Fixed Parameter Algorithms for Completion Problems on Plane Graphs - PowerPoint PPT Presentation

. . Fixed Parameter Algorithms for Completion Problems on Plane Graphs Dimitris Chatzidimitriou in collaboration with Archontia C. Giannopoulou, Spyridon Maniatis, Clément Requilé, Dimitrios M. Thilikos, Dimitris Zoros AGTAC, June 2015 Thursday, June 18, 2015 HELLENIC REPUBLIC National and Kapodistrian University of Athens

The Subgraph & Minor Isomorphism Problems The Subgraph Isomorphism Problem (S.I.) and the Minor Thursday, June 18, 2015 Plane Subgraph & Minor Completion Dimitris Chatzidimitriou (UoA) (Adler et al. 2010) (Robertson & Seymour 1995) M.I. (Eppstein 1999) ? S.I. Planar General G and H and check whether G has any subgraph or minor isomorphic to H . are two well-known NP-complete problems that accept as input two graphs Isomorphism Problem (M.I.) (also known as Minor Containment ) 2 / 31 2 O ( k ) · n O (2 O ( k ) · n + n 2 · log n ) g ( k ) · n 3 where n = | V ( G ) | and k = | V ( H ) | .

The Subgraph & Minor Isomorphism Problems The Subgraph Isomorphism Problem (S.I.) and the Minor Thursday, June 18, 2015 Plane Subgraph & Minor Completion Dimitris Chatzidimitriou (UoA) (Adler et al. 2010) (Robertson & Seymour 1995) M.I. (Eppstein 1999) ? S.I. Planar General G and H and check whether G has any subgraph or minor isomorphic to H . are two well-known NP-complete problems that accept as input two graphs Isomorphism Problem (M.I.) (also known as Minor Containment ) 2 / 31 2 O ( k ) · n O (2 O ( k ) · n + n 2 · log n ) g ( k ) · n 3 where n = | V ( G ) | and k = | V ( H ) | .

Planar and Plane Graphs no two of its edges intersect, apart from any common endpoints. points and its edges are arcs. Each plane graph can be naturally associated to a planar graph through isomorphism. planar graphs on the plane. most factorial) different up to topological isomorphism. Dimitris Chatzidimitriou (UoA) Plane Subgraph & Minor Completion Thursday, June 18, 2015 3 / 31 ⋆ A planar graph is a graph that can be embedded on the plane such that ⋆ A plane graph is a graph embedded on the plane, so that its vertices are ⋆ The plane graphs can be regarded as “drawings” or embeddings of the ⋆ A planar graph can have infinitely many embeddings but only finite (at

Planar and Plane Graphs no two of its edges intersect, apart from any common endpoints. points and its edges are arcs. Each plane graph can be naturally associated to a planar graph through isomorphism. planar graphs on the plane. most factorial) different up to topological isomorphism. Dimitris Chatzidimitriou (UoA) Plane Subgraph & Minor Completion Thursday, June 18, 2015 3 / 31 ⋆ A planar graph is a graph that can be embedded on the plane such that ⋆ A plane graph is a graph embedded on the plane, so that its vertices are ⋆ The plane graphs can be regarded as “drawings” or embeddings of the ⋆ A planar graph can have infinitely many embeddings but only finite (at

Planar and Plane Graphs no two of its edges intersect, apart from any common endpoints. points and its edges are arcs. Each plane graph can be naturally associated to a planar graph through isomorphism. planar graphs on the plane. most factorial) different up to topological isomorphism. Dimitris Chatzidimitriou (UoA) Plane Subgraph & Minor Completion Thursday, June 18, 2015 3 / 31 ⋆ A planar graph is a graph that can be embedded on the plane such that ⋆ A plane graph is a graph embedded on the plane, so that its vertices are ⋆ The plane graphs can be regarded as “drawings” or embeddings of the ⋆ A planar graph can have infinitely many embeddings but only finite (at

Planar and Plane Graphs no two of its edges intersect, apart from any common endpoints. points and its edges are arcs. Each plane graph can be naturally associated to a planar graph through isomorphism. planar graphs on the plane. most factorial) different up to topological isomorphism. Dimitris Chatzidimitriou (UoA) Plane Subgraph & Minor Completion Thursday, June 18, 2015 3 / 31 ⋆ A planar graph is a graph that can be embedded on the plane such that ⋆ A plane graph is a graph embedded on the plane, so that its vertices are ⋆ The plane graphs can be regarded as “drawings” or embeddings of the ⋆ A planar graph can have infinitely many embeddings but only finite (at

Planar and Plane Graphs cont’d For example: Dimitris Chatzidimitriou (UoA) Plane Subgraph & Minor Completion Thursday, June 18, 2015 4 / 31 G Γ 1 Γ 2 Γ 3 Here, G is a planar graph and Γ 1 , Γ 2 , and Γ 3 are planar embeddings of G . In fact, Γ 1 and Γ 2 are equivalent (topologically isomorphic) to each other but not to Γ 3 .

Completion Problems Many interesting problems, naturally parameterized by the number of new Thursday, June 18, 2015 Plane Subgraph & Minor Completion Dimitris Chatzidimitriou (UoA) ...and now we are ready to define our two main problems. have been studied a lot lately. edges ( k ), arose with the introduction of the completion operation, which have the property P ? or more of the graphs so that they will Question: Can we add some edges to one 5 / 31 Problem: Π Problem: Π - Completion Input: Graphs G 1 , . . . , G l Input: Graphs G 1 , . . . , G l Question: Do the graphs have a specified property P ?

Completion Problems Many interesting problems, naturally parameterized by the number of new Thursday, June 18, 2015 Plane Subgraph & Minor Completion Dimitris Chatzidimitriou (UoA) ...and now we are ready to define our two main problems. have been studied a lot lately. edges ( k ), arose with the introduction of the completion operation, which have the property P ? or more of the graphs so that they will Question: Can we add some edges to one 5 / 31 Problem: Π Problem: Π - Completion Input: Graphs G 1 , . . . , G l Input: Graphs G 1 , . . . , G l Question: Do the graphs have a specified property P ?

Completion Problems Many interesting problems, naturally parameterized by the number of new Thursday, June 18, 2015 Plane Subgraph & Minor Completion Dimitris Chatzidimitriou (UoA) ...and now we are ready to define our two main problems. have been studied a lot lately. edges ( k ), arose with the introduction of the completion operation, which have the property P ? or more of the graphs so that they will Question: Can we add some edges to one 5 / 31 Problem: Π Problem: Π - Completion Input: Graphs G 1 , . . . , G l Input: Graphs G 1 , . . . , G l Question: Do the graphs have a specified property P ?

The Plane Subgraph Completion Problem Plane Subgraph Completion (PSC) Dimitris Chatzidimitriou (UoA) Plane Subgraph & Minor Completion Thursday, June 18, 2015 6 / 31 Input: A “host” plane graph Γ and a “pattern” connected plane graph ∆ . Parameter: k = | V (∆) | Question: Can we add edges to Γ so that it contains a subgraph topologically isomorphic to ∆ while remaining planar ? Γ ∆

The Plane Subgraph Completion Problem Plane Subgraph Completion (PSC) Dimitris Chatzidimitriou (UoA) Plane Subgraph & Minor Completion Thursday, June 18, 2015 7 / 31 Input: A “host” plane graph Γ and a “pattern” connected plane graph ∆ . Parameter: k = | V (∆) | Question: Can we add edges to Γ so that it contains a subgraph topologically isomorphic to ∆ while remaining planar ? Γ ∆

The Plane Top. Minor Completion Problem Plane Topological Minor Completion (PTMC) Dimitris Chatzidimitriou (UoA) Plane Subgraph & Minor Completion Thursday, June 18, 2015 8 / 31 Input: A “host” plane graph Γ and a “pattern” connected plane graph ∆ . Parameter: k = | V (∆) | Question: Can we add edges to Γ so that it contains a topological minor topologically isomorphic to ∆ while remaining planar ? Γ ∆

The Plane Top. Minor Completion Problem Plane Topological Minor Completion (PTMC) Dimitris Chatzidimitriou (UoA) Plane Subgraph & Minor Completion Thursday, June 18, 2015 9 / 31 Input: A “host” plane graph Γ and a “pattern” connected plane graph ∆ . Parameter: k = | V (∆) | Question: Can we add edges to Γ so that it contains a topological minor topologically isomorphic to ∆ while remaining planar ? Γ ∆

Our Results Remark. In fact we can even solve more general problems: we can ask that its embeddings can be found in the host. Dimitris Chatzidimitriou (UoA) Plane Subgraph & Minor Completion Thursday, June 18, 2015 10 / 31 If k := | V (∆) | and n := | V (Γ) | , we give: an FPT algorithm for PSC that runs in time 2 O ( k log k ) · n 2 and an FPT algorithm for PTMC that runs in time g ( k ) · n 2 . the pattern graph ∆ be given as a planar graph and check whether any of

Our Results Remark. In fact we can even solve more general problems: we can ask that its embeddings can be found in the host. Dimitris Chatzidimitriou (UoA) Plane Subgraph & Minor Completion Thursday, June 18, 2015 10 / 31 If k := | V (∆) | and n := | V (Γ) | , we give: an FPT algorithm for PSC that runs in time 2 O ( k log k ) · n 2 and an FPT algorithm for PTMC that runs in time g ( k ) · n 2 . the pattern graph ∆ be given as a planar graph and check whether any of

First, let’s see the tools we need for the PSC -algorithm... Dimitris Chatzidimitriou (UoA) Plane Subgraph & Minor Completion Thursday, June 18, 2015 11 / 31