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Finding Triangles for Maximum Planar Subgraphs Ghent Graph Theory Workshop 2017 Parinya Chalermsook a Andreas Schmid b Ghent, August 18, 2017 a Aalto University, Espoo, Finland b Max Planck Institute for Informatics, Saarbrcken, Germany

  1. Finding Triangles for Maximum Planar Subgraphs Ghent Graph Theory Workshop 2017 Parinya Chalermsook a Andreas Schmid b Ghent, August 18, 2017 a Aalto University, Espoo, Finland b Max Planck Institute for Informatics, Saarbrücken, Germany

  2. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems The Problem Definition (Maximum Planar Subgraph Problem (MPS)) Given a graph G = ( V , E ) find a planar subgraph H = ( V , E ′ ) where E ′ ⊆ E is maximum. Ghent, August 18, 2017 2/17

  3. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems The Problem Definition (Maximum Planar Subgraph Problem (MPS)) Given a graph G = ( V , E ) find a planar subgraph H = ( V , E ′ ) where E ′ ⊆ E is maximum. NP-hard, Ghent, August 18, 2017 2/17

  4. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems The Problem Definition (Maximum Planar Subgraph Problem (MPS)) Given a graph G = ( V , E ) find a planar subgraph H = ( V , E ′ ) where E ′ ⊆ E is maximum. NP-hard, APX -hard. Ghent, August 18, 2017 2/17

  5. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems The Problem Definition (Maximum Planar Subgraph Problem (MPS)) Given a graph G = ( V , E ) find a planar subgraph H = ( V , E ′ ) where E ′ ⊆ E is maximum. NP-hard, APX -hard. Trivial: 1 3 -approximation. Ghent, August 18, 2017 2/17

  6. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems The Problem Definition (Maximum Planar Subgraph Problem (MPS)) Given a graph G = ( V , E ) find a planar subgraph H = ( V , E ′ ) where E ′ ⊆ E is maximum. NP-hard, APX -hard. Trivial: 1 3 -approximation. Best known: 4 9 -approximation. [C˘ alinescu et al. SODA 96] Ghent, August 18, 2017 2/17

  7. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems The Problem Definition (Maximum Planar Subgraph Problem (MPS)) Given a graph G = ( V , E ) find a planar subgraph H = ( V , E ′ ) where E ′ ⊆ E is maximum. NP-hard, APX -hard. Trivial: 1 3 -approximation. Best known: 4 9 -approximation. [C˘ alinescu et al. SODA 96] Many applications: Circuit design, factory layout, graph drawing problems. Ghent, August 18, 2017 2/17

  8. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Related Results 2 3 -approximation for Maximum Outerplanar Subgraph Problem. Ghent, August 18, 2017 3/17

  9. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Related Results 2 3 -approximation for Maximum Outerplanar Subgraph Problem. ( 1 3 + 1 72 ) -approximation for weighted MPS .[C˘ alinescu et al. 2003] Ghent, August 18, 2017 3/17

  10. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Related Results 2 3 -approximation for Maximum Outerplanar Subgraph Problem. ( 1 3 + 1 72 ) -approximation for weighted MPS .[C˘ alinescu et al. 2003] MPS remains NP -hard in cubic graphs.[Faria et al. 2004] Ghent, August 18, 2017 3/17

  11. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems How to beat 1/3 [C˘ alinescu et al. 96] Ghent, August 18, 2017 4/17

  12. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems How to beat 1/3 [C˘ alinescu et al. 96] Augment a spanning tree by forming edge-disjoint triangles Ghent, August 18, 2017 4/17

  13. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems How to beat 1/3 [C˘ alinescu et al. 96] Augment a spanning tree by forming edge-disjoint triangles Results in a triangular structure Ghent, August 18, 2017 4/17

  14. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems How to beat 1/3 [C˘ alinescu et al. 96] Augment a spanning tree by forming edge-disjoint triangles Results in a triangular structure 7 This gives a 18 -approximation Ghent, August 18, 2017 4/17

  15. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems How to beat 1/3 [C˘ alinescu et al. 96] Augment a spanning tree by forming edge-disjoint triangles Results in a triangular structure 7 This gives a 18 -approximation A maximum triangular structure gives a 4 9 -approximation Ghent, August 18, 2017 4/17

  16. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems How to beat 1/3 [C˘ alinescu et al. 96] Augment a spanning tree by forming edge-disjoint triangles Results in a triangular structure 7 This gives a 18 -approximation A maximum triangular structure gives a 4 9 -approximation Uses linear matroid parity algorithm as a black box. Ghent, August 18, 2017 4/17

  17. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Poranen’s Candidate Algorithm (08) Start a new component C from a single triangle in G . Ghent, August 18, 2017 5/17

  18. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Poranen’s Candidate Algorithm (08) Start a new component C from a single triangle in G . Ghent, August 18, 2017 5/17

  19. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Poranen’s Candidate Algorithm (08) Start a new component C from a single triangle in G . Find triangle t in G s.t. – t contains one vertex not in C and – t intersects C in exactly one edge Ghent, August 18, 2017 5/17

  20. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Poranen’s Candidate Algorithm (08) Start a new component C from a single triangle in G . Find triangle t in G s.t. – t contains one vertex not in C and – t intersects C in exactly one edge E ′ := E ′ ∪ E ( t ) . Ghent, August 18, 2017 5/17

  21. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Poranen’s Candidate Algorithm (08) Start a new component C from a single triangle in G . Find triangle t in G s.t. – t contains one vertex not in C and – t intersects C in exactly one edge E ′ := E ′ ∪ E ( t ) . Repeat until there are no more triangles in G . Ghent, August 18, 2017 5/17

  22. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Poranen’s Candidate Algorithm (08) Start a new component C from a single triangle in G . Find triangle t in G s.t. – t contains one vertex not in C and – t intersects C in exactly one edge E ′ := E ′ ∪ E ( t ) . Repeat until there are no more triangles in G . Ghent, August 18, 2017 5/17

  23. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Poranen’s Candidate Algorithm (08) Start a new component C from a single triangle in G . Find triangle t in G s.t. – t contains one vertex not in C and – t intersects C in exactly one edge E ′ := E ′ ∪ E ( t ) . Repeat until there are no more triangles in G . Ghent, August 18, 2017 5/17

  24. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Poranen’s Candidate Algorithm (08) Works great in practice. Ghent, August 18, 2017 6/17

  25. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Poranen’s Candidate Algorithm (08) Works great in practice. Runs in linear time in bounded degree graphs. Ghent, August 18, 2017 6/17

  26. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Poranen’s Candidate Algorithm (08) Works great in practice. Runs in linear time in bounded degree graphs. 7 Proven to give a 18 -approximation. Ghent, August 18, 2017 6/17

  27. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Poranen’s Candidate Algorithm (08) Works great in practice. Runs in linear time in bounded degree graphs. 7 Proven to give a 18 -approximation. Conjectured to give a 4 9 -approximation. Ghent, August 18, 2017 6/17

  28. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Point of This Work Is there hope to reach the 4 9 -approximation by using non edge-disjoint triangles? Ghent, August 18, 2017 7/17

  29. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Point of This Work Is there hope to reach the 4 9 -approximation by using non edge-disjoint triangles? ⇒ Start by improving over the best known greedy algorithm! Ghent, August 18, 2017 7/17

  30. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Focusing on Triangles Ghent, August 18, 2017 8/17

  31. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Focusing on Triangles We introduce a new problem that captures the previous ideas. Ghent, August 18, 2017 8/17

  32. Introduction Algorithms for MPS Maximum Planar Triangles(MPT) Match and Merge Open Problems Focusing on Triangles We introduce a new problem that captures the previous ideas. Definition (Maximum Planar Triangles Problem (MPT)) Given a graph G = ( V , E ) find a plane subgraph H = ( V , E ′ ) with a maximum number of triangular faces. Ghent, August 18, 2017 8/17

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