Bidimensionality theory and applications

Alexandros Angelopoulos

M.P.L.A.

July 1, 2014 Outline

The idea

Subexponential parametrized algorithms

PTAS / EPTAS

Kernelization

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 2/23 Flashback: separators and treewidth

S

R Q

Figure: Scheme for DP applied on graphs

 Usually try to bound the size of S√. If, for instance, we have the case of the treewidth of planar graphs (O( n)), we can design FPT algorithms with sub-exponential parameterized dependence.

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 3/23 Flashback: separators and treewidth

S

R Q

Figure: Scheme for DP applied on graphs

 Usually try to bound the size of S√. If, for instance, we have the case of the treewidth of planar graphs (O( n)), we can design FPT algorithms with sub-exponential parameterized dependence.

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 3/23 Flashback: we n treewidth

 tw is often sublinear in the solution size k (of planar graphs)  It supports a “win-win” approach: N check (roughly) if tw(G) > o(k) and if true output “NO” N else, use DP and design something that runs in 2o(k)n steps.

Remember: there exists an algorithm that, with input a graph G and an integer c either reports that tw(G) > c or outputs a tree decomposition of G of width at most 5c in 2O(c)n steps, as a result based on Bodlaender, [1].

But:  To get that tw(G) = o(k) for a sole parameter k and graph class G, we need to use our best combinatorics...

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 4/23 Flashback: we n treewidth

 tw is often sublinear in the solution size k (of planar graphs)  It supports a “win-win” approach: N check (roughly) if tw(G) > o(k) and if true output “NO” N else, use DP and design something that runs in 2o(k)n steps.

Remember: there exists an algorithm that, with input a graph G and an integer c either reports that tw(G) > c or outputs a tree decomposition of G of width at most 5c in 2O(c)n steps, as a result based on Bodlaender, [1].

But:  To get that tw(G) = o(k) for a sole parameter k and graph class G, we need to use our best combinatorics...

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 4/23 Flashback: we n treewidth

 tw is often sublinear in the solution size k (of planar graphs)  It supports a “win-win” approach: N check (roughly) if tw(G) > o(k) and if true output “NO” N else, use DP and design something that runs in 2o(k)n steps.

Remember: there exists an algorithm that, with input a graph G and an integer c either reports that tw(G) > c or outputs a tree decomposition of G of width at most 5c in 2O(c)n steps, as a result based on Bodlaender, [1].

But:  To get that tw(G) = o(k) for a sole parameter k and graph class G, we need to use our best combinatorics...

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 4/23 ...Enter Light

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 5/23 ...Enter Light

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 5/23 ...Enter Bidimensionality

Theorem 1 (Robertson et al. (1994)).

Let G be a planar graph with tw(G) ≥ 6l. Then Γl m G. Linear treewidth-grid-minor condition, LTGM

Figure: The grid graph Γ10

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 6/23 ...Enter Bidimensionality

Theorem 1 (Robertson et al. (1994)).

Let G be a planar graph with tw(G) ≥ 6l. Then Γl m G. Linear treewidth-grid-minor condition, LTGM

Figure: The grid graph Γ10

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 6/23 ...Enter Bidimensionality

Theorem 1 (Robertson et al. (1994)).

Let G be a planar graph with tw(G) ≥ 6l. Then Γl m G. Linear treewidth-grid-minor condition, LTGM

Figure: The grid graph Γ10

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 6/23 ≥ l2/2

 vc is minor closed!

Vertex Cover for grids

Figure: vc(Γl)

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 7/23 ≥ l2/2

 vc is minor closed!

Vertex Cover for grids

Figure: vc(Γl)

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 7/23  vc is minor closed!

Vertex Cover for grids

2 Figure: vc(Γl) ≥ l /2

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 7/23 Vertex Cover for grids

2 Figure: vc(Γl) ≥ l /2

 vc is minor closed!

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 7/23 N Let G be planar and vc(G) = k.

N LTGM suggests that Γ tw(G)  G 6 N As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 N Therefore, we easily proved tw(G) = Opvc(G)

planar Vertex Cover and treewidth

 vc is minor closed!

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 8/23 N LTGM suggests that Γ tw(G)  G 6 N As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 N Therefore, we easily proved tw(G) = Opvc(G)

planar Vertex Cover and treewidth

 vc is minor closed!

N Let G be planar and vc(G) = k.

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 8/23 N As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 N Therefore, we easily proved tw(G) = Opvc(G)

planar Vertex Cover and treewidth

 vc is minor closed!

N Let G be planar and vc(G) = k.

N LTGM suggests that Γ tw(G)  G 6

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 8/23 N Therefore, we easily proved tw(G) = Opvc(G)

planar Vertex Cover and treewidth

 vc is minor closed!

N Let G be planar and vc(G) = k.

N LTGM suggests that Γ tw(G)  G 6 N As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 8/23 planar Vertex Cover and treewidth

 vc is minor closed!

N Let G be planar and vc(G) = k.

N LTGM suggests that Γ tw(G)  G 6 N As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 N Therefore, we easily proved tw(G) = Opvc(G)

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 8/23 planar Vertex Cover and treewidth

 vc is minor closed!

N Let G be planar and vc(G) = k.

N LTGM suggests that Γ tw(G)  G 6 N As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 N Therefore, we easily proved tw(G) = Opvc(G)

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 8/23 Definition 2 (Demaine et al. (2004)). A parameter p is minor bidimensional if 1. it is closed under minors and 2 2. p(Γl) = Ω(l )

So what is Bidimensionality?

Take a second look: N Let G be planar and vc(G) = k.

N LTGM suggests that Γ tw(G)  G 6 N As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 N Therefore, we easily proved tw(G) = Opvc(G)

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 9/23 Definition 2 (Demaine et al. (2004)). A parameter p is minor bidimensional if 1. it is closed under minors and 2 2. p(Γl) = Ω(l )

So what is Bidimensionality?

Take a second look: N Let G be planar and vc(G) = k.

N LTGM suggests that Γ tw(G)  G 6 N As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 N Therefore, we easily proved tw(G) = Opvc(G)

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 9/23 Definition 2 (Demaine et al. (2004)). A parameter p is minor bidimensional if 1. it is closed under minors and 2 2. p(Γl) = Ω(l )

So what is Bidimensionality?

Take a second look: N Let G be planar and vc(G) = k.

N LTGM suggests that Γ tw(G)  G 6 N As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 N Therefore, we easily proved tw(G) = Opvc(G)

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 9/23 Definition 2 (Demaine et al. (2004)). A parameter p is minor bidimensional if 1. it is closed under minors and 2 2. p(Γl) = Ω(l )

So what is Bidimensionality?

Take a second look: N Let G be planar and vc(G) = k.

N LTGM suggests that Γ tw(G)  G 6 N As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 N Therefore, we easily proved tw(G) = Opvc(G)

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 9/23 Definition 2 (Demaine et al. (2004)). A parameter p is minor bidimensional if 1. it is closed under minors and 2 2. p(Γl) = Ω(l )

So what is Bidimensionality?

Take a second look: N Let G be planar and vc(G) = k.

N LTGM suggests that Γ tw(G)  G 6 N As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 N Therefore, we easily proved tw(G) = Opvc(G)

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 9/23 So what is Bidimensionality?

Take a second look: N Let G be planar and vc(G) = k.

N LTGM suggests that Γ tw(G)  G 6 N As vc is minor closed, we obtain 2 2 k = vc(G) ≥ vc(Γ tw(G) ) ≥ c · (tw(G)) = Ω(tw(G)) 6 N Therefore, we easily proved tw(G) = Opvc(G)

Definition 2 (Demaine et al. (2004)). A parameter p is minor bidimensional if 1. it is closed under minors and 2 2. p(Γl) = Ω(l )

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 9/23 So what is Bidimensionality?

Definition 3 (Demaine et al. (2004)). A parameter p is contraction bidimensional if 1. it is closed under contractions and 2 2. p(Πl) = Ω(l )

Figure: The uniformly triangulated grid graph Π9

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 10/23 So what is Bidimensionality?

Theorem 4.

Every planar graph of treewidth Ω(l) contains Πl as a contraction. Linear treewidth-triangulated-grid condition, LTTGC

Bidimensional parameters:

Feedback Vertex Set, Vertex Cover, Minimum Maximal Matching, Face Cover, Dominating Set, Edge Dominating Set, Unweighted TSP Tour, Chordal Completion, Long Path, (k, r)-Center...

...to think that prior to the theory, grid graphs (as minors or contractions) where seen more as obstructions to designing or improving FPT algorithms.

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 11/23 So what is Bidimensionality?

Theorem 4.

Every planar graph of treewidth Ω(l) contains Πl as a contraction. Linear treewidth-triangulated-grid condition, LTTGC

Bidimensional parameters:

Feedback Vertex Set, Vertex Cover, Minimum Maximal Matching, Face Cover, Dominating Set, Edge Dominating Set, Unweighted TSP Tour, Chordal Completion, Long Path, (k, r)-Center...

...to think that prior to the theory, grid graphs (as minors or contractions) where seen more as obstructions to designing or improving FPT algorithms.

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 11/23 So what is Bidimensionality?

Theorem 4.

Every planar graph of treewidth Ω(l) contains Πl as a contraction. Linear treewidth-triangulated-grid condition, LTTGC

Bidimensional parameters:

Feedback Vertex Set, Vertex Cover, Minimum Maximal Matching, Face Cover, Dominating Set, Edge Dominating Set, Unweighted TSP Tour, Chordal Completion, Long Path, (k, r)-Center...

...to think that prior to the theory, grid graphs (as minors or contractions) where seen more as obstructions to designing or improving FPT algorithms.

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 11/23 Beyond just planar graphs

Consider a graph class G. If either LTGM or LTTGC hold for the class and some parameter p is either minor or contraction closed respectively, then:

∀G ∈ G : tw(G) = O(pp(G))

LTGM: Planar graphs, H-minor free graphs (Demaine and Hajiaghayi (2005)), Bounded intersection Geometric Graph classes (Grigoriev et al. (2013))

LTTGC: Planar graphs, apex-minor free graphs (Fomin et al. (2009))

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 12/23 Beyond just planar graphs

Consider a graph class G. If either LTGM or LTTGC hold for the class and some parameter p is either minor or contraction closed respectively, then:

∀G ∈ G : tw(G) = O(pp(G))

LTGM: Planar graphs, H-minor free graphs (Demaine and Hajiaghayi (2005)), Bounded intersection Geometric Graph classes (Grigoriev et al. (2013))

LTTGC: Planar graphs, apex-minor free graphs (Fomin et al. (2009))

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The idea 12/23 Outline

The idea

Subexponential parametrized algorithms

PTAS / EPTAS

Kernelization

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• Subexponential parametrized algorithms 13/23 Subexponential parametrized algorithms

Theorem 5 (Demaine et al. (2004)). Let p be a graph parameter and G be some graph class. If 1. p is minor/contraction bidimensional 2. G satisfies the LTGM/LTTGC condition 3. checking whether p(G) ≤ k can be done by an algorithm that runs in 2O(tw(G)) · n steps, √ then p-Parameter p-Checking can be solved in 2O( k) · n steps within class G

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• Subexponential parametrized algorithms 14/23 Outline

The idea

Subexponential parametrized algorithms

PTAS / EPTAS

Kernelization

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• PTAS / EPTAS 15/23 Bidimensionality and (E)PTASs

Theorem 6 (Demaine and Hajiaghayi (2005), Fomin et al. (2010)). Let p be a vertex certifiable graph parameter and G be some graph class. If 1. p is minor/contraction bidimensional 2. p is separable 3. G satisfies the LTGM/LTTGC condition 4. There is a constant-factor approximation algorithm for the value of p 5. checking whether p(G) ≤ k can be done by an algorithm that runs in 2O(tw(G)) · n steps, then there is an EPTAS for computing p on graphs in G.

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• PTAS / EPTAS 16/23 Bidimensionality and (E)PTASs

Theorem 6 (Demaine and Hajiaghayi (2005), Fomin et al. (2010)). Let p be a vertex certifiable graph parameter and G be some graph class. If 1. p is minor/contraction bidimensional 2. p is separable 3. G satisfies the LTGM/LTTGC condition 4. There is a constant-factor approximation algorithm for the value of p 5. checking whether p(G) ≤ k can be done by an algorithm that runs in 2O(tw(G)) · n steps, then there is an EPTAS for computing p on graphs in G.

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• PTAS / EPTAS 16/23 Bidimensionality and EPTASs

[2]: Connected Dominating Set, Feedback Vertex Set (previously only PTAS known for FVS)

[6]: Cycle Packing, Vertex H-Packing, Maximum Leaf Spanning Tree, Patial r-Dominating Set (no EPTASs were previously known, even on planar graphs)

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• PTAS / EPTAS 17/23 Outline

The idea

Subexponential parametrized algorithms

PTAS / EPTAS

Kernelization

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• Kernelization 18/23 Bidimensionality and kernelization

Theorem 7. Let p be a vertex certifiable graph parameter and G be some graph class. If 1. p is minor/contraction bidimensional 2. p is separable 3. G satisfies the LTGM/LTTGC condition 4. the certifying property of p can be expressed in MSO-logic, then p-Parameter p-Checking admits a linear size kernel when resticted to G.

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• Kernelization 19/23 Bidimensionality and kernelization

Fomin et al. (2010): Edge-Dominating Set, Feedback Vertex Set, (Connected) Vertex Cover, Cycle Packing, Independent Set... admit linear kernel on H-minor free graphs

Dominating Set, {Edge, r}-Dominating Set, Induced Matching...admit linear kernel on apex-minor free graphs

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• Kernelization 20/23 That’s all folks!

Thank you!

Guest appearances: James Hetfield Porky Pig

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The end 21/23 BibliographyI

[1] H. L. Bodlaender. A linear time algorithm for finding tree-decompositions of small treewidth. In STOC, pages 226–234, 1993. [2] E. D. Demaine and M. T. Hajiaghayi. Bidimensionality: new connections between fpt algorithms and ptass. In SODA, pages 590–601, 2005. [3] E. D. Demaine and M. T. Hajiaghayi. Graphs excluding a fixed minor have grids as large as treewidth, with combinatorial and algorithmic applications through bidimensionality. In SODA, pages 682–689, 2005. [4] E. D. Demaine, F. V. Fomin, M. T. Hajiaghayi, and D. M. Thilikos. Subexponential parameterized algorithms on graphs of bounded-genus and h-minor-free graphs. In SODA, pages 830–839, 2004.

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The end 22/23 BibliographyII

[5] F. V. Fomin, P. A. Golovach, and D. M. Thilikos. Contraction bidimensionality: The accurate picture. In ESA, pages 706–717, 2009. [6] F. V. Fomin, D. Lokshtanov, V. Raman, and S. Saurabh. Bidimensionality and eptas. CoRR, abs/1005.5449, 2010. [7] F. V. Fomin, D. Lokshtanov, S. Saurabh, and D. M. Thilikos. Bidimensionality and kernels. In SODA, pages 503–510, 2010. [8] A. Grigoriev, A. Koutsonas, and D. M. Thilikos. Bidimensionality of geometric intersection graphs. CoRR, abs/1308.6166, 2013. [9] N. Robertson, P. D. Seymour, and R. Thomas. Quickly excluding a planar graph. J. Comb. Theory, Ser. B, 62(2):323–348, 1994.

Alexandros Angelopoulos (M.P.L.A.) Bidimensionality-• The end 23/23