strongly chordal graphs, dually chordal graphs [BCD’99]

tree 3 -spanner is in P for planar graphs [FK’2001]

Known Results for tree t -spanner chordal graphs [Brandstädt, Dragan, Le & Le ’02]

tree t -spanner is NP-complete for any t 4≧ tree 3-spanner admissible graphs [a Number of Authors]

cographs, complements of bipartite graphs, interval graphs, directed path graphs, split graphs, permutation graphs, convex bipartite graphs, regular bipartite graphs, distance-hereditary graphs

tree 4-spanner admissible graphs AT-free graphs [PKLMW’99], strongly chordal graphs, dually chordal graphs [BCD’99]

tree 3 -spanner is in P for planar graphs [FK’2001]

⇒ Bipartite Graphs??

Known Results for tree t -spanner bipartite graphs [Cai&Corneil ’95] tree t -spanner is NP-complete for any t 5≧ chordal graphs [Brandstädt, Dragan, Le & Le ’02]

tree t -spanner is NP-complete for any t 4≧ tree 3-spanner admissible graphs [a Number of Authors]

cographs, complements of bipartite graphs, interval graphs, directed path graphs, split graphs, permutation graphs, convex bipartite graphs, regular bipartite graphs, distance-hereditary graphs

convex bipartite interval bigraphs ⊂ ⊂ 　 bipartite ATE-free graphs ⊂ chordal bipartite graphs ⊂ bipartite graphs

This Talk

interval

rooteddirected

path

stronglychordal

chordal

weaklychordal

chordalbipartite

intervalbigraph

convex

AT-free bipartiteATE-free

bipartite

NP-C

4-Adm.

3-Adm.

This Talk

interval

rooteddirected

path

stronglychordal

chordal

weaklychordal

enhancedprobe

interval

chordalbipartite

probeinterval

intervalbigraph

convexSTS-probe

interval

AT-free bipartiteATE-free

bipartite

NP-C

4-Adm.

3-Adm.

=

This Talk

interval

rooteddirected

path

stronglychordal

chordal

weaklychordal

enhancedprobe

interval

chordalbipartite

probeinterval

intervalbigraph

convexSTS-probe

interval

AT-free bipartiteATE-free

bipartite

NP-C

4-Adm.

3-Adm.

=

7-Adm.

NP-hardness for chordal bipartite graphs

[Thm] For any t 5, the ≧ tree t-spanner problem is NP-complete for chordal bipartite graphs.

Reduction from 3SATMonotone

… (x, y, z) or (x, y, z)

NP-hardness for chordal bipartite graphs

Reduction from 3SAT Basic gadgets

Monotone

… (x, y, z) or (x, y ,z)

S1[a,b] S2[a,b] S3[a,b]

a

a’

b

b’

a b

a’ b’

S1[a,a’]

S1[a’,b’]

S1[b,b’] S2[a,a’]

S2[a’,b’]

S2[b,b’]

a b

a’ b’

NP-hardness for chordal bipartite graphs

Reduction from 3SAT Basic gadget Sk[a,b] and its spanning trees

Monotone

… (x, y, z) or (x, y ,z)

a

a’

b

b’

a

a’

b

b’a

a’

b

b’

H

with {a,b}

(2k+1)-spanner

without {a,b}

h

(2k+h)-spanner

a

a’

b

b’

without {a,b}

(2k-1)-spanner

NP-hardness for chordal bipartite graphs

Reduction from 3SAT Gadget for xi

Monotone

… (x, y, z) or (x, y ,z)

q r

sp

xi xi

xixi

xi

xi1

2 m1

2 m…

…Sk-1[]

Sk[]× ２

＝

＝

Must be selected

NP-hardness for chordal bipartite graphs

Reduction from 3SAT Gadget for Cj

Monotone

… (x, y, z) or (x, y ,z)

cj cj

djdj

Sk[]× ２＝

+ -

+ -

NP-hardness for chordal bipartite graphs

Reduction from 3SAT Gadget for C1=(x1,x2,x3) and C2=(x1,x2,x4)

Monotone

… (x, y, z) or (x, y ,z)

q r

sp

x1 x1

x1x11

21

2

Sk-2[]＝

x2 x2

x2x21

21

2

x3 x3

x3x31

21

2

x4 x4

x4x41

21

2

c1 c1

d1d1

+ -

+ -c2 c2

d2d2

+ -

+ -

Tree 3-spanner for a bipartite ATE-free graph

An ATE(Asteroidal-Triple-Edge) e1,e2,e3 [Mul97]:Any two of them there is a path from

one to the other avoids the neighborhood of the third one.

[Lamma] interval bigraphs biparti⊂te ATE-free graphs chordal bi⊂partite graphs.

e1

e3e2

Tree 3-spanner for a bipartite ATE-free graph

A maximum neighbor w of u: N(N(u))=N(w)

[Lamma] Any chordal bipartite graph has a vertex with a maximum neighbor.

u w

chordal bipartite graph⇔•bipartite graph•any cycle of length at least 6 has a chord

Tree 3-spanner for a bipartite ATE-free graph

G; connected bipartite ATE-free graph u; a vertex with maximum neighbor

For any connected component S induced by V ＼ Dk-1(u), there is w in Nk-1(u) s.t. N(w) S∩N⊇

k(u)

Su … w

Tree 3-spanner for a bipartite ATE-free graph

Construction of a tree 3-spanner of G: u; a vertex with maximum neighbor

u … w

Conclusion and open problems• Many questions remain still open. Among them:

• Can Tree 3–Spanner be decided efficientlyon general graphs??? on chordal graphs?on chordal bipartite graphs?

•Tree t–Spanner on (enhanced) probe interval graphs for t<7?

Thank you!