Tree Spanners on Chordal Graphs:Tree Spanners on Chordal Graphs:

Complexity, Algorithms, Open ProblemsComplexity, Algorithms, Open Problems

A. Brandstaedt, F.F. Dragan,

H.-O. Le and V.B. Le

University of Rostock, Germany

Kent State University, Ohio, USA

G tree 4-spanner T of G

tree t -spanner Problem • Given unweighted undirected graph G=(V,E) • Does G admit a spanning tree T =(V,U) such that

).,(),(,, uvdisttuvdistVvu GT

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G

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T

.),(, tuvdistEuv 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

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G

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4-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– tree t -spanner is Open for t=3

• 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]

Chordal Graphs• G is chordal if it has no chordless cycles of length >3 • There is no constant t [McKee, H.-O.Le]

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no tree 1-spanner

Chordal Graphs• G is chordal if it has no chordless cycles of length >3 • There is no constant t [McKee, H.-O.Le]

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no tree 2-spanner

Chordal Graphs• G is chordal if it has no chordless cycles of length >3 • There is no constant t [McKee, H.-O.Le]

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no tree 3-spanner

Chordal Graphs• G is chordal if it has no chordless cycles of length >3 • There is no constant t [McKee, H.-O.Le]

• From far away they look like trees

• there is a tree T=(V,U) (not necessarily spanning) such that

[BCD’99]

• there is a sparse (2n-2 edges) 5-spanner [PS’89]

2|),(),(|,, uvdistuvdistVvu GT

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no tree 3-spanner

These graphs are not only chordal but also are planar and 2-trees

Chordal Graphs• G is chordal if it has no chordless cycles of length >3 • There is no constant t [McKee, H.-O.Le]

• From “far away” they look like trees

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no tree 3-spanner

These graphs are not only chordal but also are planar and 2-trees

Q: What is the complexity of tree

t - spanner in chordal graphs?

This Talk

• NP-completeness results for t>3– to the best of our knowledge, this is the first hardness

result for the problem on a restricted, well-understood graph class.

• [FK01] Tree t -Spanner, t>3, is NP-complete on planar graphs, if the integer t is part of the input.

• Some easy solvable cases • Many open problems

NP-completeness results

• For any t 4, Tree t -Spanner is NP-complete on chordal graphs of diameter at most – t+1 (if t is even), – t+2 (if t is odd).

• Proof is by reduction from 3SAT. – Let F be a 3CNF formula with m clauses and n variables

jC iv

The graph obtained from H and by identifying the edge e=xy

Gadgets

],[ yxSk

The graph obtained

from and

by identifying b=d and joining

x=a with y=c.

],[ yxSk],[1 baSk ],[1 dcSk

If k=t-1, then any tree t-spanner must connect x and y in the part ],[ yxSk

Vertex Gadgets• Let

• For each variable create the graph as follows.

– Consider a clique on l+2 vertices

– For each edge create a

– Take a chordless path and connect both

to each vertex of this path

– For each edge create a

– For each edge create a

)( ivGiv

.2

:,22

:

ltl

iQ )(,,...,),( 110i

li

liiii vqqqvq

],[1 yxSt}0:{ 1 lkqqxy ki

ki

iii sss ...10 10 , l

ii qq

}0:{ 1 kssxy ki

ki

},,,{ 101000 liiii

liiii qsqsqsqsxy

],[2 yxSt],[)2( yxS lt

The graph G(F)• For each clause create the graph as follows.

– If t is even, is simply a single vertex . – If t is odd, is the graph .

• Finally, the graph G=G(F) is obtained from all and by identifying all vertices to a single vertex s, and adding the following additional edges:– connect every vertex in with every vertex in– for each literal , if then connect with

respectively, with according to the parity of t.

)( jCGjC

)( jCG ja

)( jCG ],[ 211 jjt aaS

)( ivG )( jCG0js

iQ '.,' iiQi },{ 10 l

iii qqu ji Cu iu ,ja21 and jj aa

x xy y

z z

)()(),,( zyxzyxzyxF 1a 2a

zs

ysxs

t=4

NP-completeness

• G(F) is chordal, and diam(G(F)) is at most t+1 if t is even, and at most t+2 if t is odd.

• G(F) admits a tree t-spanner if and only if F is satisfiable.

Diameter at most Complexity

t + 2, t 5 odd NP-C

t + 1, t 4 even NP-C

Efficient solvable cases• For any even integer t, every chordal graph of diameter at most t-1

admits a tree t-spanner, and such a tree spanner can be constructed in linear time.

• For any odd integer t, every chordal graph of diameter at most t-2 admits a tree t-spanner, and such a tree spanner can be constructed in linear time.

Diameter at most Complexity

t + 2, t 5 odd NP-C

t + 1, t 4 even NP-C

t - 1, t 2 even linear

t - 2, t 3 odd linear

• Any BFS-tree started at a central vertex is such a spanner.

• A central vertex of a chordal graph can be found in linear time [CD’94].

• For any chordal graph,

2)(2)()(2 GradGdiamGrad

Q: can we improve this to t-1?

t-1 question (t is odd)• chordal graphs of diameter at most t-1 (t is odd) admit tree t-spanners

if and only if chordal graphs of diameter 2 admit tree 3-spanners. – First we reduce this problem to the existence of a tree (2rad(G)-1)-

spanner in a chordal graph of diameter 2rad(G)-2.

– Then, any such graph has an m-convex two-set M such that

– Therefore….

.,2)(),( VvGradMvdist .,2)(),( VvGradMvdist

2r2

G

M

Tree 3-spanners in chordal graphs of diam=2.

• Unfortunately, the reduction above (from arbitrary odd t to t=3) is of no direct use for general chordal graphs because not every chordal graph of diameter at most 2 admits a tree 3-spanner.– We used this reduction to obtain some results for planar chordal graphs

and k-trees (k<4).

• A chordal graph G of diameter at most 2 admits a tree 3-spanner if and only if there is a vertex v in G such that any connected component of the second neighborhood of v has a dominating vertex in N(v).

Tree 3-spanners in chordal graphs of diam=2.

• Unfortunately, the reduction above (from arbitrary odd t to t=3) is of no direct use for general chordal graphs because not every chordal graph of diameter at most 2 admits a tree 3-spanner.– We used this reduction to obtain some results for planar chordal graphs

and k-trees (k<4).

• A chordal graph G of diameter at most 2 admits a tree 3-spanner if and only if there is a vertex v in G such that any connected component of the second neighborhood of v has a dominating vertex in N(v).

• For a given chordal graph G=(V,E) of diameter at most 2, the Tree 3–Spanner can be decided in O(|V| |E|) time. Moreover, a tree 3-spanner of G, if it exists, can be constructed within the same time bound.

• OQ: Can that reduction and the result above be combined to solve the “t-1 question”?

Conclusion and open problems

• Many questions remain still open. Among them:• Can Tree 3–Spanner be decided efficiently on chordal graphs?• Can Tree (2r(G)-1)-- Spanner be decided efficiently on chordal graphs of diameter 2r(G)-2? • What is the complexity of Tree t–Spanner for chordal graphs of diameter at most t ?.

Diameter at most Complexity

t + 2, t 5 odd NP-C

t + 1, t 4 even NP-C

t + 1, t 3 odd ?

t, t 3 ?

t - 1, t 5 odd ?

t - 1, t = 3 polynomial

t - 1, t 2 even linear

t - 2, t 3 odd linear

Thank You