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On Hard Instances of the Minimum-Weight Triangulation Problem Sndor P. Fekete, Andreas Haas, Yannic Lieder, Eike Niehs, Michael Perk, Victoria Sack , and Christian Scheffer Department of Computer Science TU Braunschweig What is a

  1. On Hard Instances of the Minimum-Weight Triangulation Problem Sándor P. Fekete, Andreas Haas, Yannic Lieder, Eike Niehs, Michael Perk, Victoria Sack , and Christian Scheffer Department of Computer Science TU Braunschweig

  2. What is a Minimum-Weight Triangulation? Given: A set of points in � Wanted: A triangulation of that minimizes the length of its edges On Hard MWT Instances | Slide 2 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  3. Solving an MWT-Instance Between 1979 and 2008  1979/80: simple polygons (DP) 𝑃(𝑜 � ) [Gil79,Kli80] On Hard MWT Instances | Slide 3 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  4. Solving an MWT-Instance Between 1979 and 2008  1979/80: simple polygons (DP) 𝑃(𝑜 � ) [Gil79,Kli80] 1989: diamond property  [DJ89, DMS01] On Hard MWT Instances | Slide 4 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  5. Solving an MWT-Instance Between 1979 and 2008  1979/80: simple polygons (DP) 𝑃(𝑜 � ) [Gil79,Kli80] 1989: diamond property  [DJ89, DMS01]  1997: LMT-skeleton [DM96, DKM97] On Hard MWT Instances | Slide 5 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  6. Solving an MWT-Instance Between 1979 and 2008  1979/80: simple polygons (DP) 𝑃(𝑜 � ) [Gil79,Kli80] 1989: diamond property  [DJ89, DMS01]  1997: LMT-skeleton [DM96, DKM97]  2005/06: DP for polygons with 𝑙 inner points  𝑃(6 � 𝑜 � log 𝑜) [HO06] 𝑃(𝑜 � 4 � 𝑙) [GBL05a]  𝑃(𝑜 � 𝑙! 𝑙) [GBL05b]  On Hard MWT Instances | Slide 6 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  7. MWT is NP-Hard  2008: Proof by Mulzer and Rote [MR08]  Reduction from PLANAR 1-IN-3-SAT all images from [MR08] On Hard MWT Instances | Slide 7 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  8. hard SAT-instances The Gap  millions of points  hardware as limiting factor ? triangulate up to 30 Mio points in < 4min [Haa18] MWT is NP-hard On Hard MWT Instances | Slide 8 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  9. The IP △ � △ △ s.t. boundary component △ △∈�(�) On Hard MWT Instances | Slide 9 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  10. The IP △ � △ △ s.t. boundary component △ △∈�(�) On Hard MWT Instances | Slide 10 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  11. The IP △ � △ △ s.t. boundary component △ △∈�(�) On Hard MWT Instances | Slide 11 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  12. The IP △ � △ △ s.t. boundary component △ △∈�(�) On Hard MWT Instances | Slide 12 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  13. The IP △ � △ △ s.t. boundary component △ △∈�(�) △∈� � (�) △ antennas △∈� � (�) △ On Hard MWT Instances | Slide 13 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  14. The IP △ � △ △ s.t. boundary component △ △∈�(�) △∈� � (�) △ antennas △∈� � (�) △ On Hard MWT Instances | Slide 14 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  15. The IP △ � △ △ s.t. boundary component △ △∈�(�) △∈� � (�) △ antennas △∈� � (�) △ On Hard MWT Instances | Slide 15 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  16. The IP △ � △ △ s.t. boundary component △ △∈�(�) △∈� � (�) △ antennas △∈� � (�) △ On Hard MWT Instances | Slide 16 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  17. The IP △ � △ △ s.t. boundary component △ △∈�(�) △∈� � (�) △ antennas △∈� � (�) △ On Hard MWT Instances | Slide 17 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  18. The IP △ � △ △ s.t. boundary component △ △∈�(�) △∈� � (�) △ antennas △∈� � (�) △ On Hard MWT Instances | Slide 18 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  19. The IP △ � △ △ s.t. boundary component △ △∈�(�) △∈� � (�) △ antennas △∈� � (�) △ inner edges △∈� � � △ ∈� � (�) △ △ △ On Hard MWT Instances | Slide 19 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  20. The IP △ � △ △ s.t. boundary component △ △∈�(�) △∈� � (�) △ antennas △∈� � (�) △ inner edges △∈� � � △ ∈� � (�) △ △ △ On Hard MWT Instances | Slide 20 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  21. The IP △ � △ △ s.t. boundary component △ △∈�(�) △∈� � (�) △ antennas △∈� � (�) △ inner edges △∈� � � △ ∈� � (�) △ △ △ On Hard MWT Instances | Slide 21 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  22. The IP △ � △ △ s.t. boundary component △ △∈�(�) △∈� � (�) △ antennas △∈� � (�) △ inner edges △∈� � � △ ∈� � (�) △ △ △ On Hard MWT Instances | Slide 22 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  23. The IP △ � △ △ s.t. boundary component △ △∈�(�) △∈� � (�) △ antennas △∈� � (�) △ inner edges △∈� � � △ ∈� � (�) △ △ △ On Hard MWT Instances | Slide 23 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  24. What Makes an MWT-Instance Practically Hard? 1. After Diamond-test 1 complex face, and LMT-skeleton… 2. faces with many inner � else points, else 𝑃 6 � 𝑜 � log 𝑜 / 𝑃(𝑜 � 4 � 𝑙) / 𝑃(𝑜 � 𝑙! 𝑙) 3. LP-relaxation of the IP must have fractional solution, else easy to solve from [BKM96] from [YY14] On Hard MWT Instances | Slide 24 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  25. Experimental Setup: Evolver create initial instance construct small polygons in simple faces generate/mutate offspring evaluate fitness select next generation On Hard MWT Instances | Slide 25 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  26. Experimental Setup: Evolver construct small polygons in simple faces On Hard MWT Instances | Slide 26 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  27. Experimental Setup: Pertubation On Hard MWT Instances | Slide 27 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  28. Experimental Setup: Pertubation On Hard MWT Instances | Slide 28 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  29. Experimental Setup: Pertubation On Hard MWT Instances | Slide 29 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  30. Experimental Setup: Pertubation On Hard MWT Instances | Slide 30 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  31. Experimental Setup: Pertubation On Hard MWT Instances | Slide 31 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  32. Experimental Setup: Pertubation On Hard MWT Instances | Slide 32 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  33. Experimental Setup: Pertubation On Hard MWT Instances | Slide 33 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  34. Experimental Results: Runtime  created ~17,000 instances  fixed size of 306 points No instance had a fractional LP solution! On Hard MWT Instances | Slide 34 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  35. Experimental Results: Scaling On Hard MWT Instances | Slide 35 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  36. Experimental Results: Stability On Hard MWT Instances | Slide 36 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

  37. Conclusion What makes an MWT instance practically hard? On Hard MWT Instances | Slide 37 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack , C. Scheffer

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