B. Li F. Moataz N. Nisse K. Suchan

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Minimum Size Tree Decompositions

1, Inria, France2, University Nice Sophia Antipolis, CNRS, I3S, France3, Institute of Applied Mathematics, CAS, Beijing, China4, FIC, Universidad Adolfo Ibanez, Santiago, Chile5, WMS, AGH, Univ. of Science and Technology, Krakow, Poland

1,2,3 2,1 1,2 4,5

Thank you to B. Li for her slides

LAGOS, Praia das Fontes, May 12th 2015

Motivations to Study Tree Decompositions

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Dynamic programming based on tree decomposition

[B. Courcelle, Information and Computation, 1990]

Many NP-hard problems in general graphslinear in graphs of bounded treewidth

All problems expressible in monadic second order logic are linear solvable in graphs of bounded treewidth

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Characterizations for small treewidthis a forest [Folklore]

no -minor in

no following minors in

[Wald and Colbourn, Networks, 1983]

[Arnborg and Proskurowski, Discrete Math. 1990]

Graphs of treewidth at most 4 can be reduced to empty by at most 6 reduction rules. [Sanders, Discrete Math., 1995]

e.g. outerplanar graphs, series parallel graphs

Our goal: Explore other parameters to understand better and compute tree decomposition for algorithmic applications

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Tree-Decomposition ofa pair

-

For

is a tree:

-there is s.t.

-induces a subtree in

For

called bag,

It’s called path decomposition if T is a path.

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Tree-Decomposition of

Size of a tree decomposition, the number of bags

Width 3Size 6

Minimum size tree decompositionof width at most k

Width 3

Size 4

Minimum length path decomposition

of width at most k

Treewidth, is the minimum width.

Width of a tree decomposition, the size of the largest bag

Width 3

Length 4

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Related Work

[D. Dereniowski et al., CoRR, 2013]

NP-hard to compute for any fixed

-

Polynomial algorithms for computing -

in general graphs

NP-hard to compute for any fixed

in connected graphs-

Minimum length path decomposition of width at most k

Our contributions

NP-hard to compute for any fixed-

General approach for computing

- Polynomial algorithm for computing

- Polynomial algorithms for computing

-

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in general graphs (resp. connected graphs)

for any fixed

in the class of forests and 2-connected outerplanar graphs

(resp. )

Minimum size tree decomposition of width at most k

NP-hard in planar graphs with -

Reduction from 3-partition

Instance: a list of positive integers ,

NP-hardness of computing

Question: is there a partition of into sets s.t.

for each

Construct a graph s.t.

Yes for 3-partition

of width at most 4 and size at most

has a tree decomposition

based on [D. Dereniowski et al., CoRR, 2013]

Construction of Graph

and copies of copies of

and copies of copies of

edges

disjoint union of and

[D. Dereniowski et al., CoRR, 2013]

Yes for 3-partitionof width at most 4 and length at most s

has a path decomposition

copies of

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Any tree decomposition of

is a path decomposition

of width 4 and size s

Path-Decomposition & Tree-Decomposition of

is NP-hard

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NP-hardness of computing If computing is NP-hard in general graphs,

then computing is NP-hard in connected graphs.

has a tree decomposition of width and size

has a tree decomposition of width and size

NP-hard to compute for any fixedin general graphs (resp. connected graphs)

(resp. )

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General Approach for Computing

find a set of vertices s.t.

containing as a leaf bag

Given and

there is a tree decomposition of width and size

is called a k-potential leaf

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General Approach for Computing

If there is a -time algorithm that, computes a k-potential leaf of

then can be computed in

for any

be the the class of graphs of treewidthLet

in

Prove by induction on

Find a k-potential leaf

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Polynomial Algorithm for computing Characterization of all 2-potential leaves in the class of graphs of treewidth 2:

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Polynomial in Trees

Characterization of 3-potential leaves in trees

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Polynomial in 2-connected outerplanar graphs

3-potential leaves in 2-connected outerplanar graphs are

Future Works

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- Compute in the class of graphs of treewidth 2 or 3

is a 3-potential leaf of but not 3-potential leaf of

It seems more tricky.

Future Works

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- Compute in trees

There are 5-potential leaf that are not connected

It is different when

Future Works

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Obrigado !