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1/12 Introduction tw t P.m.c. in planar graphs Algorithm Conclusion Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor Fomin Ioan Todinca Yngve Villanger University of Bergen, Universit´ e d’Orl´ eans Workshop on Graph Decompositions, CIRM, October 19th, 2010
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Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

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Page 1: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

1/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Exact algorithm for the Maximum Induced PlanarSubgraph Problem

Fedor Fomin Ioan Todinca Yngve Villanger

University of Bergen, Universite d’Orleans

Workshop on Graph Decompositions, CIRM, October 19th,2010

Page 2: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

2/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Outline of the result

Theorem

There is an O(1.73601n) algorithm for the Max InducedPlanar Subgraph problem.

Ingredients:

1. [Fomin, Villanger, STACS 2010]: an O(1.73601n · nt+3)algorithm for the Max Induced Subgraph ofTheewidth ≤ t

2. [Robertson, Seymour 1986; Fomin, Thilikos 2006]: planargraphs have treewidth O(

√n)

3. Combinatorial results on minimal triangulations and potentialmaximal cliques of planar graphs.

4. An algorithm putting everything together

Page 3: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

2/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Outline of the result

Theorem

There is an O(1.73601n) algorithm for the Max InducedPlanar Subgraph problem.

Ingredients:

1. [Fomin, Villanger, STACS 2010]: an O(1.73601n · nt+3)algorithm for the Max Induced Subgraph ofTheewidth ≤ t

2. [Robertson, Seymour 1986; Fomin, Thilikos 2006]: planargraphs have treewidth O(

√n)

3. Combinatorial results on minimal triangulations and potentialmaximal cliques of planar graphs.

4. An algorithm putting everything together

Page 4: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

2/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Outline of the result

Theorem

There is an O(1.73601n) algorithm for the Max InducedPlanar Subgraph problem.

Ingredients:

1. [Fomin, Villanger, STACS 2010]: an O(1.73601n · nt+3)algorithm for the Max Induced Subgraph ofTheewidth ≤ t

2. [Robertson, Seymour 1986; Fomin, Thilikos 2006]: planargraphs have treewidth O(

√n)

3. Combinatorial results on minimal triangulations and potentialmaximal cliques of planar graphs.

4. An algorithm putting everything together

Page 5: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

3/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Motivation and related work

Exact algorithms for NP-hard problems

• [Godel, 1959]: ”how strongly in general the number of stepsin finite combinatorial problems can be reduced with respectto simple exhaustive search?”

• Nice combinatorics, nice algorithmic techniques

Max Induced Subgraph with Property Π

• Max Independent Set [Moon, Moser, 1965; Fomin,Grandoni, Kratsch 2009]

• Max Feedback Vertex Set [Razgon 2006; Fomin,Villanger 2010]

• Max Induced Subgraph of treewidth ≤ t [Fomin,Villanger 2010]

Page 6: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

4/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Max Induced Subgraph of tw ≤ t

[Fomin, Villanger 2010]F an induced subgraph of G , HF a minimal triangulation of F .

• There is a minimal triangulation HG of G such that HF is aninduced subgraph of G ; we say thay HF and HG arecompatible

• Consequence: for every bag ΩG of HG , ΩG ∩ V (F ) iscontained is some bag ΩF of HF

• Fix a minimal triangulation HG . One can compute a maximuminduced subgraph F of G s.t. tw(F ) ≤ t and F has anoptimal triangulation compatible with HG in O(nt+cst) time

• Dynamic programming over all minimal triangulations HG

using potential maximal cliques: O(1.73601n · nt+cst) time

Page 7: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

4/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Max Induced Subgraph of tw ≤ t

[Fomin, Villanger 2010]F an induced subgraph of G , HF a minimal triangulation of F .

• There is a minimal triangulation HG of G such that HF is aninduced subgraph of G ; we say thay HF and HG arecompatible

• Consequence: for every bag ΩG of HG , ΩG ∩ V (F ) iscontained is some bag ΩF of HF

• Fix a minimal triangulation HG . One can compute a maximuminduced subgraph F of G s.t. tw(F ) ≤ t and F has anoptimal triangulation compatible with HG in O(nt+cst) time

• Dynamic programming over all minimal triangulations HG

using potential maximal cliques: O(1.73601n · nt+cst) time

Page 8: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

4/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Max Induced Subgraph of tw ≤ t

[Fomin, Villanger 2010]F an induced subgraph of G , HF a minimal triangulation of F .

• There is a minimal triangulation HG of G such that HF is aninduced subgraph of G ; we say thay HF and HG arecompatible

• Consequence: for every bag ΩG of HG , ΩG ∩ V (F ) iscontained is some bag ΩF of HF

• Fix a minimal triangulation HG .

One can compute a maximuminduced subgraph F of G s.t. tw(F ) ≤ t and F has anoptimal triangulation compatible with HG in O(nt+cst) time

• Dynamic programming over all minimal triangulations HG

using potential maximal cliques: O(1.73601n · nt+cst) time

Page 9: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

4/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Max Induced Subgraph of tw ≤ t

[Fomin, Villanger 2010]F an induced subgraph of G , HF a minimal triangulation of F .

• There is a minimal triangulation HG of G such that HF is aninduced subgraph of G ; we say thay HF and HG arecompatible

• Consequence: for every bag ΩG of HG , ΩG ∩ V (F ) iscontained is some bag ΩF of HF

• Fix a minimal triangulation HG . One can compute a maximuminduced subgraph F of G s.t. tw(F ) ≤ t and F has anoptimal triangulation compatible with HG in O(nt+cst) time

• Dynamic programming over all minimal triangulations HG

using potential maximal cliques: O(1.73601n · nt+cst) time

Page 10: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

4/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Max Induced Subgraph of tw ≤ t

[Fomin, Villanger 2010]F an induced subgraph of G , HF a minimal triangulation of F .

• There is a minimal triangulation HG of G such that HF is aninduced subgraph of G ; we say thay HF and HG arecompatible

• Consequence: for every bag ΩG of HG , ΩG ∩ V (F ) iscontained is some bag ΩF of HF

• Fix a minimal triangulation HG . One can compute a maximuminduced subgraph F of G s.t. tw(F ) ≤ t and F has anoptimal triangulation compatible with HG in O(nt+cst) time

• Dynamic programming over all minimal triangulations HG

using potential maximal cliques: O(1.73601n · nt+cst) time

Page 11: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

5/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Computing F for a fixed tree decomposition of G

• α(S ,W ,C ): the size ofthe largest partial solutionintersecting S in W

• α(S ,W ,C ) =maxW ′⊂Ω(W ′⊕

α(S1,W1,C1)⊕α(S2,W2,C2))

• Running time: O(nt+cst)

• Also constructs a minimaltriangulation of F

Page 12: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

5/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Computing F for a fixed tree decomposition of G

• α(S ,W ,C ): the size ofthe largest partial solutionintersecting S in W

• α(S ,W ,C ) =

maxW ′⊂Ω(W ′⊕α(S1,W1,C1)⊕α(S2,W2,C2))

• Running time: O(nt+cst)

• Also constructs a minimaltriangulation of F

Page 13: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

5/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Computing F for a fixed tree decomposition of G

• α(S ,W ,C ): the size ofthe largest partial solutionintersecting S in W

• α(S ,W ,C ) =maxW ′⊂Ω(W ′⊕

α(S1,W1,C1)⊕α(S2,W2,C2))

• Running time: O(nt+cst)

• Also constructs a minimaltriangulation of F

Page 14: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

5/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Computing F for a fixed tree decomposition of G

• α(S ,W ,C ): the size ofthe largest partial solutionintersecting S in W

• α(S ,W ,C ) =maxW ′⊂Ω(W ′⊕

α(S1,W1,C1)⊕α(S2,W2,C2))

• Running time: O(nt+cst)

• Also constructs a minimaltriangulation of F

Page 15: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

5/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Computing F for a fixed tree decomposition of G

• α(S ,W ,C ): the size ofthe largest partial solutionintersecting S in W

• α(S ,W ,C ) =maxW ′⊂Ω(W ′⊕

α(S1,W1,C1)⊕α(S2,W2,C2))

• Running time: O(nt+cst)

• Also constructs a minimaltriangulation of F

Page 16: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

5/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Computing F for a fixed tree decomposition of G

• α(S ,W ,C ): the size ofthe largest partial solutionintersecting S in W

• α(S ,W ,C ) =maxW ′⊂Ω(W ′⊕

α(S1,W1,C1)⊕α(S2,W2,C2))

• Running time: O(nt+cst)

• Also constructs a minimaltriangulation of F

Page 17: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

6/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Browsing through all minimal triangulations of G

Definition

A vertex subset ΩG of G is a potential maximal clique if thereexists a minimal triangulation HG such that ΩG is a maximalclique of HG .

• One can ”browse” through all minimal triangulations of agraph, in time O∗(]p.m.c .) [Bouchitte, Todinca 2001, Fomin,Kratsch, Todinca 2008]

• An n-vertex graph has O∗(1.73601n) potential maximalcliques [Fomin Villanger 2010]

A Maximum Induced Subgraph of tw ≤ t can be computedin O(1.73601n · nt+cst) time [Fomin Villanger 2010].

Page 18: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

7/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Towards an extension to the Max Induced PlanarGraph problem

• Good news: planar graphs have treewidth at most3.182

√n [Fomin, Thilikos 2006];

an O(]p.m.c . · nc√

n+cst) = O(]p.m.c. · 2o(n)) = O(1.73601n)algorithm?

• Bad news: in the algorithm of Fomin and Villanger, even whenwe ”glue” two planar graphs, the result might not be planar.

We need more tools for gluing partial solutions. Recall that we gluealong potential maximal cliques of the target (planar) graph F .

Page 19: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

7/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Towards an extension to the Max Induced PlanarGraph problem

• Good news: planar graphs have treewidth at most3.182

√n [Fomin, Thilikos 2006];

an O(]p.m.c . · nc√

n+cst) = O(]p.m.c. · 2o(n)) = O(1.73601n)algorithm?

• Bad news: in the algorithm of Fomin and Villanger, even whenwe ”glue” two planar graphs, the result might not be planar.

We need more tools for gluing partial solutions. Recall that we gluealong potential maximal cliques of the target (planar) graph F .

Page 20: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

7/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Towards an extension to the Max Induced PlanarGraph problem

• Good news: planar graphs have treewidth at most3.182

√n [Fomin, Thilikos 2006];

an O(]p.m.c . · nc√

n+cst) = O(]p.m.c. · 2o(n)) = O(1.73601n)algorithm?

• Bad news: in the algorithm of Fomin and Villanger, even whenwe ”glue” two planar graphs, the result might not be planar.

We need more tools for gluing partial solutions. Recall that we gluealong potential maximal cliques of the target (planar) graph F .

Page 21: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

8/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Potential maximal cliques in planar graphs

(a)

u

zyx

fe

b

dc

a

(b)

c d

fe

ba

t

• A potential maximal clique ΩF of a plane graph F forms aθ-structure [Bouchitte, Mazoit, Todinca 2003; . . . ]

• The neighbourhoods of the components of F −ΩF correspondto pairwise non-crossing Jordan curves

• Conversely, if we draw planar pieces ”inside” these curves wepreserve planarity

Page 22: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

8/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Potential maximal cliques in planar graphs

(a)

u

zyx

fe

dc

ba

(b)

c d

fe

ba

t

• A potential maximal clique ΩF of a plane graph F forms a

θ-structure [Bouchitte, Mazoit, Todinca 2003; . . . ]

• The neighbourhoods of the components of F −ΩF correspondto pairwise non-crossing Jordan curves

• Conversely, if we draw planar pieces ”inside” these curves wepreserve planarity

Page 23: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

8/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Potential maximal cliques in planar graphs

(a)

u

zyx

fe

dc

ba

(b)

c d

fe

ba

t

• A potential maximal clique ΩF of a plane graph F forms aθ-structure [Bouchitte, Mazoit, Todinca 2003; . . . ]

• The neighbourhoods of the components of F −ΩF correspondto pairwise non-crossing Jordan curves

• Conversely, if we draw planar pieces ”inside” these curves wepreserve planarity

Page 24: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

8/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Potential maximal cliques in planar graphs

(a)

t

u

c d

fe

ba

zyx

fe

dc

a b

(b)

• A potential maximal clique ΩF of a plane graph F forms aθ-structure [Bouchitte, Mazoit, Todinca 2003; . . . ]

• The neighbourhoods of the components of F −ΩF correspondto pairwise non-crossing Jordan curves

• Conversely, if we draw planar pieces ”inside” these curves wepreserve planarity

Page 25: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

8/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Potential maximal cliques in planar graphs

(b)(a)

t

u

c d

fe

ba

zyx

fe

dc

a b

• A potential maximal clique ΩF of a plane graph F forms aθ-structure [Bouchitte, Mazoit, Todinca 2003; . . . ]

• The neighbourhoods of the components of F −ΩF correspondto pairwise non-crossing Jordan curves

• Conversely, if we draw planar pieces ”inside” these curves wepreserve planarity

Page 26: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

9/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

θ-structures and neighborhood assignments

c

a b

e f

d

S1 = [e, a, b, f ]S2 = [e, c, d , b, f ]S3 = [e, a, b, d , c]

We can avoid geometry.

• θ-structures θ(ΩF ) on ΩF : three totally ordered subsets,sharing the extremities and forming three cyclic orderings Si .

• neighboorhood assignment [θ(ΩF )]:• assigns to each cyclic ordering Si of θ(ΩF ) a set of pairwise

non-crossing subsets of Si .• for each edge xy of F [ΩF ], the neighborhood x , y is

assigned to some Si .

Page 27: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

9/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

θ-structures and neighborhood assignments

c

a b

e f

d

S1 : b, f , a, b, f

We can avoid geometry.

• θ-structures θ(ΩF ) on ΩF : three totally ordered subsets,sharing the extremities and forming three cyclic orderings Si .

• neighboorhood assignment [θ(ΩF )]:• assigns to each cyclic ordering Si of θ(ΩF ) a set of pairwise

non-crossing subsets of Si .• for each edge xy of F [ΩF ], the neighborhood x , y is

assigned to some Si .

Page 28: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

9/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

θ-structures and neighborhood assignments

c

a b

e f

d

S2 : d , b, f , e, c , d , f

We can avoid geometry.

• θ-structures θ(ΩF ) on ΩF : three totally ordered subsets,sharing the extremities and forming three cyclic orderings Si .

• neighboorhood assignment [θ(ΩF )]:• assigns to each cyclic ordering Si of θ(ΩF ) a set of pairwise

non-crossing subsets of Si .• for each edge xy of F [ΩF ], the neighborhood x , y is

assigned to some Si .

Page 29: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

9/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

θ-structures and neighborhood assignments

c

a b

e f

d S3 :a, c , e, a, c, a, b, d , c

We can avoid geometry.

• θ-structures θ(ΩF ) on ΩF : three totally ordered subsets,sharing the extremities and forming three cyclic orderings Si .

• neighboorhood assignment [θ(ΩF )]:• assigns to each cyclic ordering Si of θ(ΩF ) a set of pairwise

non-crossing subsets of Si .• for each edge xy of F [ΩF ], the neighborhood x , y is

assigned to some Si .

Page 30: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

10/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Counting neighborhood assignements

Theorem

Over all subets ΩF of size at most c√

n, over all possibleθ-structures θ(ΩF ), there are 2o(n) possible (partial) neighborhoodassignements.

• there are( nc√

n

)possible subsets ΩF

• for each ΩF , there are 2o(n) posible θ-structures

• for a fixed θ-structure θ(ΩF ), for each cyclic ordering Si thenumber of posssible neighborhood assignements on Si isupper bounded by the Catalan numberCN(|Si |) ≤ 4|Si | [Kreweras 1972].

Page 31: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

11/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

The algorithm

Maximum partial solution for

• (S ,C ,ΩG )

• a θ-structure θ(ΩF )

• a (partial) neighborhoodassignment

Dynamic programming over allp.m.c. ΩG of G and on allpossible (partial) neighborhoodassignments [θ(ΩF )], over all”small” subsets ΩF ....and many other details!

Running time: O(]p.m.c. ·(]neighborhood assignments)3) =O(1.73601n)

Page 32: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

11/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

The algorithm

Maximum partial solution for

• (S ,C ,ΩG )

• a θ-structure θ(ΩF )

• a (partial) neighborhoodassignment

Dynamic programming over allp.m.c. ΩG of G and on allpossible (partial) neighborhoodassignments [θ(ΩF )], over all”small” subsets ΩF ....and many other details!

Running time: O(]p.m.c. ·(]neighborhood assignments)3) =O(1.73601n)

Page 33: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

11/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

The algorithm

Maximum partial solution for

• (S ,C ,ΩG )

• a θ-structure θ(ΩF )

• a (partial) neighborhoodassignment

Dynamic programming over allp.m.c. ΩG of G and on allpossible (partial) neighborhoodassignments [θ(ΩF )], over all”small” subsets ΩF ....and many other details!

Running time: O(]p.m.c. ·(]neighborhood assignments)3) =O(1.73601n)

Page 34: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

11/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

The algorithm

Maximum partial solution for

• (S ,C ,ΩG )

• a θ-structure θ(ΩF )

• a (partial) neighborhoodassignment

Dynamic programming over allp.m.c. ΩG of G and on allpossible (partial) neighborhoodassignments [θ(ΩF )], over all”small” subsets ΩF .

...and many other details!

Running time: O(]p.m.c. ·(]neighborhood assignments)3) =O(1.73601n)

Page 35: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

11/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

The algorithm

Maximum partial solution for

• (S ,C ,ΩG )

• a θ-structure θ(ΩF )

• a (partial) neighborhoodassignment

Dynamic programming over allp.m.c. ΩG of G and on allpossible (partial) neighborhoodassignments [θ(ΩF )], over all”small” subsets ΩF ....and many other details!

Running time: O(]p.m.c. ·(]neighborhood assignments)3) =O(1.73601n)

Page 36: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

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Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

The algorithm

Maximum partial solution for

• (S ,C ,ΩG )

• a θ-structure θ(ΩF )

• a (partial) neighborhoodassignment

Dynamic programming over allp.m.c. ΩG of G and on allpossible (partial) neighborhoodassignments [θ(ΩF )], over all”small” subsets ΩF ....and many other details!

Running time: O(]p.m.c. ·(]neighborhood assignments)3) =O(1.73601n)

Page 37: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

12/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Conclusion and open questions

An O(1.73601n) algorithm for the Max Induced PlanarSubgraph problem.

• Max Induced Subgraph With Property Π?• Bounded genus?• Excluded minors?• Bounded degeneracy?

• Combinatorial questions• What is the maximum number of minimal separators in an

n-vertex graph?• The same for potential maximal cliques? (The current upper

bound O(1.73601n) does not seem tight).

• Thank you! Your questions?

Page 38: Exact algorithm for the Maximum Induced Planar Subgraph ...paul/ANR/CIRM-TALKS-2010/Todinca-cirm-2010.pdf · Exact algorithm for the Maximum Induced Planar Subgraph Problem Fedor

12/12

Introduction tw ≤ t P.m.c. in planar graphs Algorithm Conclusion

Conclusion and open questions

An O(1.73601n) algorithm for the Max Induced PlanarSubgraph problem.

• Max Induced Subgraph With Property Π?• Bounded genus?• Excluded minors?• Bounded degeneracy?

• Combinatorial questions• What is the maximum number of minimal separators in an

n-vertex graph?• The same for potential maximal cliques? (The current upper

bound O(1.73601n) does not seem tight).

• Thank you! Your questions?