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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!