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Treewidth Some problems parametrized by treewidth Results Further research The role of planarity in connectivity problems parameterized by treewidth Julien Baste and Ignasi Sau 1/22 Julien Baste and Ignasi Sau The role of planarity in

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Treewidth Some problems parametrized by treewidth Results Further research

The role of planarity in connectivity problems parameterized by treewidth

Julien Baste and Ignasi Sau

1/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research Definition Courcelle

Tree decomposition

Definition (Tree decomposition) Let G = (V , E) be a graph, a tree-decomposition of width w of G is a pair (T, σ), where T is a tree and σ = {Bt|Bt ⊆ V , t ∈ V (T)} such that :

t∈V (T) Bt = V (G),

For every edge {u, v} ∈ E(G) there is a t ∈ V (T) such that {u, v} ⊆ Bt, Bi ∩ Bk ⊆ Bj for all {i, j, k} ⊆ V (T) such that j lies on the path i, . . . , k in T, |Bt| ≤ w + 1 for every t ∈ V (T). The set Bt are called bag.

2/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research Definition Courcelle

Tree decomposition

Definition (treewidth) The treewidth of G noted tw is the minimal value w such that there is a tree-decomposition of G of width w. An optimal tree decomposition is a tree-decomposition of width tw.

[Roberson and Seymour.] 3/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research Definition Courcelle

Courcelle Theorem

Theorem (Courcelle) Each graph property definable in monadic second-order logic can be decided in time f (tw) · n for some function f where n is the number of vertices of the input graph and tw its treewidth.

[Courcelle]

In this theorem the function f is a tower of exponential.

4/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research A simple problem Some more complicated problems Sparse graphs Optimal algorithms What now?

A simple problem

Vertex Cover Input: A graph G = (V , E) and an integer k Question: Can we find X ⊆ V such that for all (x, y) ∈ E, x ∈ X

r y ∈ X?

We have algorithm in time 2O(tw) · nO(1). It is the best possible under ETH.

[Impagliazzo, Paturi, Zane. JCSS 01] 5/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research A simple problem Some more complicated problems Sparse graphs Optimal algorithms What now?

Some more complicated problems

Longest Path Input: A graph G = (V , E) and an interger k. Question: Does there exist a simple path of length k in G ? The best algorithm known in general graphs is in time 2O(tw log tw) · nO(1).

6/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research A simple problem Some more complicated problems Sparse graphs Optimal algorithms What now?

Some more complicated problems

Longest Path Input: A graph G = (V , E) and an interger k. Question: Does there exist a simple path of length k in G ? The best algorithm known in general graphs is in time 2O(tw log tw) · nO(1). We have other examples of such problems : Steiner Tree Connected vertex cover ...

6/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research A simple problem Some more complicated problems Sparse graphs Optimal algorithms What now?

Some more complicated problems

Longest Path Input: A graph G = (V , E) and an interger k. Question: Does there exist a simple path of length k in G ? The best algorithm known in general graphs is in time 2O(tw log tw) · nO(1). We have other examples of such problems : Steiner Tree Connected vertex cover ... These problems are called connectivity problems.

6/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research A simple problem Some more complicated problems Sparse graphs Optimal algorithms What now?

Meanwhile in sparse graphs

Based on catalan structures, There are algorithms for these problems computing in time 2O(tw) · nO(1) for some sparse graphs. In planar graphs.

[Dorn, Penninkx, Bodlaender, and Fomin. ESA 2005]

In graph of bounded genus.

[Ru´ e, Sau, and Thilikos. ICALP 2010]

H-minor free graph.

[Dorn, Fomin, and Thilikos. SODA 2008] 7/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research A simple problem Some more complicated problems Sparse graphs Optimal algorithms What now?

New algorithms

We recently have two new algorithms for many connectivity problems Cut & Count algorithm :

[Cygan, Nederlof, Pilipczuk, Pilipczuk, Van Rooij and Wojtaszczyk. FOCS 11]

Give randomize algorithm in time 2O(tw) · nO(1). Give connectivity problems that cannot be solved in time 2o(tw log tw) · nO(1)

8/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research A simple problem Some more complicated problems Sparse graphs Optimal algorithms What now?

New algorithms

We recently have two new algorithms for many connectivity problems Cut & Count algorithm :

[Cygan, Nederlof, Pilipczuk, Pilipczuk, Van Rooij and Wojtaszczyk. FOCS 11]

Give randomize algorithm in time 2O(tw) · nO(1). Give connectivity problems that cannot be solved in time 2o(tw log tw) · nO(1)

Rank based algorithm :

[Bodlaender, Cygan, Kratsh, and Nederlof. ICALP 2013]

Give deterministic in time 2O(tw) · nO(1).

8/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research A simple problem Some more complicated problems Sparse graphs Optimal algorithms What now?

What now?

Type 1 Type 2 2O(tw) · nO(1) in general graphs no 2o(tw log tw) · nO(1) in general graphs

9/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research A simple problem Some more complicated problems Sparse graphs Optimal algorithms What now?

What now?

Type 1 Type 3 Type 2 2O(tw) · nO(1) in general graphs no 2o(tw log tw) · nO(1) in general graphs no 2o(tw log tw) · nO(1) in general graphs 2O(tw) · nO(1) no 2o(tw log tw) · nO(1) in planar graphs in planar graphs

10/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research Type 1 Type 2 Type 3

Problem of type 1

Type 1: Problems that can be solved in time 2O(tw) · nO(1) on general graphs. 3-colorability Input: A graph G = (V , E). Question: Is there a color function c : V → {1, 2, 3} such that for all {x, y} ∈ E, c(x) = c(y). Theorem Planar 3-colorability cannot be solved in time 2o(tw) · nO(1) unless the ETH fails.

11/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research Type 1 Type 2 Type 3

Problem of type 2

Type 2: No algorithm in time 2o(tw log tw) · nO(1) on general graphs An algorithm in time 2O(tw) · nO(1) when restricted to planar graphs. We find some problems of type 2: Cycle packing Max Cycle Cover Maximally Disconnected Dominating Set Maximally Disconnected Feedback Vertex Set

12/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research Type 1 Type 2 Type 3

Problem of type 3

Type 3: An algorithm in time 2O(tw log tw) · nO(1) on general graphs. No algorithm in time 2o(tw log tw) · nO(1) even when restricted to planar graphs. Monochromatic Disjoint Paths Input: A graph G = (V , E) of treewidth tw, a color function γ : V → {0, . . . , tw}, m ∈ N and N = {Ni = {si, ti}|i ∈ {1, . . . , m}, si, ti ∈ V } Question: Does G contain m pairwise vertex-disjoint monochro- matic paths from si to ti, i ∈ {1, . . . , m}?

13/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research Type 1 Type 2 Type 3

k × k hitting set

k × k hitting set Input: A family of sets S1, S2, . . . , Sm ⊆ [k] × [k], such that each set contains at most one element from each row of [k] × [k] Question: Is there a set S containing exactly one element of each row such that S ∩ Si = ∅ for any 1 ≤ i ≤ m ?

[Lokshtanov, Marx, Saurabh. SODA 2011]

Theorem k × k hitting set cannot be solved in time 2o(k log k) unless the ETH fails.

14/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research Type 1 Type 2 Type 3

s1 CS v1,0 v1,1 v1,2 v1,m−1 v1,m t1 s2 CS v2,0 v2,1 v2,2 v2,m−1 v2,m t2 s3 CS v3,0 v3,1 v3,2 v3,m−1 v3,m t3 sk CS vk,0 vk,1 vk,2 vk,m−1 vk,m tk SET1 SET1 SET1 SET2 SET2 SET2 SETm SETm SETm SET1 SET2 SETm

15/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research Type 1 Type 2 Type 3

c1 c2 c3 sr ck vr,0 The gadget. sr CC vr,0 The representation.

Figure : Color selection gadget

16/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research Type 1 Type 2 Type 3

sr,i v tr,i u The gadget. u EE v The representation.

Figure : Expel gadget

17/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research Type 1 Type 2 Type 3

v1,i−1 a1,i v1,i v2,i−1 a2,i v2,i v3,i−1 a3,i v3,i vk,i−1 ak,i vk,i w1,i,2 EE w2,i,1 w2,i,2 EE w3,i,1 w3,i,2 wk,i,1 The gadget. v1,i−1 v1,i v2,i−1 v2,i v3,i−1 v3,i vk,i−1 vk,i SET SET SET SET The representation.

Figure : Set gadget

18/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research Type 1 Type 2 Type 3

Given a solution of Monochromatic Disjoint Paths, we have a solution of k × k hitting set.

19/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research Type 1 Type 2 Type 3

Given a solution of k × k hitting set, we have a solution of Monochromatic Disjoint Paths.

20/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research

Further research

Is there a problem that can be solved in 2o(tw) in planar graphs?

21/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research

Further research

Is there a problem that can be solved in 2o(tw) in planar graphs? What is the complexity of Planar Disjoint Paths?

There is an algorithm for Disjoint Paths in time 2O(tw log tw) · nO(1). We prove that we cannot solve Planar Disjoint Paths in time 2o(tw) · nO(1).

21/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research

Further research

Is there a problem that can be solved in 2o(tw) in planar graphs? What is the complexity of Planar Disjoint Paths?

There is an algorithm for Disjoint Paths in time 2O(tw log tw) · nO(1). We prove that we cannot solve Planar Disjoint Paths in time 2o(tw) · nO(1).

What is the complexity of Subgraph Isomorphism when parametrized by treewidth?

We prove that we cannot solve Planar Subgraph Isomorphism in time 2o(tw log tw) · nO(1).

21/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research

Further research

Is there a problem that can be solved in 2o(tw) in planar graphs? What is the complexity of Planar Disjoint Paths?

There is an algorithm for Disjoint Paths in time 2O(tw log tw) · nO(1). We prove that we cannot solve Planar Disjoint Paths in time 2o(tw) · nO(1).

What is the complexity of Subgraph Isomorphism when parametrized by treewidth?

We prove that we cannot solve Planar Subgraph Isomorphism in time 2o(tw log tw) · nO(1).

What is the lower bound of 2-edge-connected when parametrized by treewidth?

21/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized

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Treewidth Some problems parametrized by treewidth Results Further research

Thanks

22/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized