Tree%20Spanners%20for%20Bipartite%20Graphs%20and%20Probe%20Interval%20Graphs

1
Tree Spanners for Bipartite Graphs and Probe
Interval Graphs

• Andreas Brandstädt1, Feodor Dragan2,
• Oanh Le1, Van Bang Le1, and Ryuhei Uehara3

1 Universität Rostock
2 Kent State University
3 Komazawa University
2
Tree Spanners for Bipartite Graphs and Probe
Interval Graphs

• Andreas Brandstädt1, Feodor Dragan2,
• Oanh Le1, Van Bang Le1, and Ryuhei Uehara3

1 Universität Rostock
2 Kent State University
3 Komazawa University
3
Tree Spanner
x
G
T
x
y
y

• Spanning tree T is a tree t-spanner iff
• dT (x,y) ?t dG (x,y)
• for all x and y in V.

4
Tree Spanner
G
T

• Spanning tree T is a tree t-spanner iff

dT (x,y) ? t dG (x,y) for all x,y
in E.
5
Tree Spanner
G
T

• Spanning tree T is a tree 6-spanner.

6
Tree Spanner
G
T

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

7
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

8
Known Results for tree t -spanner

• general graphs CaiCorneil95
• a linear time algorithm for t 2 (t1 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 t3

9
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 PKLMW99,
• strongly chordal graphs, dually chordal graphs
  BCD99
• tree 3 -spanner is in P for planar graphs
  FK2001

10
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 PKLMW99,
• strongly chordal graphs, dually chordal graphs
  BCD99
• tree 3 -spanner is in P for planar graphs
  FK2001

? Bipartite Graphs??
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Known Results for tree t -spanner

• bipartite graphs CaiCorneil 95
• tree t -spanner is NP-complete for any t ?5
• chordal graphs Brandstädt, Dragan, Le Le
  02tree 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

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This Talk
weakly chordal
bipartite
chordal bipartite
chordal
NP-C
strongly chordal
AT-free
bipartite ATE-free
4-Adm.
rooted directed path
interval bigraph
3-Adm.
convex
interval
13
This Talk
weakly chordal
bipartite
chordal bipartite
chordal
NP-C
strongly chordal
AT-free
bipartite ATE-free
enhanced probe interval
probe interval
4-Adm.
rooted directed path
interval bigraph
3-Adm.
STS-probe interval

convex
interval
14
This Talk
weakly chordal
bipartite
chordal bipartite
chordal
NP-C
strongly chordal
AT-free
bipartite ATE-free
7-Adm.
enhanced probe interval
probe interval
4-Adm.
rooted directed path
interval bigraph
3-Adm.
STS-probe interval

convex
interval
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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 3SAT

Monotone
(x, y, z) or (x, y, z)
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NP-hardness for chordal bipartite graphs

• Reduction from 3SAT
• Basic gadgets

Monotone
(x, y, z) or (x, y ,z)
S1a,b
S2a,b
S3a,b
a
b
a
b
a
b
S2a,a
S2b,b
S1a,a
S1b,b
a
b
a
b
a
b
S2a,b
S1a,b
17
NP-hardness for chordal bipartite graphs

• Reduction from 3SAT
• Basic gadget Ska,b and its spanning trees

Monotone
(x, y, z) or (x, y ,z)
h
a
b
a
b
a
b
H
a
b
a
b
a
b
a
b
with a,b
without a,b
without a,b
a
b
(2kh)-spanner
(2k-1)-spanner
(2k1)-spanner
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NP-hardness for chordal bipartite graphs

• Reduction from 3SAT
• Gadget for xi

Monotone
(x, y, z) or (x, y ,z)

q
r
xi
xi
xi
2
m
1
Sk-1

Sk2


xi
m
s
p
xi
xi
1
2
Must be selected
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NP-hardness for chordal bipartite graphs

• Reduction from 3SAT
• Gadget for Cj

Monotone
(x, y, z) or (x, y ,z)
-

cj
cj
Sk2

-

dj
dj
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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)
2
1
2
1
2
1
2
1
x2
x2
x4
x4
q
r
x1
x1
x3
x3
1
2
1
2
1
2
1
2
s
p
x1
x1
x2
x2
x3
x3
x4
x4

-

-
c1
c1
c2
c2
Sk-2

-

-

d1
d1
d2
d2
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Tree 3-spanner for a bipartite ATE-free graph
e1

• 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 ? bipartite ATE-free
  graphs ? chordal bipartite graphs.

e3
e2
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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

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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)?SnNk(u)

w

u
S
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Tree 3-spanner for a bipartite ATE-free graph

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

w

u
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Conclusion and open problems

• Many questions remain still open. Among them
• Can Tree 3Spanner be decided efficiently
• on general graphs???
• on chordal graphs?
• on chordal bipartite graphs?
• Tree tSpanner on (enhanced) probe interval
  graphs for tlt7?

Thank you!